CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>PageRank - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy","wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"e67174a6-d5fd-4404-8ac4-ec49492f634d","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"PageRank","wgTitle":"PageRank","wgCurRevisionId":1281628054,"wgRevisionId":1281628054,"wgArticleId":26334893,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Webarchive template wayback links","CS1: long volume value","Articles with short description","Short description is different from Wikidata","All articles with unsourced statements","Articles with unsourced statements from June 2013","All articles lacking reliable references","Articles lacking reliable references from October 2012","Articles with unsourced statements from November 2015","Articles with example MATLAB/Octave code","Google Search","Search engine optimization","Reputation management","Internet search algorithms","American inventions","Crowdsourcing","Markov models","Link analysis","Graph algorithms"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"PageRank","wgRelevantArticleId":26334893,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":70000,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q184316","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGELevelingUpEnabledForUser":false}; RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","ext.pygments":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","ext.pygments.view","mediawiki.page.media","site","mediawiki.page.ready","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.pygments%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediamessages.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.21"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/f/fb/PageRanks-Example.svg/1200px-PageRanks-Example.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="967"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/f/fb/PageRanks-Example.svg/800px-PageRanks-Example.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="645"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/f/fb/PageRanks-Example.svg/640px-PageRanks-Example.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="516"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="PageRank - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/PageRank"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=PageRank&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/PageRank"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-PageRank rootpage-PageRank skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" title="Main menu" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Main menu </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z">Main page</a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia">Contents</a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events">Current events</a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x">Random article</a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works">About Wikipedia</a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia">Contact us</a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia">Help</a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia">Learn to edit</a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors">Community portal</a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r">Recent changes</a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia">Upload file</a></li><li id="n-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages">Special pages</a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"> Search </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Appearance </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en" class="">Donate</a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=PageRank" title="You are encouraged to create an account and log in; however, it is not mandatory" class="">Create account</a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=PageRank" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class="">Log in</a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Personal tools </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en">Donate</a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=PageRank" title="You are encouraged to create an account and log in; however, it is not mandatory"> Create account</a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=PageRank" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"> Log in</a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing">learn more</a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y">Contributions</a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n">Talk</a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Description" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Description"> <div class="vector-toc-text"> 1 Description </div> </a> <ul id="toc-Description-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 2 History </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algorithm" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Algorithm"> <div class="vector-toc-text"> 3 Algorithm </div> </a> <button aria-controls="toc-Algorithm-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Algorithm subsection </button> <ul id="toc-Algorithm-sublist" class="vector-toc-list"> <li id="toc-Simplified_algorithm" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simplified_algorithm"> <div class="vector-toc-text"> 3.1 Simplified algorithm </div> </a> <ul id="toc-Simplified_algorithm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Damping_factor" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Damping_factor"> <div class="vector-toc-text"> 3.2 Damping factor </div> </a> <ul id="toc-Damping_factor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computation"> <div class="vector-toc-text"> 3.3 Computation </div> </a> <ul id="toc-Computation-sublist" class="vector-toc-list"> <li id="toc-Iterative" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Iterative"> <div class="vector-toc-text"> 3.3.1 Iterative </div> </a> <ul id="toc-Iterative-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Power_method" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Power_method"> <div class="vector-toc-text"> 3.3.2 Power method </div> </a> <ul id="toc-Power_method-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Implementation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Implementation"> <div class="vector-toc-text"> 3.4 Implementation </div> </a> <ul id="toc-Implementation-sublist" class="vector-toc-list"> <li id="toc-Python" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Python"> <div class="vector-toc-text"> 3.4.1 Python </div> </a> <ul id="toc-Python-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Variations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Variations"> <div class="vector-toc-text"> 4 Variations </div> </a> <button aria-controls="toc-Variations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Variations subsection </button> <ul id="toc-Variations-sublist" class="vector-toc-list"> <li id="toc-PageRank_of_an_undirected_graph" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#PageRank_of_an_undirected_graph"> <div class="vector-toc-text"> 4.1 PageRank of an undirected graph </div> </a> <ul id="toc-PageRank_of_an_undirected_graph-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ranking_objects_of_two_kinds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ranking_objects_of_two_kinds"> <div class="vector-toc-text"> 4.2 Ranking objects of two kinds </div> </a> <ul id="toc-Ranking_objects_of_two_kinds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Distributed_algorithm_for_PageRank_computation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distributed_algorithm_for_PageRank_computation"> <div class="vector-toc-text"> 4.3 Distributed algorithm for PageRank computation </div> </a> <ul id="toc-Distributed_algorithm_for_PageRank_computation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Google_Toolbar" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Google_Toolbar"> <div class="vector-toc-text"> 4.4 Google Toolbar </div> </a> <ul id="toc-Google_Toolbar-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-SERP_rank" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#SERP_rank"> <div class="vector-toc-text"> 4.5 SERP rank </div> </a> <ul id="toc-SERP_rank-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Google_directory_PageRank" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Google_directory_PageRank"> <div class="vector-toc-text"> 4.6 Google directory PageRank </div> </a> <ul id="toc-Google_directory_PageRank-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-False_or_spoofed_PageRank" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#False_or_spoofed_PageRank"> <div class="vector-toc-text"> 4.7 False or spoofed PageRank </div> </a> <ul id="toc-False_or_spoofed_PageRank-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Manipulating_PageRank" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Manipulating_PageRank"> <div class="vector-toc-text"> 4.8 Manipulating PageRank </div> </a> <ul id="toc-Manipulating_PageRank-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Directed_Surfer_Model" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Directed_Surfer_Model"> <div class="vector-toc-text"> 4.9 Directed Surfer Model </div> </a> <ul id="toc-Directed_Surfer_Model-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_uses" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Other_uses"> <div class="vector-toc-text"> 5 Other uses </div> </a> <button aria-controls="toc-Other_uses-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Other uses subsection </button> <ul id="toc-Other_uses-sublist" class="vector-toc-list"> <li id="toc-Scientific_research_and_academia" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scientific_research_and_academia"> <div class="vector-toc-text"> 5.1 Scientific research and academia </div> </a> <ul id="toc-Scientific_research_and_academia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Internet_use" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Internet_use"> <div class="vector-toc-text"> 5.2 Internet use </div> </a> <ul id="toc-Internet_use-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_applications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_applications"> <div class="vector-toc-text"> 5.3 Other applications </div> </a> <ul id="toc-Other_applications-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-nofollow" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#nofollow"> <div class="vector-toc-text"> 6 nofollow </div> </a> <ul id="toc-nofollow-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 7 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 8 References </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle References subsection </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> 8.1 Citations </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> 8.2 Sources </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relevant_patents" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relevant_patents"> <div class="vector-toc-text"> 9 Relevant patents </div> </a> <ul id="toc-Relevant_patents-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 10 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">PageRank</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 43 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-43" aria-hidden="true" > 43 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A8%D9%8A%D8%AC_%D8%B1%D8%A7%D9%86%D9%83_(%D8%AE%D9%88%D8%A7%D8%B1%D8%B2%D9%85%D9%8A%D8%A9)" title="بيج رانك (خوارزمية) – Arabic" lang="ar" hreflang="ar" data-title="بيج رانك (خوارزمية)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Page_Rank" title="Page Rank – Azerbaijani" lang="az" hreflang="az" data-title="Page Rank" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A7%87%E0%A6%9C%E0%A6%B0%E2%80%8D%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%82%E0%A6%95" title="পেজর‍্যাংক – Bangla" lang="bn" hreflang="bn" data-title="পেজর‍্যাংক" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/PageRank" title="PageRank – Bulgarian" lang="bg" hreflang="bg" data-title="PageRank" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/PageRank" title="PageRank – Catalan" lang="ca" hreflang="ca" data-title="PageRank" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/PageRank" title="PageRank – Czech" lang="cs" hreflang="cs" data-title="PageRank" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/PageRank" title="PageRank – Danish" lang="da" hreflang="da" data-title="PageRank" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/PageRank" title="PageRank – German" lang="de" hreflang="de" data-title="PageRank" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BB%CE%B3%CF%8C%CF%81%CE%B9%CE%B8%CE%BC%CE%BF%CF%82_PageRank" title="Αλγόριθμος PageRank – Greek" lang="el" hreflang="el" data-title="Αλγόριθμος PageRank" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/PageRank" title="PageRank – Spanish" lang="es" hreflang="es" data-title="PageRank" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/PageRank" title="PageRank – Basque" lang="eu" hreflang="eu" data-title="PageRank" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%BE%DB%8C%D8%AC%E2%80%8C%D8%B1%D9%86%DA%A9" title="پیج‌رنک – Persian" lang="fa" hreflang="fa" data-title="پیج‌رنک" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/PageRank" title="PageRank – French" lang="fr" hreflang="fr" data-title="PageRank" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%AA%E0%AB%87%E0%AA%9C_%E0%AA%B0%E0%AB%87%E0%AA%A8%E0%AB%8D%E0%AA%95" title="પેજ રેન્ક – Gujarati" lang="gu" hreflang="gu" data-title="પેજ રેન્ક" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target">ગુજરાતી</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8E%98%EC%9D%B4%EC%A7%80%EB%9E%AD%ED%81%AC" title="페이지랭크 – Korean" lang="ko" hreflang="ko" data-title="페이지랭크" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/PageRank" title="PageRank – Armenian" lang="hy" hreflang="hy" data-title="PageRank" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%87%E0%A4%9C%E0%A4%B0%E0%A5%88%E0%A4%82%E0%A4%95" title="पेजरैंक – Hindi" lang="hi" hreflang="hi" data-title="पेजरैंक" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/PageRank" title="PageRank – Indonesian" lang="id" hreflang="id" data-title="PageRank" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/PageRank" title="PageRank – Italian" lang="it" hreflang="it" data-title="PageRank" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/PageRank" title="PageRank – Hebrew" lang="he" hreflang="he" data-title="PageRank" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AA%E0%B3%87%E0%B2%9C%E0%B3%8D%E2%80%8D%E0%B2%B0%E0%B3%8D%E0%B2%AF%E0%B2%BE%E0%B2%82%E0%B2%95%E0%B3%8D" title="ಪೇಜ್‍ರ್ಯಾಂಕ್ – Kannada" lang="kn" hreflang="kn" data-title="ಪೇಜ್‍ರ್ಯಾಂಕ್" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/PageRank" title="PageRank – Kazakh" lang="kk" hreflang="kk" data-title="PageRank" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/PageRank" title="PageRank – Lithuanian" lang="lt" hreflang="lt" data-title="PageRank" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/PageRank" title="PageRank – Hungarian" lang="hu" hreflang="hu" data-title="PageRank" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/PageRank" title="PageRank – Malay" lang="ms" hreflang="ms" data-title="PageRank" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/PageRank" title="PageRank – Dutch" lang="nl" hreflang="nl" data-title="PageRank" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%9A%E3%83%BC%E3%82%B8%E3%83%A9%E3%83%B3%E3%82%AF" title="ページランク – Japanese" lang="ja" hreflang="ja" data-title="ページランク" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/PageRank" title="PageRank – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="PageRank" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/PageRank" title="PageRank – Polish" lang="pl" hreflang="pl" data-title="PageRank" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/PageRank" title="PageRank – Portuguese" lang="pt" hreflang="pt" data-title="PageRank" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/PageRank" title="PageRank – Romanian" lang="ro" hreflang="ro" data-title="PageRank" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/PageRank" title="PageRank – Russian" lang="ru" hreflang="ru" data-title="PageRank" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/PageRank" title="PageRank – Slovak" lang="sk" hreflang="sk" data-title="PageRank" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%B5%D1%98%D1%9F_%D1%80%D0%B0%D0%BD%D0%BA" title="Пејџ ранк – Serbian" lang="sr" hreflang="sr" data-title="Пејџ ранк" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/PageRank" title="PageRank – Finnish" lang="fi" hreflang="fi" data-title="PageRank" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Pagerank" title="Pagerank – Swedish" lang="sv" hreflang="sv" data-title="Pagerank" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AF%87%E0%AE%9C%E0%AF%8D_%E0%AE%A4%E0%AE%B0%E0%AE%B5%E0%AE%B0%E0%AE%BF%E0%AE%9A%E0%AF%88" title="பேஜ் தரவரிசை – Tamil" lang="ta" hreflang="ta" data-title="பேஜ் தரவரிசை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%9E%E0%B8%88%E0%B9%81%E0%B8%A3%E0%B8%87%E0%B8%81%E0%B9%8C" title="เพจแรงก์ – Thai" lang="th" hreflang="th" data-title="เพจแรงก์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/PageRank" title="PageRank – Turkish" lang="tr" hreflang="tr" data-title="PageRank" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/PageRank" title="PageRank – Ukrainian" lang="uk" hreflang="uk" data-title="PageRank" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/PageRank" title="PageRank – Vietnamese" lang="vi" hreflang="vi" data-title="PageRank" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/PageRank" title="PageRank – Cantonese" lang="yue" hreflang="yue" data-title="PageRank" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/PageRank" title="PageRank – Chinese" lang="zh" hreflang="zh" data-title="PageRank" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q184316#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/PageRank" title="View the content page [c]" accesskey="c">Article</a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:PageRank" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t">Talk</a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" >English </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/PageRank">Read</a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=PageRank&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=PageRank&action=history" title="Past revisions of this page [h]" accesskey="h">View history</a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" >Tools </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/PageRank">Read</a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=PageRank&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=PageRank&action=history">View history</a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/PageRank" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j">What links here</a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/PageRank" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k">Related changes</a></li><li id="t-upload" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u">Upload file</a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=PageRank&oldid=1281628054" title="Permanent link to this revision of this page">Permanent link</a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=PageRank&action=info" title="More information about this page">Page information</a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=PageRank&id=1281628054&wpFormIdentifier=titleform" title="Information on how to cite this page">Cite this page</a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FPageRank">Get shortened URL</a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FPageRank">Download QR code</a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=PageRank&action=show-download-screen" title="Download this page as a PDF file">Download as PDF</a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=PageRank&printable=yes" title="Printable version of this page [p]" accesskey="p">Printable version</a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:PageRank" hreflang="en">Wikimedia Commons</a></li><li class="wb-otherproject-link wb-otherproject-wikiquote mw-list-item"><a href="https://en.wikiquote.org/wiki/PageRank" hreflang="en">Wikiquote</a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q184316" title="Structured data on this page hosted by Wikidata [g]" accesskey="g">Wikidata item</a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Algorithm used by Google Search to rank web pages</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Page_rank_animation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Page_rank_animation.gif/500px-Page_rank_animation.gif" decoding="async" width="500" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Page_rank_animation.gif/750px-Page_rank_animation.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Page_rank_animation.gif/1000px-Page_rank_animation.gif 2x" data-file-width="1500" data-file-height="750" /></a><figcaption>An animation of the PageRank algorithm running on a small network of pages. The size of the nodes represents the perceived importance of the page, and arrows represent hyperlinks.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:PageRanks-Example.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/PageRanks-Example.svg/330px-PageRanks-Example.svg.png" decoding="async" width="300" height="242" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/PageRanks-Example.svg/500px-PageRanks-Example.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/PageRanks-Example.svg/600px-PageRanks-Example.svg.png 2x" data-file-width="758" data-file-height="611" /></a><figcaption>A simple illustration of the Pagerank algorithm. The percentage shows the perceived importance, and the arrows represent hyperlinks.</figcaption></figure> PageRank (PR) is an <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> used by <a href="/wiki/Google_Search" title="Google Search">Google Search</a> to <a href="/wiki/Ranking" title="Ranking">rank</a> <a href="/wiki/Webpages" class="mw-redirect" title="Webpages">web pages</a> in their <a href="/wiki/Search_engine" title="Search engine">search engine</a> results. It is named after both the term "web page" and co-founder <a href="/wiki/Larry_Page" title="Larry Page">Larry Page</a>. PageRank is a way of measuring the importance of website pages. According to Google: <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote">PageRank works by counting the number and quality of links to a page to determine a rough estimate of how important the website is. The underlying assumption is that more important websites are likely to receive more links from other websites.<a href="#cite_note-1">[1]</a></blockquote> Currently, PageRank is not the only algorithm used by Google to order search results, but it is the first algorithm that was used by the company, and it is the best known.<a href="#cite_note-:1-2">[2]</a><a href="#cite_note-3">[3]</a> As of September 24, 2019, all patents associated with PageRank have expired.<a href="#cite_note-4">[4]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Description">Description</h2>[<a href="/w/index.php?title=PageRank&action=edit&section=1" title="Edit section: Description">edit</a>]</div> PageRank is a <a href="/wiki/Network_theory#Link_analysis" title="Network theory">link analysis</a> algorithm and it assigns a numerical <a href="/wiki/Weighting" title="Weighting">weighting</a> to each element of a <a href="/wiki/Hyperlink" title="Hyperlink">hyperlinked</a> <a href="/wiki/Set_(computer_science)" class="mw-redirect" title="Set (computer science)">set</a> of documents, such as the <a href="/wiki/World_Wide_Web" title="World Wide Web">World Wide Web</a>, with the purpose of "measuring" its relative importance within the set. The <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> may be applied to any collection of entities with <a href="/wiki/Reciprocal_link" class="mw-redirect" title="Reciprocal link">reciprocal</a> quotations and references. The numerical weight that it assigns to any given element E is referred to as the PageRank of E and denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PR(E).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PR(E).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a1cf2cf5d0fab047535790b2e5962369f081f74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.741ex; height:2.843ex;" alt="{\displaystyle PR(E).}" /> A PageRank results from a mathematical algorithm based on the <a href="/wiki/Webgraph" title="Webgraph">Webgraph</a>, created by all World Wide Web pages as nodes and <a href="/wiki/Hyperlink" title="Hyperlink">hyperlinks</a> as edges, taking into consideration authority hubs such as <a href="/wiki/Cnn.com" class="mw-redirect" title="Cnn.com">cnn.com</a> or <a href="/wiki/Mayoclinic.org" class="mw-redirect" title="Mayoclinic.org">mayoclinic.org</a>. The rank value indicates an importance of a particular page. A hyperlink to a page counts as a vote of support. The PageRank of a page is defined <a href="/wiki/Recursion" title="Recursion">recursively</a> and depends on the number and PageRank metric of all pages that link to it ("<a href="/wiki/Incoming_link" class="mw-redirect" title="Incoming link">incoming links</a>"). A page that is linked to by many pages with high PageRank receives a high rank itself. Numerous academic papers concerning PageRank have been published since Page and Brin's original paper.<a href="#cite_note-originalpaper-5">[5]</a> In practice, the PageRank concept may be vulnerable to manipulation. Research has been conducted into identifying falsely influenced PageRank rankings. The goal is to find an effective means of ignoring links from documents with falsely influenced PageRank.<a href="#cite_note-6">[6]</a> Other link-based ranking algorithms for Web pages include the <a href="/wiki/HITS_algorithm" title="HITS algorithm">HITS algorithm</a> invented by <a href="/wiki/Jon_Kleinberg" title="Jon Kleinberg">Jon Kleinberg</a> (used by <a href="/wiki/Teoma" title="Teoma">Teoma</a> and now <a href="/wiki/Ask.com" title="Ask.com">Ask.com</a>), the IBM <a href="/wiki/CLEVER_project" title="CLEVER project">CLEVER project</a>, the <a href="/wiki/TrustRank" title="TrustRank">TrustRank</a> algorithm, the <a href="/wiki/Google_Hummingbird" title="Google Hummingbird">Hummingbird</a> algorithm,<a href="#cite_note-7">[7]</a> and the <a href="/wiki/SALSA_algorithm" title="SALSA algorithm">SALSA algorithm</a>.<a href="#cite_note-8">[8]</a> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=PageRank&action=edit&section=2" title="Edit section: History">edit</a>]</div> The <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> problem behind PageRank's algorithm was independently rediscovered and reused in many scoring problems. In 1895, <a href="/wiki/Edmund_Landau" title="Edmund Landau">Edmund Landau</a> suggested using it for determining the winner of a chess tournament.<a href="#cite_note-9">[9]</a><a href="#cite_note-10">[10]</a> The eigenvalue problem was also suggested in 1976 by Gabriel Pinski and Francis Narin, who worked on <a href="/wiki/Scientometrics" title="Scientometrics">scientometrics</a> ranking scientific journals,<a href="#cite_note-11">[11]</a> in 1977 by <a href="/wiki/Thomas_Saaty" class="mw-redirect" title="Thomas Saaty">Thomas Saaty</a> in his concept of <a href="/wiki/Analytic_Hierarchy_Process" class="mw-redirect" title="Analytic Hierarchy Process">Analytic Hierarchy Process</a> which weighted alternative choices,<a href="#cite_note-12">[12]</a> and in 1995 by Bradley Love and Steven Sloman as a <a href="/wiki/Cognitive_model" title="Cognitive model">cognitive model</a> for concepts, the centrality algorithm.<a href="#cite_note-13">[13]</a><a href="#cite_note-bradloveblog-14">[14]</a> A search engine called "<a href="/wiki/RankDex" class="mw-redirect" title="RankDex">RankDex</a>" from IDD Information Services, designed by <a href="/wiki/Robin_Li" title="Robin Li">Robin Li</a> in 1996, developed a strategy for site-scoring and page-ranking.<a href="#cite_note-15">[15]</a> Li referred to his search mechanism as "link analysis," which involved ranking the popularity of a web site based on how many other sites had linked to it.<a href="#cite_note-nytimes-16">[16]</a> RankDex, the first search engine with page-ranking and site-scoring algorithms, was launched in 1996.<a href="#cite_note-rankdex-17">[17]</a> Li filed a patent for the technology in RankDex in 1997; it was granted in 1999.<a href="#cite_note-18">[18]</a> He later used it when he founded <a href="/wiki/Baidu" title="Baidu">Baidu</a> in China in 2000.<a href="#cite_note-19">[19]</a><a href="#cite_note-20">[20]</a> Google founder <a href="/wiki/Larry_Page" title="Larry Page">Larry Page</a> referenced Li's work as a citation in some of his U.S. patents for PageRank.<a href="#cite_note-21">[21]</a><a href="#cite_note-rankdex-17">[17]</a><a href="#cite_note-22">[22]</a> Larry Page and <a href="/wiki/Sergey_Brin" title="Sergey Brin">Sergey Brin</a> developed PageRank at <a href="/wiki/Stanford_University" title="Stanford University">Stanford University</a> in 1996 as part of a research project about a new kind of search engine. An interview with <a href="/wiki/H%C3%A9ctor_Garc%C3%ADa-Molina" title="Héctor García-Molina">Héctor García-Molina</a>, Stanford Computer Science professor and advisor to Sergey,<a href="#cite_note-23">[23]</a> provides background into the development of the page-rank algorithm.<a href="#cite_note-24">[24]</a> Sergey Brin had the idea that information on the web could be ordered in a hierarchy by "link popularity": a page ranks higher as there are more links to it.<a href="#cite_note-gpower-25">[25]</a> The system was developed with the help of Scott Hassan and Alan Steremberg, both of whom were cited by Page and Brin as being critical to the development of Google.<a href="#cite_note-originalpaper-5">[5]</a> <a href="/wiki/Rajeev_Motwani" title="Rajeev Motwani">Rajeev Motwani</a> and <a href="/wiki/Terry_Winograd" title="Terry Winograd">Terry Winograd</a> co-authored with Page and Brin the first paper about the project, describing PageRank and the initial prototype of the <a href="/wiki/Google_Search" title="Google Search">Google search engine</a>, published in 1998.<a href="#cite_note-originalpaper-5">[5]</a> Shortly after, Page and Brin founded <a href="/wiki/Google_Inc." class="mw-redirect" title="Google Inc.">Google Inc.</a>, the company behind the Google search engine. While just one of many factors that determine the ranking of Google search results, PageRank continues to provide the basis for all of Google's web-search tools.<a href="#cite_note-googletechnology-26">[26]</a> The name "PageRank" plays on the name of developer Larry Page, as well as of the concept of a <a href="/wiki/Web_page" title="Web page">web page</a>.<a href="#cite_note-27">[27]</a><a href="#cite_note-28">[28]</a> The word is a trademark of Google, and the PageRank process has been <a href="/wiki/Software_patent" title="Software patent">patented</a> (<a rel="nofollow" class="external text" href="https://patents.google.com/patent/US6285999">U.S. patent 6,285,999</a>). However, the patent is assigned to Stanford University and not to Google. Google has exclusive license rights on the patent from Stanford University. The university received 1.8 million shares of Google in exchange for use of the patent; it sold the shares in 2005 for $336 million.<a href="#cite_note-29">[29]</a><a href="#cite_note-30">[30]</a> PageRank was influenced by <a href="/wiki/Citation_analysis" title="Citation analysis">citation analysis</a>, early developed by <a href="/wiki/Eugene_Garfield" title="Eugene Garfield">Eugene Garfield</a> in the 1950s at the University of Pennsylvania, and by <a href="/wiki/Hyper_Search" title="Hyper Search">Hyper Search</a>, developed by <a href="/wiki/Massimo_Marchiori" title="Massimo Marchiori">Massimo Marchiori</a> at the <a href="/wiki/University_of_Padua" title="University of Padua">University of Padua</a>. In the same year PageRank was introduced (1998), <a href="/wiki/Jon_Kleinberg" title="Jon Kleinberg">Jon Kleinberg</a> published his work on <a href="/wiki/HITS_algorithm" title="HITS algorithm">HITS</a>. Google's founders cite Garfield, Marchiori, and Kleinberg in their original papers.<a href="#cite_note-originalpaper-5">[5]</a><a href="#cite_note-:0-31">[31]</a> <div class="mw-heading mw-heading2"><h2 id="Algorithm">Algorithm</h2>[<a href="/w/index.php?title=PageRank&action=edit&section=3" title="Edit section: Algorithm">edit</a>]</div> The PageRank algorithm outputs a <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> used to represent the likelihood that a person randomly clicking on links will arrive at any particular page. PageRank can be calculated for collections of documents of any size. It is assumed in several research papers that the distribution is evenly divided among all documents in the collection at the beginning of the computational process. The PageRank computations require several passes, called "iterations", through the collection to adjust approximate PageRank values to more closely reflect the theoretical true value. A probability is expressed as a numeric value between 0 and 1. A 0.5 probability is commonly expressed as a "50% chance" of something happening. Hence, a document with a PageRank of 0.5 means there is a 50% chance that a person clicking on a random link will be directed to said document. <div class="mw-heading mw-heading3"><h3 id="Simplified_algorithm">Simplified algorithm</h3>[<a href="/w/index.php?title=PageRank&action=edit&section=4" title="Edit section: Simplified algorithm">edit</a>]</div> Assume a small universe of four web pages: A, B, C, and D. Links from a page to itself are ignored. Multiple outbound links from one page to another page are treated as a single link. PageRank is initialized to the same value for all pages. In the original form of PageRank, the sum of PageRank over all pages was the total number of pages on the web at that time, so each page in this example would have an initial value of 1. However, later versions of PageRank, and the remainder of this section, assume a <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> between 0 and 1. Hence the initial value for each page in this example is 0.25. The PageRank transferred from a given page to the targets of its outbound links upon the next iteration is divided equally among all outbound links. If the only links in the system were from pages B, C, and D to A, each link would transfer 0.25 PageRank to A upon the next iteration, for a total of 0.75. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PR(A)=PR(B)+PR(C)+PR(D).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PR(A)=PR(B)+PR(C)+PR(D).\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a20704b3f4eaf53c5fa51896bd3e89b1c56f93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.286ex; height:2.843ex;" alt="{\displaystyle PR(A)=PR(B)+PR(C)+PR(D).\,}" /></dd></dl> Suppose instead that page B had a link to pages C and A, page C had a link to page A, and page D had links to all three pages. Thus, upon the first iteration, page B would transfer half of its existing value (0.125) to page A and the other half (0.125) to page C. Page C would transfer all of its existing value (0.25) to the only page it links to, A. Since D had three outbound links, it would transfer one third of its existing value, or approximately 0.083, to A. At the completion of this iteration, page A will have a PageRank of approximately 0.458. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PR(A)={\frac {PR(B)}{2}}+{\frac {PR(C)}{1}}+{\frac {PR(D)}{3}}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> <mn>1</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>.</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PR(A)={\frac {PR(B)}{2}}+{\frac {PR(C)}{1}}+{\frac {PR(D)}{3}}.\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88496ec65433d4cd8f2618a0462823ff2a04cfca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:40.794ex; height:5.676ex;" alt="{\displaystyle PR(A)={\frac {PR(B)}{2}}+{\frac {PR(C)}{1}}+{\frac {PR(D)}{3}}.\,}" /></dd></dl> In other words, the PageRank conferred by an outbound link is equal to the document's own PageRank score divided by the number of outbound links L( ). <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PR(A)={\frac {PR(B)}{L(B)}}+{\frac {PR(C)}{L(C)}}+{\frac {PR(D)}{L(D)}}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PR(A)={\frac {PR(B)}{L(B)}}+{\frac {PR(C)}{L(C)}}+{\frac {PR(D)}{L(D)}}.\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/179ce06124adc01074d09b5ba728733d5380902a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.794ex; height:6.509ex;" alt="{\displaystyle PR(A)={\frac {PR(B)}{L(B)}}+{\frac {PR(C)}{L(C)}}+{\frac {PR(D)}{L(D)}}.\,}" /></dd></dl> In the general case, the PageRank value for any page u can be expressed as: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PR(u)=\sum _{v\in B_{u}}{\frac {PR(v)}{L(v)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mo>∈</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PR(u)=\sum _{v\in B_{u}}{\frac {PR(v)}{L(v)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a8c0a807f62331cc1740dd6c0f28ac1809926c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:21.485ex; height:7.176ex;" alt="{\displaystyle PR(u)=\sum _{v\in B_{u}}{\frac {PR(v)}{L(v)}}}" />,</dd></dl> i.e. the PageRank value for a page u is dependent on the PageRank values for each page v contained in the set Bu (the set containing all pages linking to page u), divided by the number L(v) of links from page v. <div class="mw-heading mw-heading3"><h3 id="Damping_factor">Damping factor</h3>[<a href="/w/index.php?title=PageRank&action=edit&section=5" title="Edit section: Damping factor">edit</a>]</div> The PageRank theory holds that an imaginary surfer who is randomly clicking on links will eventually stop clicking. The probability, at any step, that the person will continue following links is a damping factor d. The probability that they instead jump to any random page is 1 - d. Various studies have tested different damping factors, but it is generally assumed that the damping factor will be set around 0.85.<a href="#cite_note-originalpaper-5">[5]</a> The damping factor is subtracted from 1 (and in some variations of the algorithm, the result is divided by the number of documents (N) in the collection) and this term is then added to the product of the damping factor and the sum of the incoming PageRank scores. That is, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PR(A)={1-d \over N}+d\left({\frac {PR(B)}{L(B)}}+{\frac {PR(C)}{L(C)}}+{\frac {PR(D)}{L(D)}}+\,\cdots \right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>d</mi> </mrow> <mi>N</mi> </mfrac> </mrow> <mo>+</mo> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mspace width="thinmathspace"></mspace> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PR(A)={1-d \over N}+d\left({\frac {PR(B)}{L(B)}}+{\frac {PR(C)}{L(C)}}+{\frac {PR(D)}{L(D)}}+\,\cdots \right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c3da6d608ba21cac0bbfc96e59615ffe8f33360" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:60.664ex; height:6.509ex;" alt="{\displaystyle PR(A)={1-d \over N}+d\left({\frac {PR(B)}{L(B)}}+{\frac {PR(C)}{L(C)}}+{\frac {PR(D)}{L(D)}}+\,\cdots \right).}" /></dd></dl> So any page's PageRank is derived in large part from the PageRanks of other pages. The damping factor adjusts the derived value downward. The original paper, however, gave the following formula, which has led to some confusion: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PR(A)=1-d+d\left({\frac {PR(B)}{L(B)}}+{\frac {PR(C)}{L(C)}}+{\frac {PR(D)}{L(D)}}+\,\cdots \right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>d</mi> <mo>+</mo> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mspace width="thinmathspace"></mspace> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PR(A)=1-d+d\left({\frac {PR(B)}{L(B)}}+{\frac {PR(C)}{L(C)}}+{\frac {PR(D)}{L(D)}}+\,\cdots \right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af3909afbed9793d94f41bf99ec98d75d5f1335a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:59.828ex; height:6.509ex;" alt="{\displaystyle PR(A)=1-d+d\left({\frac {PR(B)}{L(B)}}+{\frac {PR(C)}{L(C)}}+{\frac {PR(D)}{L(D)}}+\,\cdots \right).}" /></dd></dl> The difference between them is that the PageRank values in the first formula sum to one, while in the second formula each PageRank is multiplied by N and the sum becomes N. A statement in Page and Brin's paper that "the sum of all PageRanks is one"<a href="#cite_note-originalpaper-5">[5]</a> and claims by other Google employees<a href="#cite_note-32">[32]</a> support the first variant of the formula above. Page and Brin confused the two formulas in their most popular paper "The Anatomy of a Large-Scale Hypertextual Web Search Engine", where they mistakenly claimed that the latter formula formed a probability distribution over web pages.<a href="#cite_note-originalpaper-5">[5]</a> Google recalculates PageRank scores each time it crawls the Web and rebuilds its index. As Google increases the number of documents in its collection, the initial approximation of PageRank decreases for all documents. The formula uses a model of a random surfer who reaches their target site after several clicks, then switches to a random page. The PageRank value of a page reflects the chance that the random surfer will land on that page by clicking on a link. It can be understood as a <a href="/wiki/Markov_chain" title="Markov chain">Markov chain</a> in which the states are pages, and the transitions are the links between pages – all of which are all equally probable. If a page has no links to other pages, it becomes a sink and therefore terminates the random surfing process. If the random surfer arrives at a sink page, it picks another <a href="/wiki/Uniform_Resource_Locator" class="mw-redirect" title="Uniform Resource Locator">URL</a> at random and continues surfing again. When calculating PageRank, pages with no outbound links are assumed to link out to all other pages in the collection. Their PageRank scores are therefore divided evenly among all other pages. In other words, to be fair with pages that are not sinks, these random transitions are added to all nodes in the Web. This residual probability, d, is usually set to 0.85, estimated from the frequency that an average surfer uses his or her browser's bookmark feature. So, the equation is as follows: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PR(p_{i})={\frac {1-d}{N}}+d\sum _{p_{j}\in M(p_{i})}{\frac {PR(p_{j})}{L(p_{j})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>d</mi> </mrow> <mi>N</mi> </mfrac> </mrow> <mo>+</mo> <mi>d</mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∈</mo> <mi>M</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PR(p_{i})={\frac {1-d}{N}}+d\sum _{p_{j}\in M(p_{i})}{\frac {PR(p_{j})}{L(p_{j})}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f853c33de82a94b16ff0ea7e7a7346620c0ea04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:36.601ex; height:7.676ex;" alt="{\displaystyle PR(p_{i})={\frac {1-d}{N}}+d\sum _{p_{j}\in M(p_{i})}{\frac {PR(p_{j})}{L(p_{j})}}}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1},p_{2},...,p_{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1},p_{2},...,p_{N}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87490def075e50a6067f1450f1918466480a6e37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:13.601ex; height:2.009ex;" alt="{\displaystyle p_{1},p_{2},...,p_{N}}" /> are the pages under consideration, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(p_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(p_{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/733ef9718364811dbb28a8c06383db62d2cd7837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.221ex; height:2.843ex;" alt="{\displaystyle M(p_{i})}" /> is the set of pages that link to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(p_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(p_{j})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bca5899d25ec773204cb98f65202899fa7deca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.471ex; height:3.009ex;" alt="{\displaystyle L(p_{j})}" /> is the number of outbound links on page <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/499e0821b28c43e9bc2a6360b937de535057bc62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:2.169ex; height:2.343ex;" alt="{\displaystyle p_{j}}" />, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" /> is the total number of pages. The PageRank values are the entries of the dominant right <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvector</a> of the modified <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a> rescaled so that each column adds up to one. This makes PageRank a particularly elegant metric: the eigenvector is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ={\begin{bmatrix}PR(p_{1})\\PR(p_{2})\\\vdots \\PR(p_{N})\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ={\begin{bmatrix}PR(p_{1})\\PR(p_{2})\\\vdots \\PR(p_{N})\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1da019838e787e9868c41ec97e7ad8ef8110fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:17.133ex; height:14.176ex;" alt="{\displaystyle \mathbf {R} ={\begin{bmatrix}PR(p_{1})\\PR(p_{2})\\\vdots \\PR(p_{N})\end{bmatrix}}}" /></dd></dl> where R is the solution of the equation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ={\begin{bmatrix}{(1-d)/N}\\{(1-d)/N}\\\vdots \\{(1-d)/N}\end{bmatrix}}+d{\begin{bmatrix}\ell (p_{1},p_{1})&\ell (p_{1},p_{2})&\cdots &\ell (p_{1},p_{N})\\\ell (p_{2},p_{1})&\ddots &&\vdots \\\vdots &&\ell (p_{i},p_{j})&\\\ell (p_{N},p_{1})&\cdots &&\ell (p_{N},p_{N})\end{bmatrix}}\mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>ℓ</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>ℓ</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>ℓ</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>ℓ</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd></mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd></mtd> <mtd> <mi>ℓ</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd></mtd> </mtr> <mtr> <mtd> <mi>ℓ</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd></mtd> <mtd> <mi>ℓ</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ={\begin{bmatrix}{(1-d)/N}\\{(1-d)/N}\\\vdots \\{(1-d)/N}\end{bmatrix}}+d{\begin{bmatrix}\ell (p_{1},p_{1})&\ell (p_{1},p_{2})&\cdots &\ell (p_{1},p_{N})\\\ell (p_{2},p_{1})&\ddots &&\vdots \\\vdots &&\ell (p_{i},p_{j})&\\\ell (p_{N},p_{1})&\cdots &&\ell (p_{N},p_{N})\end{bmatrix}}\mathbf {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/135fc270ee538cdfc96ddc308bfc57a24a894c5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:70.641ex; height:15.843ex;" alt="{\displaystyle \mathbf {R} ={\begin{bmatrix}{(1-d)/N}\\{(1-d)/N}\\\vdots \\{(1-d)/N}\end{bmatrix}}+d{\begin{bmatrix}\ell (p_{1},p_{1})&\ell (p_{1},p_{2})&\cdots &\ell (p_{1},p_{N})\\\ell (p_{2},p_{1})&\ddots &&\vdots \\\vdots &&\ell (p_{i},p_{j})&\\\ell (p_{N},p_{1})&\cdots &&\ell (p_{N},p_{N})\end{bmatrix}}\mathbf {R} }" /></dd></dl> where the adjacency function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell (p_{i},p_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell (p_{i},p_{j})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/145afef13f07d4c827ddfd6498bc221b59549fef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.861ex; height:3.009ex;" alt="{\displaystyle \ell (p_{i},p_{j})}" /> is the ratio between number of links outbound from page j to page i to the total number of outbound links of page j. The adjacency function is 0 if page <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/499e0821b28c43e9bc2a6360b937de535057bc62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:2.169ex; height:2.343ex;" alt="{\displaystyle p_{j}}" /> does not link to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}" />, and normalized such that, for each j <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{N}\ell (p_{i},p_{j})=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mi>ℓ</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{N}\ell (p_{i},p_{j})=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d3a3d9a326269615735f64d32d5ef6334296c22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.864ex; height:7.343ex;" alt="{\displaystyle \sum _{i=1}^{N}\ell (p_{i},p_{j})=1}" />,</dd></dl> i.e. the elements of each column sum up to 1, so the matrix is a <a href="/wiki/Stochastic_matrix" title="Stochastic matrix">stochastic matrix</a> (for more details see the <a href="#Computation">computation</a> section below). Thus this is a variant of the <a href="/wiki/Eigenvector_centrality" title="Eigenvector centrality">eigenvector centrality</a> measure used commonly in <a href="/wiki/Network_theory" title="Network theory">network analysis</a>. Because of the large <a href="/wiki/Eigengap" title="Eigengap">eigengap</a> of the modified adjacency matrix above,<a href="#cite_note-33">[33]</a> the values of the PageRank eigenvector can be approximated to within a high degree of accuracy within only a few iterations. Google's founders, in their original paper,<a href="#cite_note-:0-31">[31]</a> reported that the PageRank algorithm for a network consisting of 322 million links (in-edges and out-edges) converges to within a tolerable limit in 52 iterations. The convergence in a network of half the above size took approximately 45 iterations. Through this data, they concluded the algorithm can be scaled very well and that the scaling factor for extremely large networks would be roughly linear in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/317ab5292da7c7935aec01a570461fe0613b21d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.754ex; height:2.509ex;" alt="{\displaystyle \log n}" />, where n is the size of the network. As a result of <a href="/wiki/Markov_process" class="mw-redirect" title="Markov process">Markov theory</a>, it can be shown that the PageRank of a page is the probability of arriving at that page after a large number of clicks. This happens to equal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82182729910430d9930cc0cbf96a3f43050b381c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.172ex; height:2.676ex;" alt="{\displaystyle t^{-1}}" /> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" /> is the <a href="/wiki/Expected_value" title="Expected value">expectation</a> of the number of clicks (or random jumps) required to get from the page back to itself. One main disadvantage of PageRank is that it favors older pages. A new page, even a very good one, will not have many links unless it is part of an existing site (a site being a densely connected set of pages, such as <a href="/wiki/Wikipedia" title="Wikipedia">Wikipedia</a>). Several strategies have been proposed to accelerate the computation of PageRank.<a href="#cite_note-34">[34]</a> Various strategies to manipulate PageRank have been employed in concerted efforts to improve search results rankings and monetize advertising links. These strategies have severely impacted the reliability of the PageRank concept,[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] which purports to determine which documents are actually highly valued by the Web community. Since December 2007, when it started actively penalizing sites selling paid text links, Google has combatted <a href="/wiki/Link_farm" title="Link farm">link farms</a> and other schemes designed to artificially inflate PageRank. How Google identifies link farms and other PageRank manipulation tools is among Google's <a href="/wiki/Trade_secret" title="Trade secret">trade secrets</a>. <div class="mw-heading mw-heading3"><h3 id="Computation">Computation</h3>[<a href="/w/index.php?title=PageRank&action=edit&section=6" title="Edit section: Computation">edit</a>]</div> PageRank can be computed either iteratively or algebraically. The iterative method can be viewed as the <a href="/wiki/Power_iteration" title="Power iteration">power iteration</a> method <a href="#cite_note-35">[35]</a><a href="#cite_note-36">[36]</a> or the power method. The basic mathematical operations performed are identical. <div class="mw-heading mw-heading4"><h4 id="Iterative">Iterative</h4>[<a href="/w/index.php?title=PageRank&action=edit&section=7" title="Edit section: Iterative">edit</a>]</div> At <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43469ec032d858feae5aa87029e22eaaf0109e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}" />, an initial probability distribution is assumed, usually <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PR(p_{i};0)={\frac {1}{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>;</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PR(p_{i};0)={\frac {1}{N}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a88d72ae8cfe827f8872d60bbe5905a8165b587" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.482ex; height:5.176ex;" alt="{\displaystyle PR(p_{i};0)={\frac {1}{N}}}" />.</dd></dl> where N is the total number of pages, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i};0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>;</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i};0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c611e79a0a1f63a956c549973b67bb7ab180e9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.255ex; height:2.509ex;" alt="{\displaystyle p_{i};0}" /> is page i at time 0. At each time step, the computation, as detailed above, yields <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PR(p_{i};t+1)={\frac {1-d}{N}}+d\sum _{p_{j}\in M(p_{i})}{\frac {PR(p_{j};t)}{L(p_{j})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>;</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>d</mi> </mrow> <mi>N</mi> </mfrac> </mrow> <mo>+</mo> <mi>d</mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∈</mo> <mi>M</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>;</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PR(p_{i};t+1)={\frac {1-d}{N}}+d\sum _{p_{j}\in M(p_{i})}{\frac {PR(p_{j};t)}{L(p_{j})}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb0f1469218a064274fd4691143e9ce64639dc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:44.351ex; height:7.676ex;" alt="{\displaystyle PR(p_{i};t+1)={\frac {1-d}{N}}+d\sum _{p_{j}\in M(p_{i})}{\frac {PR(p_{j};t)}{L(p_{j})}}}" /></dd></dl> where d is the damping factor, or in matrix notation <style data-mw-deduplicate="TemplateStyles:r1266403038">.mw-parser-output table.numblk{border-collapse:collapse;border:none;margin-top:0;margin-right:0;margin-bottom:0}.mw-parser-output table.numblk>tbody>tr>td{vertical-align:middle;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2){width:99%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table{border-collapse:collapse;margin:0;border:none;width:100%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:first-child,.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:last-child{padding:0 0.4ex}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:nth-child(2){width:100%;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{padding:0}.mw-parser-output table.numblk>tbody>tr>td:last-child{font-weight:bold}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child{font-weight:unset}.mw-parser-output table.numblk>tbody>tr>td:last-child::before{content:"("}.mw-parser-output table.numblk>tbody>tr>td:last-child::after{content:")"}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::before,.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::after{content:none}.mw-parser-output table.numblk>tbody>tr>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:none;border-right:none;border-bottom:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:thin solid;border-right:thin solid;border-bottom:thin solid}.mw-parser-output table.numblk:target{color:var(--color-base,#202122);background-color:#cfe8fd}@media screen{html.skin-theme-clientpref-night .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}</style><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} (t+1)=d{\mathcal {M}}\mathbf {R} (t)+{\frac {1-d}{N}}\mathbf {1} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>d</mi> </mrow> <mi>N</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} (t+1)=d{\mathcal {M}}\mathbf {R} (t)+{\frac {1-d}{N}}\mathbf {1} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96265e6c41318e793194287f36b5f929075bb876" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.644ex; height:5.343ex;" alt="{\displaystyle \mathbf {R} (t+1)=d{\mathcal {M}}\mathbf {R} (t)+{\frac {1-d}{N}}\mathbf {1} }" />,</td> <td></td> <td class="nowrap">1</td></tr></tbody></table> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} _{i}(t)=PR(p_{i};t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mi>R</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>;</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} _{i}(t)=PR(p_{i};t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2af6be614047773cafa5e1dd486240587e92b78a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.712ex; height:2.843ex;" alt="{\displaystyle \mathbf {R} _{i}(t)=PR(p_{i};t)}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/235ffc0f1788b720aef5caa7b97246a84421fd0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {1} }" /> is the column vector of length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" /> containing only ones. The matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc2abebd45ec020509a0ec548b67c9a2cb7cecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.176ex;" alt="{\displaystyle {\mathcal {M}}}" /> is defined as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{ij}={\begin{cases}1/L(p_{j}),&{\mbox{if }}j{\mbox{ links to }}i\ \\0,&{\mbox{otherwise}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>L</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> links to </mtext> </mstyle> </mrow> <mi>i</mi> <mtext> </mtext> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>otherwise</mtext> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{ij}={\begin{cases}1/L(p_{j}),&{\mbox{if }}j{\mbox{ links to }}i\ \\0,&{\mbox{otherwise}}\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e82b446a376633a386b10668703a4547f167d1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.446ex; height:6.176ex;" alt="{\displaystyle {\mathcal {M}}_{ij}={\begin{cases}1/L(p_{j}),&{\mbox{if }}j{\mbox{ links to }}i\ \\0,&{\mbox{otherwise}}\end{cases}}}" /></dd></dl> i.e., <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}:=(K^{-1}A)^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo>:=</mo> <mo stretchy="false">(</mo> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>A</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}:=(K^{-1}A)^{T}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f08514130e2e26a62ae8ff7645b19b4cab08daf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.904ex; height:3.176ex;" alt="{\displaystyle {\mathcal {M}}:=(K^{-1}A)^{T}}" />,</dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /> denotes the <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a> of the graph and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" /> is the diagonal matrix with the outdegrees in the diagonal. The probability calculation is made for each page at a time point, then repeated for the next time point. The computation ends when for some small <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" /> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {R} (t+1)-\mathbf {R} (t)|<\epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {R} (t+1)-\mathbf {R} (t)|<\epsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c60e37b238eed8b1bac59fe2841ecc718b8804a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.484ex; height:2.843ex;" alt="{\displaystyle |\mathbf {R} (t+1)-\mathbf {R} (t)|<\epsilon }" />,</dd></dl> i.e., when convergence is assumed. <div class="mw-heading mw-heading4"><h4 id="Power_method">Power method</h4>[<a href="/w/index.php?title=PageRank&action=edit&section=8" title="Edit section: Power method">edit</a>]</div> If the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc2abebd45ec020509a0ec548b67c9a2cb7cecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.176ex;" alt="{\displaystyle {\mathcal {M}}}" /> is a transition probability, i.e., column-stochastic and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }" /> is a probability distribution (i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {R} |=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {R} |=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3beb98543fc9b650f79d0e6769366bec65c0276c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.558ex; height:2.843ex;" alt="{\displaystyle |\mathbf {R} |=1}" />, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} \mathbf {R} =\mathbf {1} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} \mathbf {R} =\mathbf {1} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff32d3dadac699f47604b69356c606dddd5cce41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.195ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} \mathbf {R} =\mathbf {1} }" /> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7f22b39d51f780fc02859059c1757c606b9de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.757ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} }" /> is matrix of all ones), then equation (<a href="#math_2">2</a>) is equivalent to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} =\left(d{\mathcal {M}}+{\frac {1-d}{N}}\mathbf {E} \right)\mathbf {R} =:{\widehat {\mathcal {M}}}\mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>d</mi> </mrow> <mi>N</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>=:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} =\left(d{\mathcal {M}}+{\frac {1-d}{N}}\mathbf {E} \right)\mathbf {R} =:{\widehat {\mathcal {M}}}\mathbf {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb5144828734e99d7894bfaa1ee4c7d13c1021eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.201ex; height:6.176ex;" alt="{\displaystyle \mathbf {R} =\left(d{\mathcal {M}}+{\frac {1-d}{N}}\mathbf {E} \right)\mathbf {R} =:{\widehat {\mathcal {M}}}\mathbf {R} }" />.</td> <td></td> <td class="nowrap">3</td></tr></tbody></table> Hence PageRank <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }" /> is the principal eigenvector of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mathcal {M}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathcal {M}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7219d2c5c7b91b51336ce174e75e327118bc06f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.009ex; width:2.889ex; height:3.009ex;" alt="{\displaystyle {\widehat {\mathcal {M}}}}" />. A fast and easy way to compute this is using the <a href="/wiki/Power_iteration" title="Power iteration">power method</a>: starting with an arbitrary vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f7176643e6d36fa7674dc79fdff1a4daa068f5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.301ex; height:2.843ex;" alt="{\displaystyle x(0)}" />, the operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mathcal {M}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathcal {M}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7219d2c5c7b91b51336ce174e75e327118bc06f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.009ex; width:2.889ex; height:3.009ex;" alt="{\displaystyle {\widehat {\mathcal {M}}}}" /> is applied in succession, i.e., <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t+1)={\widehat {\mathcal {M}}}x(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t+1)={\widehat {\mathcal {M}}}x(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f06088bcf96afd1716cf954eeb4786682a0ab46e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.939ex; height:3.509ex;" alt="{\displaystyle x(t+1)={\widehat {\mathcal {M}}}x(t)}" />,</dd></dl> until <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x(t+1)-x(t)|<\epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x(t+1)-x(t)|<\epsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6da4f6374e7e362fa12683af73b0c777125aba4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.137ex; height:2.843ex;" alt="{\displaystyle |x(t+1)-x(t)|<\epsilon }" />.</dd></dl> Note that in equation (<a href="#math_3">3</a>) the matrix on the right-hand side in the parenthesis can be interpreted as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1-d}{N}}\mathbf {E} =(1-d)\mathbf {P} \mathbf {1} ^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>d</mi> </mrow> <mi>N</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1-d}{N}}\mathbf {E} =(1-d)\mathbf {P} \mathbf {1} ^{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/645336287367348454ada56dc36d0779eb00f851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.928ex; height:5.343ex;" alt="{\displaystyle {\frac {1-d}{N}}\mathbf {E} =(1-d)\mathbf {P} \mathbf {1} ^{t}}" />,</dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c250ef2a112c86b93c637dfa288c6d7f34ac3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }" /> is an initial probability distribution. n the current case <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} :={\frac {1}{N}}\mathbf {1} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} :={\frac {1}{N}}\mathbf {1} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69b312f0f87863d243f1b536de21937d37bf0180" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.808ex; height:5.176ex;" alt="{\displaystyle \mathbf {P} :={\frac {1}{N}}\mathbf {1} }" />.</dd></dl> Finally, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc2abebd45ec020509a0ec548b67c9a2cb7cecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.176ex;" alt="{\displaystyle {\mathcal {M}}}" /> has columns with only zero values, they should be replaced with the initial probability vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c250ef2a112c86b93c637dfa288c6d7f34ac3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }" />. In other words, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}^{\prime }:={\mathcal {M}}+{\mathcal {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}^{\prime }:={\mathcal {M}}+{\mathcal {D}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7507bea0b6e13daa6502f1c26e0425efbd456353" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.643ex; height:2.676ex;" alt="{\displaystyle {\mathcal {M}}^{\prime }:={\mathcal {M}}+{\mathcal {D}}}" />,</dd></dl> where the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3277962e1959c3241fb1b70c7f0ac6dcefebd966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle {\mathcal {D}}}" /> is defined as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}:=\mathbf {P} \mathbf {D} ^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}:=\mathbf {P} \mathbf {D} ^{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dec37bc118b4d3331649b6d2103784454442548" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.24ex; height:2.509ex;" alt="{\displaystyle {\mathcal {D}}:=\mathbf {P} \mathbf {D} ^{t}}" />,</dd></dl> with <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {D} _{i}={\begin{cases}1,&{\mbox{if }}L(p_{i})=0\ \\0,&{\mbox{otherwise}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mi>L</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mtext> </mtext> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>otherwise</mtext> </mstyle> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {D} _{i}={\begin{cases}1,&{\mbox{if }}L(p_{i})=0\ \\0,&{\mbox{otherwise}}\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f6762e541780bcec1fe8c9a4581c6553f9ec903" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.716ex; height:6.176ex;" alt="{\displaystyle \mathbf {D} _{i}={\begin{cases}1,&{\mbox{if }}L(p_{i})=0\ \\0,&{\mbox{otherwise}}\end{cases}}}" /></dd></dl> In this case, the above two computations using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc2abebd45ec020509a0ec548b67c9a2cb7cecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.176ex;" alt="{\displaystyle {\mathcal {M}}}" /> only give the same PageRank if their results are normalized: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} _{\textrm {power}}={\frac {\mathbf {R} _{\textrm {iterative}}}{|\mathbf {R} _{\textrm {iterative}}|}}={\frac {\mathbf {R} _{\textrm {algebraic}}}{|\mathbf {R} _{\textrm {algebraic}}|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>power</mtext> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>iterative</mtext> </mrow> </mrow> </msub> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>iterative</mtext> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>algebraic</mtext> </mrow> </mrow> </msub> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>algebraic</mtext> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} _{\textrm {power}}={\frac {\mathbf {R} _{\textrm {iterative}}}{|\mathbf {R} _{\textrm {iterative}}|}}={\frac {\mathbf {R} _{\textrm {algebraic}}}{|\mathbf {R} _{\textrm {algebraic}}|}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60cd59b033a85159606117df3e5af2f5010da9c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.848ex; height:6.343ex;" alt="{\displaystyle \mathbf {R} _{\textrm {power}}={\frac {\mathbf {R} _{\textrm {iterative}}}{|\mathbf {R} _{\textrm {iterative}}|}}={\frac {\mathbf {R} _{\textrm {algebraic}}}{|\mathbf {R} _{\textrm {algebraic}}|}}}" />.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Implementation">Implementation</h3>[<a href="/w/index.php?title=PageRank&action=edit&section=9" title="Edit section: Implementation">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Python"><a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a></h4>[<a href="/w/index.php?title=PageRank&action=edit&section=10" title="Edit section: Python">edit</a>]</div> <div class="mw-highlight mw-highlight-lang-numpy mw-content-ltr" dir="ltr"><pre>import numpy as np def pagerank(M, d: float = 0.85): """PageRank algorithm with explicit number of iterations. Returns ranking of nodes (pages) in the adjacency matrix. Parameters ---------- M : numpy array adjacency matrix where M_i,j represents the link from 'j' to 'i', such that for all 'j' sum(i, M_i,j) = 1 d : float, optional damping factor, by default 0.85 Returns ------- numpy array a vector of ranks such that v_i is the i-th rank from [0, 1], """ N = M.shape[1] w = np.ones(N) / N M_hat = d * M v = M_hat @ w + (1 - d) / N while(np.linalg.norm(w - v) >= 1e-10): w = v v = M_hat @ w + (1 - d) / N return v M = np.array([[0, 0, 0, .25], [0, 0, 0, .5], [1, 0.5, 0, .25], [0, 0.5, 1, 0]]) v = pagerank(M, 0.85) </pre></div> <div class="mw-heading mw-heading2"><h2 id="Variations">Variations</h2>[<a href="/w/index.php?title=PageRank&action=edit&section=11" title="Edit section: Variations">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="PageRank_of_an_undirected_graph">PageRank of an undirected graph</h3>[<a href="/w/index.php?title=PageRank&action=edit&section=12" title="Edit section: PageRank of an undirected graph">edit</a>]</div> The PageRank of an undirected <a href="/wiki/Graph_(data_structure)" class="mw-redirect" title="Graph (data structure)">graph</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> is statistically close to the <a href="/wiki/Degree_distribution" title="Degree distribution">degree distribution</a> of the graph <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" />,<a href="#cite_note-37">[37]</a> but they are generally not identical: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> is the PageRank vector defined above, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" /> is the degree distribution vector <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D={1 \over 2|E|}{\begin{bmatrix}\deg(p_{1})\\\deg(p_{2})\\\vdots \\\deg(p_{N})\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D={1 \over 2|E|}{\begin{bmatrix}\deg(p_{1})\\\deg(p_{2})\\\vdots \\\deg(p_{N})\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c23dae3eff4fd2e328e6ef3299e14344674972b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:22.1ex; height:14.176ex;" alt="{\displaystyle D={1 \over 2|E|}{\begin{bmatrix}\deg(p_{1})\\\deg(p_{2})\\\vdots \\\deg(p_{N})\end{bmatrix}}}" /></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg(p_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \deg(p_{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33cda9ae5483b54746019bd1027c98475882a6d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.266ex; height:2.843ex;" alt="{\displaystyle \deg(p_{i})}" /> denotes the degree of vertex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}" />, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" /> is the edge-set of the graph, then, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y={1 \over N}\mathbf {1} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y={1 \over N}\mathbf {1} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39b16e8f78f22ae0e578b255804b6e07ae23b5e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.108ex; height:5.176ex;" alt="{\displaystyle Y={1 \over N}\mathbf {1} }" />,<a href="#cite_note-38">[38]</a> shows that: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {1-d \over 1+d}\|Y-D\|_{1}\leq \|R-D\|_{1}\leq \|Y-D\|_{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>d</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>d</mi> </mrow> </mfrac> </mrow> <mo fence="false" stretchy="false">‖</mo> <mi>Y</mi> <mo>−</mo> <mi>D</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>R</mi> <mo>−</mo> <mi>D</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>Y</mi> <mo>−</mo> <mi>D</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {1-d \over 1+d}\|Y-D\|_{1}\leq \|R-D\|_{1}\leq \|Y-D\|_{1},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f2616581ca7f7b33dca40b0b15c969fb5e0bb28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:42.641ex; height:5.676ex;" alt="{\displaystyle {1-d \over 1+d}\|Y-D\|_{1}\leq \|R-D\|_{1}\leq \|Y-D\|_{1},}" /> that is, the PageRank of an undirected graph equals to the degree distribution vector if and only if the graph is regular, i.e., every vertex has the same degree. <div class="mw-heading mw-heading3"><h3 id="Ranking_objects_of_two_kinds">Ranking objects of two kinds</h3>[<a href="/w/index.php?title=PageRank&action=edit&section=13" title="Edit section: Ranking objects of two kinds">edit</a>]</div> A generalization of PageRank for the case of ranking two interacting groups of objects was described by Daugulis.<a href="#cite_note-39">[39]</a> In applications it may be necessary to model systems having objects of two kinds where a weighted relation is defined on object pairs. This leads to considering <a href="/wiki/Bipartite_graphs" class="mw-redirect" title="Bipartite graphs">bipartite graphs</a>. For such graphs two related positive or nonnegative irreducible matrices corresponding to vertex partition sets can be defined. One can compute rankings of objects in both groups as eigenvectors corresponding to the maximal positive eigenvalues of these matrices. Normed eigenvectors exist and are unique by the Perron or Perron–Frobenius theorem. Example: consumers and products. The relation weight is the product consumption rate. <div class="mw-heading mw-heading3"><h3 id="Distributed_algorithm_for_PageRank_computation">Distributed algorithm for PageRank computation</h3>[<a href="/w/index.php?title=PageRank&action=edit&section=14" title="Edit section: Distributed algorithm for PageRank computation">edit</a>]</div> Sarma et al. describe two <a href="/wiki/Random_walk" title="Random walk">random walk</a>-based <a href="/wiki/Distributed_algorithms" class="mw-redirect" title="Distributed algorithms">distributed algorithms</a> for computing PageRank of nodes in a network.<a href="#cite_note-40">[40]</a> One algorithm takes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(\log n/\epsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(\log n/\epsilon )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64956675fad27462a0cc44acd4f908edf48c6932" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.443ex; height:2.843ex;" alt="{\displaystyle O(\log n/\epsilon )}" /> rounds with high probability on any graph (directed or undirected), where n is the network size and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" /> is the reset probability (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\epsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57f9b07affe80ff61cdc4f2e47977c8421a59c73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.947ex; height:2.343ex;" alt="{\displaystyle 1-\epsilon }" />, which is called the damping factor) used in the PageRank computation. They also present a faster algorithm that takes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O({\sqrt {\log n}}/\epsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O({\sqrt {\log n}}/\epsilon )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c707fb5dbfab344e9b20d6f26bd00732474263dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.766ex; height:3.509ex;" alt="{\displaystyle O({\sqrt {\log n}}/\epsilon )}" /> rounds in undirected graphs. In both algorithms, each node processes and sends a number of bits per round that are polylogarithmic in n, the network size. <div class="mw-heading mw-heading3"><h3 id="Google_Toolbar">Google Toolbar</h3>[<a href="/w/index.php?title=PageRank&action=edit&section=15" title="Edit section: Google Toolbar">edit</a>]</div> The <a href="/wiki/Google_Toolbar" title="Google Toolbar">Google Toolbar</a> long had a PageRank feature which displayed a visited page's PageRank as a whole number between 0 (least popular) and 10 (most popular). Google had not disclosed the specific method for determining a Toolbar PageRank value, which was to be considered only a rough indication of the value of a website. The "Toolbar Pagerank" was available for verified site maintainers through the Google Webmaster Tools interface. However, on October 15, 2009, a Google employee confirmed that the company had removed PageRank from its Webmaster Tools section, saying that "We've been telling people for a long time that they shouldn't focus on PageRank so much. Many site owners seem to think it's the most important <a href="/wiki/Search_Engine_Optimization_Metrics" class="mw-redirect" title="Search Engine Optimization Metrics">metric</a> for them to track, which is simply not true."<a href="#cite_note-Moskwa-41">[41]</a> The "Toolbar Pagerank" was updated very infrequently. It was last updated in November 2013. In October 2014 Matt Cutts announced that another visible pagerank update would not be coming.<a href="#cite_note-42">[42]</a> In March 2016 Google announced it would no longer support this feature, and the underlying API would soon cease to operate.<a href="#cite_note-43">[43]</a> On April 15, 2016, Google turned off display of PageRank Data in Google Toolbar,<a href="#cite_note-44">[44]</a> though the PageRank continued to be used internally to rank content in search results.<a href="#cite_note-45">[45]</a> <div class="mw-heading mw-heading3"><h3 id="SERP_rank">SERP rank</h3>[<a href="/w/index.php?title=PageRank&action=edit&section=16" title="Edit section: SERP rank">edit</a>]</div> The <a href="/wiki/Search_engine_results_page" title="Search engine results page">search engine results page</a> (SERP) is the actual result returned by a search engine in response to a keyword query. The SERP consists of a list of links to web pages with associated text snippets, paid ads, featured snippets, and Q&A. The SERP rank of a web page refers to the placement of the corresponding link on the SERP, where higher placement means higher SERP rank. The SERP rank of a web page is a function not only of its PageRank, but of a relatively large and continuously adjusted set of factors (over 200).<a href="#cite_note-46">[46]</a>[<a href="/wiki/Wikipedia:Reliable_sources" title="Wikipedia:Reliable sources">unreliable source?</a>] <a href="/wiki/Search_engine_optimization" title="Search engine optimization">Search engine optimization</a> (SEO) is aimed at influencing the SERP rank for a website or a set of web pages. Positioning of a webpage on Google SERPs for a keyword depends on relevance and reputation, also known as authority and popularity. PageRank is Google's indication of its assessment of the reputation of a webpage: It is non-keyword specific. Google uses a combination of webpage and website authority to determine the overall authority of a webpage competing for a keyword.<a href="#cite_note-47">[47]</a> The PageRank of the HomePage of a website is the best indication Google offers for website authority.<a href="#cite_note-48">[48]</a> After the introduction of <a href="/wiki/Google_Places" class="mw-redirect" title="Google Places">Google Places</a> into the mainstream organic SERP, numerous other factors in addition to PageRank affect ranking a business in Local Business Results.<a href="#cite_note-49">[49]</a> When Google elaborated on the reasons for PageRank deprecation at Q&A #March 2016, they announced Links and Content as the Top Ranking Factors. RankBrain had earlier in October 2015 been announced as the #3 Ranking Factor, so the Top 3 Factors have been confirmed officially by Google.<a href="#cite_note-50">[50]</a> <div class="mw-heading mw-heading3"><h3 id="Google_directory_PageRank">Google directory PageRank</h3>[<a href="/w/index.php?title=PageRank&action=edit&section=17" title="Edit section: Google directory PageRank">edit</a>]</div> The <a href="/wiki/Google_Directory" title="Google Directory">Google Directory</a> PageRank was an 8-unit measurement. Unlike the Google Toolbar, which shows a numeric PageRank value upon mouseover of the green bar, the Google Directory only displayed the bar, never the numeric values. Google Directory was closed on July 20, 2011.<a href="#cite_note-51">[51]</a> <div class="mw-heading mw-heading3"><h3 id="False_or_spoofed_PageRank">False or spoofed PageRank</h3>[<a href="/w/index.php?title=PageRank&action=edit&section=18" title="Edit section: False or spoofed PageRank">edit</a>]</div> It was known that the PageRank shown in the Toolbar could easily be <a href="/wiki/Website_spoofing" title="Website spoofing">spoofed</a>. Redirection from one page to another, either via a <a href="/wiki/HTTP_302" title="HTTP 302">HTTP 302</a> response or a "Refresh" <a href="/wiki/Meta_tag" class="mw-redirect" title="Meta tag">meta tag</a>, caused the source page to acquire the PageRank of the destination page. Hence, a new page with PR 0 and no incoming links could have acquired PR 10 by redirecting to the Google home page. Spoofing can usually be detected by performing a Google search for a source URL; if the URL of an entirely different site is displayed in the results, the latter URL may represent the destination of a redirection. <div class="mw-heading mw-heading3"><h3 id="Manipulating_PageRank">Manipulating PageRank</h3>[<a href="/w/index.php?title=PageRank&action=edit&section=19" title="Edit section: Manipulating PageRank">edit</a>]</div> For <a href="/wiki/Search_engine_optimization" title="Search engine optimization">search engine optimization</a> purposes, some companies offer to sell high PageRank links to webmasters.<a href="#cite_note-Cutts-0414-52">[52]</a> As links from higher-PR pages are believed to be more valuable, they tend to be more expensive. It can be an effective and viable marketing strategy to buy link advertisements on content pages of quality and relevant sites to drive traffic and increase a webmaster's link popularity. However, Google has publicly warned webmasters that if they are or were discovered to be selling links for the purpose of conferring PageRank and reputation, their links will be devalued (ignored in the calculation of other pages' PageRanks). The practice of buying and selling <a href="#cite_note-53">[53]</a> is intensely debated across the Webmaster community. Google advised webmasters to use the <a href="/wiki/Nofollow" title="Nofollow">nofollow</a> <a href="/wiki/HTML_attribute" title="HTML attribute">HTML attribute</a> value on paid links. According to <a href="/wiki/Matt_Cutts" title="Matt Cutts">Matt Cutts</a>, Google is concerned about webmasters who try to <a href="/wiki/Game_the_system" class="mw-redirect" title="Game the system">game the system</a>, and thereby reduce the quality and relevance of Google search results.<a href="#cite_note-Cutts-0414-52">[52]</a> In 2019, Google offered a new type of tags that do not pass PageRank and thus do not have value for SEO link manipulation: rel="ugc" as a tag for user-generated content, such as comments; and rel="sponsored" tag for advertisements or other types of sponsored content.<a href="#cite_note-54">[54]</a> Even though PageRank has become less important for SEO purposes, the existence of back-links from more popular websites continues to push a webpage higher up in search rankings.<a href="#cite_note-55">[55]</a> <div class="mw-heading mw-heading3"><h3 id="Directed_Surfer_Model">Directed Surfer Model</h3>[<a href="/w/index.php?title=PageRank&action=edit&section=20" title="Edit section: Directed Surfer Model">edit</a>]</div> A more intelligent surfer that probabilistically hops from page to page depending on the content of the pages and query terms the surfer is looking for. This model is based on a query-dependent PageRank score of a page which as the name suggests is also a function of query. When given a multiple-term query, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=\{q1,q2,\cdots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>q</mi> <mn>1</mn> <mo>,</mo> <mi>q</mi> <mn>2</mn> <mo>,</mo> <mo>⋯</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=\{q1,q2,\cdots \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/def250a18328bba5b10b1a25dd38ca0a0086cae4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.517ex; height:2.843ex;" alt="{\displaystyle Q=\{q1,q2,\cdots \}}" />, the surfer selects a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" /> according to some probability distribution, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(q)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7076229c299e38dcffbc3bca2144e5f64fac9537" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.624ex; height:2.843ex;" alt="{\displaystyle P(q)}" />, and uses that term to guide its behavior for a large number of steps. It then selects another term according to the distribution to determine its behavior, and so on. The resulting distribution over visited web pages is QD-PageRank.<a href="#cite_note-PedMat-56">[56]</a> <div class="mw-heading mw-heading2"><h2 id="Other_uses">Other uses</h2>[<a href="/w/index.php?title=PageRank&action=edit&section=21" title="Edit section: Other uses">edit</a>]</div> The mathematics of PageRank are entirely general and apply to any graph or network in any domain. Thus, PageRank is now regularly used in bibliometrics, social and information network analysis, and for link prediction and recommendation. It is used for systems analysis of road networks, and in biology, chemistry, neuroscience, and physics.<a href="#cite_note-57">[57]</a> <div class="mw-heading mw-heading3"><h3 id="Scientific_research_and_academia">Scientific research and academia</h3>[<a href="/w/index.php?title=PageRank&action=edit&section=22" title="Edit section: Scientific research and academia">edit</a>]</div> PageRank has been used to quantify the scientific impact of researchers. The underlying citation and collaboration networks are used in conjunction with pagerank algorithm in order to come up with a ranking system for individual publications which propagates to individual authors. The new index known as pagerank-index (Pi) is demonstrated to be fairer compared to h-index in the context of many drawbacks exhibited by h-index.<a href="#cite_note-Senanayake2015-58">[58]</a> For the analysis of protein networks in biology PageRank is also a useful tool.<a href="#cite_note-59">[59]</a><a href="#cite_note-60">[60]</a> In any ecosystem, a modified version of PageRank may be used to determine species that are essential to the continuing health of the environment.<a href="#cite_note-61">[61]</a> A similar newer use of PageRank is to rank academic doctoral programs based on their records of placing their graduates in faculty positions. In PageRank terms, academic departments link to each other by hiring their faculty from each other (and from themselves).<a href="#cite_note-62">[62]</a> A version of PageRank has recently been proposed as a replacement for the traditional <a href="/wiki/Institute_for_Scientific_Information" title="Institute for Scientific Information">Institute for Scientific Information</a> (ISI) <a href="/wiki/Impact_factor" title="Impact factor">impact factor</a>,<a href="#cite_note-63">[63]</a> and implemented at <a href="/wiki/Eigenfactor" title="Eigenfactor">Eigenfactor</a> as well as at <a href="/wiki/SCImago" class="mw-redirect" title="SCImago">SCImago</a>. Instead of merely counting total citations to a journal, the "importance" of each citation is determined in a PageRank fashion. In <a href="/wiki/Neuroscience" title="Neuroscience">neuroscience</a>, the PageRank of a <a href="/wiki/Neuron" title="Neuron">neuron</a> in a neural network has been found to correlate with its relative firing rate.<a href="#cite_note-64">[64]</a> <div class="mw-heading mw-heading3"><h3 id="Internet_use">Internet use</h3>[<a href="/w/index.php?title=PageRank&action=edit&section=23" title="Edit section: Internet use">edit</a>]</div> Personalized PageRank is used by <a href="/wiki/Twitter" title="Twitter">Twitter</a> to present users with other accounts they may wish to follow.<a href="#cite_note-twitterwtf-65">[65]</a> <a href="/wiki/Swiftype" title="Swiftype">Swiftype</a>'s site search product builds a "PageRank that's specific to individual websites" by looking at each website's signals of importance and prioritizing content based on factors such as number of links from the home page.<a href="#cite_note-66">[66]</a> A <a href="/wiki/Web_crawler" title="Web crawler">Web crawler</a> may use PageRank as one of a number of importance metrics it uses to determine which URL to visit during a crawl of the web. One of the early working papers<a href="#cite_note-67">[67]</a> that were used in the creation of Google is Efficient crawling through URL ordering,<a href="#cite_note-68">[68]</a> which discusses the use of a number of different importance metrics to determine how deeply, and how much of a site Google will crawl. PageRank is presented as one of a number of these importance metrics, though there are others listed such as the number of inbound and outbound links for a URL, and the distance from the root directory on a site to the URL. The PageRank may also be used as a methodology to measure the apparent impact of a community like the <a href="/wiki/Blogosphere" title="Blogosphere">Blogosphere</a> on the overall Web itself. This approach uses therefore the PageRank to measure the distribution of attention in reflection of the <a href="/wiki/Scale-free_network" title="Scale-free network">Scale-free network</a> paradigm.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading3"><h3 id="Other_applications">Other applications</h3>[<a href="/w/index.php?title=PageRank&action=edit&section=24" title="Edit section: Other applications">edit</a>]</div> In 2005, in a pilot study in Pakistan, Structural Deep Democracy, SD2<a href="#cite_note-69">[69]</a><a href="#cite_note-70">[70]</a> was used for leadership selection in a sustainable agriculture group called Contact Youth. SD2 uses PageRank for the processing of the transitive proxy votes, with the additional constraints of mandating at least two initial proxies per voter, and all voters are proxy candidates. More complex variants can be built on top of SD2, such as adding specialist proxies and direct votes for specific issues, but SD2 as the underlying umbrella system, mandates that generalist proxies should always be used. In sport the PageRank algorithm has been used to rank the performance of: teams in the National Football League (NFL) in the USA;<a href="#cite_note-71">[71]</a> individual soccer players;<a href="#cite_note-72">[72]</a> and athletes in the Diamond League.<a href="#cite_note-73">[73]</a> PageRank has been used to rank spaces or streets to predict how many people (pedestrians or vehicles) come to the individual spaces or streets.<a href="#cite_note-74">[74]</a><a href="#cite_note-75">[75]</a> In <a href="/wiki/Lexical_semantics" title="Lexical semantics">lexical semantics</a> it has been used to perform <a href="/wiki/Word_Sense_Disambiguation" class="mw-redirect" title="Word Sense Disambiguation">Word Sense Disambiguation</a>,<a href="#cite_note-76">[76]</a> <a href="/wiki/Semantic_similarity" title="Semantic similarity">Semantic similarity</a>,<a href="#cite_note-77">[77]</a> and also to automatically rank <a href="/wiki/WordNet" title="WordNet">WordNet</a> <a href="/wiki/Synsets" class="mw-redirect" title="Synsets">synsets</a> according to how strongly they possess a given semantic property, such as positivity or negativity.<a href="#cite_note-78">[78]</a> How a traffic system changes its operational mode can be described by transitions between quasi-stationary states in correlation structures of traffic flow. PageRank has been used to identify and explore the dominant states among these quasi-stationary states in traffic systems.<a href="#cite_note-79">[79]</a> <div class="mw-heading mw-heading2"><h2 id="nofollow">nofollow</h2>[<a href="/w/index.php?title=PageRank&action=edit&section=25" title="Edit section: nofollow">edit</a>]</div> In early 2005, Google implemented a new value, "<a href="/wiki/Nofollow" title="Nofollow">nofollow</a>",<a href="#cite_note-80">[80]</a> for the <a href="/wiki/Semantic_link" class="mw-redirect" title="Semantic link">rel</a> attribute of HTML link and anchor elements, so that website developers and <a href="/wiki/Blog" title="Blog">bloggers</a> can make links that Google will not consider for the purposes of PageRank—they are links that no longer constitute a "vote" in the PageRank system. The nofollow relationship was added in an attempt to help combat <a href="/wiki/Spamdexing" title="Spamdexing">spamdexing</a>. As an example, people could previously create many message-board posts with links to their website to artificially inflate their PageRank. With the nofollow value, message-board administrators can modify their code to automatically insert "rel='nofollow'" to all hyperlinks in posts, thus preventing PageRank from being affected by those particular posts. This method of avoidance, however, also has various drawbacks, such as reducing the link value of legitimate comments. (See: <a href="/wiki/Spam_in_blogs#nofollow" title="Spam in blogs">Spam in blogs#nofollow</a>) In an effort to manually control the flow of PageRank among pages within a website, many webmasters practice what is known as PageRank Sculpting<a href="#cite_note-81">[81]</a>—which is the act of strategically placing the nofollow attribute on certain internal links of a website in order to funnel PageRank towards those pages the webmaster deemed most important. This tactic had been used since the inception of the nofollow attribute, but may no longer be effective since Google announced that blocking PageRank transfer with nofollow does not redirect that PageRank to other links.<a href="#cite_note-82">[82]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=PageRank&action=edit&section=26" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Attention_inequality" title="Attention inequality">Attention inequality</a></li> <li><a href="/wiki/CheiRank" title="CheiRank">CheiRank</a></li> <li><a href="/wiki/Domain_authority" title="Domain authority">Domain authority</a></li> <li><a href="/wiki/EigenTrust" title="EigenTrust">EigenTrust</a> — a decentralized PageRank algorithm</li> <li><a href="/wiki/Google_bombing" title="Google bombing">Google bombing</a></li> <li><a href="/wiki/Google_Hummingbird" title="Google Hummingbird">Google Hummingbird</a></li> <li><a href="/wiki/Google_matrix" title="Google matrix">Google matrix</a></li> <li><a href="/wiki/Google_Panda" title="Google Panda">Google Panda</a></li> <li><a href="/wiki/Google_Penguin" title="Google Penguin">Google Penguin</a></li> <li><a href="/wiki/Google_Search" title="Google Search">Google Search</a></li> <li><a href="/wiki/Hilltop_algorithm" title="Hilltop algorithm">Hilltop algorithm</a></li> <li><a href="/wiki/Katz_centrality" title="Katz centrality">Katz centrality</a> – a 1953 scheme closely related to pagerank</li> <li><a href="/wiki/Link_building" title="Link building">Link building</a></li> <li><a href="/wiki/Search_engine_optimization" title="Search engine optimization">Search engine optimization</a></li> <li><a href="/wiki/SimRank" title="SimRank">SimRank</a> — a measure of object-to-object similarity based on random-surfer model</li> <li><a href="/wiki/TrustRank" title="TrustRank">TrustRank</a></li> <li><a href="/wiki/VisualRank" title="VisualRank">VisualRank</a> - Google's application of PageRank to image-search</li> <li><a href="/wiki/Webgraph" title="Webgraph">Webgraph</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=PageRank&action=edit&section=27" title="Edit section: References">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3>[<a href="/w/index.php?title=PageRank&action=edit&section=28" title="Edit section: Citations">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 32em;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20111104131332/https://www.google.com/competition/howgooglesearchworks.html">"Facts about Google and Competition"</a>. Archived from <a rel="nofollow" class="external text" href="https://www.google.com/competition/howgooglesearchworks.html">the original</a> on 4 November 2011. Retrieved 12 July 2014.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Facts+about+Google+and+Competition&rft_id=https%3A%2F%2Fwww.google.com%2Fcompetition%2Fhowgooglesearchworks.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-:1-2"><a href="#cite_ref-:1_2-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSullivan2007" class="citation web cs1">Sullivan, Danny (2007-04-26). <a rel="nofollow" class="external text" href="http://searchengineland.com/what-is-google-pagerank-a-guide-for-searchers-webmasters-11068">"What Is Google PageRank? A Guide For Searchers & Webmasters"</a>. Search Engine Land. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160703031514/http://searchengineland.com/what-is-google-pagerank-a-guide-for-searchers-webmasters-11068">Archived</a> from the original on 2016-07-03.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Search+Engine+Land&rft.atitle=What+Is+Google+PageRank%3F+A+Guide+For+Searchers+%26+Webmasters&rft.date=2007-04-26&rft.aulast=Sullivan&rft.aufirst=Danny&rft_id=http%3A%2F%2Fsearchengineland.com%2Fwhat-is-google-pagerank-a-guide-for-searchers-webmasters-11068&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCutts" class="citation web cs1">Cutts, Matt. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130702063520/https://www.google.com/competition/howgooglesearchworks.html">"Algorithms Rank Relevant Results Higher"</a>. Archived from <a rel="nofollow" class="external text" href="https://www.google.com/competition/howgooglesearchworks.html">the original</a> on July 2, 2013. Retrieved 19 October 2015.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Algorithms+Rank+Relevant+Results+Higher&rft.aulast=Cutts&rft.aufirst=Matt&rft_id=https%3A%2F%2Fwww.google.com%2Fcompetition%2Fhowgooglesearchworks.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://patents.google.com/patent/US7058628B1/en">"US7058628B1 - Method for node ranking in a linked database - Google Patents"</a>. <a href="/wiki/Google_Patents" title="Google Patents">Google Patents</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200116103001/https://patents.google.com/patent/US7058628B1/en">Archived</a> from the original on January 16, 2020. Retrieved September 14, 2019.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Google+Patents&rft.atitle=US7058628B1+-+Method+for+node+ranking+in+a+linked+database+-+Google+Patents&rft_id=https%3A%2F%2Fpatents.google.com%2Fpatent%2FUS7058628B1%2Fen&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-originalpaper-5">^ <a href="#cite_ref-originalpaper_5-0">a</a> <a href="#cite_ref-originalpaper_5-1">b</a> <a href="#cite_ref-originalpaper_5-2">c</a> <a href="#cite_ref-originalpaper_5-3">d</a> <a href="#cite_ref-originalpaper_5-4">e</a> <a href="#cite_ref-originalpaper_5-5">f</a> <a href="#cite_ref-originalpaper_5-6">g</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBrinPage1998" class="citation journal cs1"><a href="/wiki/Sergey_Brin" title="Sergey Brin">Brin, S.</a>; <a href="/wiki/Larry_Page" title="Larry Page">Page, L.</a> (1998). <a rel="nofollow" class="external text" href="http://infolab.stanford.edu/pub/papers/google.pdf">"The anatomy of a large-scale hypertextual Web search engine"</a> (PDF). Computer Networks and ISDN Systems. 30 (1–7): 107–117. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.115.5930">10.1.1.115.5930</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0169-7552%2898%2900110-X">10.1016/S0169-7552(98)00110-X</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0169-7552">0169-7552</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7587743">7587743</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150927004511/http://infolab.stanford.edu/pub/papers/google.pdf">Archived</a> (PDF) from the original on 2015-09-27.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computer+Networks+and+ISDN+Systems&rft.atitle=The+anatomy+of+a+large-scale+hypertextual+Web+search+engine&rft.volume=30&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E107-%3C%2Fspan%3E117&rft.date=1998&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.115.5930%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A7587743%23id-name%3DS2CID&rft.issn=0169-7552&rft_id=info%3Adoi%2F10.1016%2FS0169-7552%2898%2900110-X&rft.aulast=Brin&rft.aufirst=S.&rft.au=Page%2C+L.&rft_id=http%3A%2F%2Finfolab.stanford.edu%2Fpub%2Fpapers%2Fgoogle.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGyöngyiBerkhinGarcia-MolinaPedersen2006" class="citation cs2">Gyöngyi, Zoltán; Berkhin, Pavel; Garcia-Molina, Hector; Pedersen, Jan (2006), "Link spam detection based on mass estimation", <a rel="nofollow" class="external text" href="http://ilpubs.stanford.edu:8090/697/1/2005-33.pdf">Proceedings of the 32nd International Conference on Very Large Data Bases (VLDB '06, Seoul, Korea)</a> (PDF), pp. 439–450, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141203194914/http://ilpubs.stanford.edu:8090/697/1/2005-33.pdf">archived</a> (PDF) from the original on 2014-12-03</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Link+spam+detection+based+on+mass+estimation&rft.btitle=Proceedings+of+the+32nd+International+Conference+on+Very+Large+Data+Bases+%28VLDB+%2706%2C+Seoul%2C+Korea%29&rft.pages=%3Cspan+class%3D%22nowrap%22%3E439-%3C%2Fspan%3E450&rft.date=2006&rft.aulast=Gy%C3%B6ngyi&rft.aufirst=Zolt%C3%A1n&rft.au=Berkhin%2C+Pavel&rft.au=Garcia-Molina%2C+Hector&rft.au=Pedersen%2C+Jan&rft_id=http%3A%2F%2Filpubs.stanford.edu%3A8090%2F697%2F1%2F2005-33.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988">. </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://searchengineland.com/google-hummingbird-172816">"FAQ: All About The New Google "Hummingbird" Algorithm"</a>. Search Engine Land. 26 September 2013. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20181223110045/https://searchengineland.com/google-hummingbird-172816">Archived</a> from the original on 23 December 2018. Retrieved 18 December 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Search+Engine+Land&rft.atitle=FAQ%3A+All+About+The+New+Google+%22Hummingbird%22+Algorithm&rft.date=2013-09-26&rft_id=https%3A%2F%2Fsearchengineland.com%2Fgoogle-hummingbird-172816&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWang" class="citation web cs1">Wang, Ziyang. <a rel="nofollow" class="external text" href="https://cs.nyu.edu/media/publications/TR2003-846.pdf">"Improved Link-Based Algorithms for Ranking Web Pages"</a> (PDF). cs.nyu.edu. New York University, Department of Computer Science. Retrieved 7 August 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=cs.nyu.edu&rft.atitle=Improved+Link-Based+Algorithms+for+Ranking+Web+Pages&rft.aulast=Wang&rft.aufirst=Ziyang&rft_id=https%3A%2F%2Fcs.nyu.edu%2Fmedia%2Fpublications%2FTR2003-846.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLandau1895" class="citation journal cs1">Landau, Edmund (1895). "Zur relativen Wertbemessung der Turnierresultate". Deutsches Wochenschach. 11 (42): 51–54.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Deutsches+Wochenschach&rft.atitle=Zur+relativen+Wertbemessung+der+Turnierresultate&rft.volume=11&rft.issue=42&rft.pages=%3Cspan+class%3D%22nowrap%22%3E51-%3C%2Fspan%3E54&rft.date=1895&rft.aulast=Landau&rft.aufirst=Edmund&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSinnZiegler2022" class="citation arxiv cs1">Sinn, Rainer; Ziegler, Günter M. (2022-10-31). "Landau on Chess Tournaments and Google's PageRank". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2210.17300">2210.17300</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.HO">math.HO</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Landau+on+Chess+Tournaments+and+Google%27s+PageRank&rft.date=2022-10-31&rft_id=info%3Aarxiv%2F2210.17300&rft.aulast=Sinn&rft.aufirst=Rainer&rft.au=Ziegler%2C+G%C3%BCnter+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGabriel_PinskiFrancis_Narin1976" class="citation journal cs1">Gabriel Pinski & Francis Narin (1976). "Citation influence for journal aggregates of scientific publications: Theory, with application to the literature of physics". Information Processing & Management. 12 (5): 297–312. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0306-4573%2876%2990048-0">10.1016/0306-4573(76)90048-0</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Information+Processing+%26+Management&rft.atitle=Citation+influence+for+journal+aggregates+of+scientific+publications%3A+Theory%2C+with+application+to+the+literature+of+physics&rft.volume=12&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E297-%3C%2Fspan%3E312&rft.date=1976&rft_id=info%3Adoi%2F10.1016%2F0306-4573%2876%2990048-0&rft.au=Gabriel+Pinski&rft.au=Francis+Narin&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFThomas_Saaty1977" class="citation journal cs1">Thomas Saaty (1977). "A scaling method for priorities in hierarchical structures". Journal of Mathematical Psychology. 15 (3): 234–281. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-2496%2877%2990033-5">10.1016/0022-2496(77)90033-5</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F101787">10338.dmlcz/101787</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Mathematical+Psychology&rft.atitle=A+scaling+method+for+priorities+in+hierarchical+structures&rft.volume=15&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E234-%3C%2Fspan%3E281&rft.date=1977&rft_id=info%3Ahdl%2F10338.dmlcz%2F101787&rft_id=info%3Adoi%2F10.1016%2F0022-2496%2877%2990033-5&rft.au=Thomas+Saaty&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBradley_C._LoveSteven_A._Sloman" class="citation book cs1">Bradley C. Love & Steven A. Sloman. <a rel="nofollow" class="external text" href="http://bradlove.org/papers/love_sloman_1995.pdf">"Mutability and the determinants of conceptual transformability"</a> (PDF). Proceedings of the Seventeenth Annual Conference of the Cognitive Science Society. pp. 654–659. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171223215954/http://bradlove.org/papers/love_sloman_1995.pdf">Archived</a> (PDF) from the original on 2017-12-23. Retrieved 2017-12-23.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Mutability+and+the+determinants+of+conceptual+transformability&rft.btitle=Proceedings+of+the+Seventeenth+Annual+Conference+of+the+Cognitive+Science+Society&rft.pages=%3Cspan+class%3D%22nowrap%22%3E654-%3C%2Fspan%3E659&rft.au=Bradley+C.+Love&rft.au=Steven+A.+Sloman&rft_id=http%3A%2F%2Fbradlove.org%2Fpapers%2Flove_sloman_1995.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-bradloveblog-14"><a href="#cite_ref-bradloveblog_14-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://bradlove.org/blog/cogsci-page-rank">"How a CogSci undergrad invented PageRank three years before Google"</a>. bradlove.org. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171211160207/http://bradlove.org/blog/cogsci-page-rank">Archived</a> from the original on 2017-12-11. Retrieved 2017-12-23.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=How+a+CogSci+undergrad+invented+PageRank+three+years+before+Google&rft.pub=bradlove.org&rft_id=http%3A%2F%2Fbradlove.org%2Fblog%2Fcogsci-page-rank&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLi2002" class="citation journal cs1">Li, Yanhong (August 6, 2002). "Toward a qualitative search engine". IEEE Internet Computing. 2 (4): 24–29. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F4236.707687">10.1109/4236.707687</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Internet+Computing&rft.atitle=Toward+a+qualitative+search+engine&rft.volume=2&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E24-%3C%2Fspan%3E29&rft.date=2002-08-06&rft_id=info%3Adoi%2F10.1109%2F4236.707687&rft.aulast=Li&rft.aufirst=Yanhong&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-nytimes-16"><a href="#cite_ref-nytimes_16-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation news cs1"><a rel="nofollow" class="external text" href="https://www.nytimes.com/2006/09/17/business/yourmoney/17baidu.html">"The Rise of Baidu (That's Chinese for Google)"</a>. <a href="/wiki/The_New_York_Times" title="The New York Times">The New York Times</a>. 17 September 2006. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190627071550/https://www.nytimes.com/2006/09/17/business/yourmoney/17baidu.html">Archived</a> from the original on 27 June 2019. Retrieved 16 June 2019.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+New+York+Times&rft.atitle=The+Rise+of+Baidu+%28That%27s+Chinese+for+Google%29&rft.date=2006-09-17&rft_id=https%3A%2F%2Fwww.nytimes.com%2F2006%2F09%2F17%2Fbusiness%2Fyourmoney%2F17baidu.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-rankdex-17">^ <a href="#cite_ref-rankdex_17-0">a</a> <a href="#cite_ref-rankdex_17-1">b</a> <a rel="nofollow" class="external text" href="http://www.rankdex.com/about.html">"About: RankDex"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150525015816/http://www.rankdex.com/about.html">Archived</a> 2015-05-25 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, <a href="/wiki/RankDex" class="mw-redirect" title="RankDex">RankDex</a>; accessed 3 May 2014. </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> USPTO, <a rel="nofollow" class="external text" href="https://www.google.com/patents?hl=en&lr=&vid=USPAT5920859&id=x04ZAAAAEBAJ&oi=fnd&dq=yanhong+li&printsec=abstract#v=onepage&q=yanhong%20li&f=false">"Hypertext Document Retrieval System and Method"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20111205225726/http://www.google.com/patents?hl=en&lr=&vid=USPAT5920859&id=x04ZAAAAEBAJ&oi=fnd&dq=yanhong+li&printsec=abstract#v=onepage&q=yanhong%20li&f=false">Archived</a> 2011-12-05 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, U.S. Patent number: 5920859, Inventor: Yanhong Li, Filing date: Feb 5, 1997, Issue date: Jul 6, 1999 </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> Greenberg, Andy, <a rel="nofollow" class="external text" href="https://www.forbes.com/forbes/2009/1005/technology-baidu-robin-li-man-whos-beating-google_2.html">"The Man Who's Beating Google"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130308094445/http://www.forbes.com/forbes/2009/1005/technology-baidu-robin-li-man-whos-beating-google_2.html">Archived</a> 2013-03-08 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Forbes magazine, October 05, 2009 </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <a rel="nofollow" class="external text" href="http://www.rankdex.com/about.html">"About: RankDex"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120120002301/http://www.rankdex.com/about.html">Archived</a> 2012-01-20 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, rankdex.com </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://patents.google.com/patent/US6285999">"Method for node ranking in a linked database"</a>. Google Patents. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151015185034/http://www.google.com/patents/US6285999">Archived</a> from the original on 15 October 2015. Retrieved 19 October 2015.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Method+for+node+ranking+in+a+linked+database&rft.pub=Google+Patents&rft_id=https%3A%2F%2Fpatents.google.com%2Fpatent%2FUS6285999&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAltucher2011" class="citation web cs1">Altucher, James (March 18, 2011). <a rel="nofollow" class="external text" href="https://www.forbes.com/sites/jamesaltucher/2011/03/18/10-unusual-things-about-google-also-the-worst-vc-decision-i-ever-made/">"10 Unusual Things About Google"</a>. <a href="/wiki/Forbes" title="Forbes">Forbes</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190616133656/https://www.forbes.com/sites/jamesaltucher/2011/03/18/10-unusual-things-about-google-also-the-worst-vc-decision-i-ever-made/">Archived</a> from the original on 16 June 2019. Retrieved 16 June 2019.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Forbes&rft.atitle=10+Unusual+Things+About+Google&rft.date=2011-03-18&rft.aulast=Altucher&rft.aufirst=James&rft_id=https%3A%2F%2Fwww.forbes.com%2Fsites%2Fjamesaltucher%2F2011%2F03%2F18%2F10-unusual-things-about-google-also-the-worst-vc-decision-i-ever-made%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGreg_Wientjes" class="citation web cs1">Greg Wientjes. <a rel="nofollow" class="external text" href="https://www.podomatic.com/podcasts/gwientjes/episodes/2015-09-05T06_05_37-07_00">"Hector Garcia-Molina: Stanford Computer Science Professor and Advisor to Sergey"</a>. pp. minutes 25.45-32.50, 34.00 – 38.20. Retrieved 2019-12-06.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Hector+Garcia-Molina%3A+Stanford+Computer+Science+Professor+and+Advisor+to+Sergey&rft.pages=minutes+25.45-32.50%2C+%3Cspan+class%3D%22nowrap%22%3E34.00+-%3C%2Fspan%3E+%3Cspan+class%3D%22nowrap%22%3E38.20%3C%2Fspan%3E&rft.au=Greg+Wientjes&rft_id=https%3A%2F%2Fwww.podomatic.com%2Fpodcasts%2Fgwientjes%2Fepisodes%2F2015-09-05T06_05_37-07_00&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> Page, Larry, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://ilpubs.stanford.edu:8090/422/1/1999-66.pdf">"PageRank: Bringing Order to the Web"</a> (PDF). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090126204112/http://ilpubs.stanford.edu:8090/422/1/1999-66.pdf">Archived</a> (PDF) from the original on January 26, 2009. Retrieved 2022-10-06.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=PageRank%3A+Bringing+Order+to+the+Web&rft_id=http%3A%2F%2Filpubs.stanford.edu%3A8090%2F422%2F1%2F1999-66.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988">, Stanford Digital Library Project, talk. August 18, 1997 (archived 2002) </li> <li id="cite_note-gpower-25"><a href="#cite_ref-gpower_25-0">^</a> <a rel="nofollow" class="external text" href="http://www.iicm.tugraz.at/Ressourcen/Papers/dangers_google.pdf">187-page study from Graz University, Austria</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140116234734/http://www.iicm.tugraz.at/Ressourcen/Papers/dangers_google.pdf">Archived</a> 2014-01-16 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, includes the note that also human brains are used when determining the page rank in Google. </li> <li id="cite_note-googletechnology-26"><a href="#cite_ref-googletechnology_26-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.google.com/technology/">"Our products and services"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080623233116/http://www.google.com/technology/">Archived</a> from the original on 2008-06-23. Retrieved 2011-05-27.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Our+products+and+services&rft_id=https%3A%2F%2Fwww.google.com%2Ftechnology%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDavid_ViseMark_Malseed2005" class="citation book cs1">David Vise & Mark Malseed (2005). <a rel="nofollow" class="external text" href="https://archive.org/details/googlestory00vise/page/37">The Google Story</a>. Delacorte Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/googlestory00vise/page/37">37</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-553-80457-7" title="Special:BookSources/978-0-553-80457-7"><bdi>978-0-553-80457-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Google+Story&rft.pages=37&rft.pub=Delacorte+Press&rft.date=2005&rft.isbn=978-0-553-80457-7&rft.au=David+Vise&rft.au=Mark+Malseed&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgooglestory00vise%2Fpage%2F37&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20010715123343/https://www.google.com/press/funfacts.html">"Google Press Center: Fun Facts"</a>. Archived from <a rel="nofollow" class="external text" href="https://www.google.com/press/funfacts.html">the original</a> on 2001-07-15.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Google+Press+Center%3A+Fun+Facts&rft_id=https%3A%2F%2Fwww.google.com%2Fpress%2Ffunfacts.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLisa_M._Krieger2005" class="citation web cs1">Lisa M. Krieger (1 December 2005). <a rel="nofollow" class="external text" href="http://www.redorbit.com/news/education/318480/stanford_earns_336_million_off_google_stock/">"Stanford Earns $336 Million Off Google Stock"</a>. San Jose Mercury News. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090408084918/http://www.redorbit.com/news/education/318480/stanford_earns_336_million_off_google_stock/">Archived</a> from the original on 8 April 2009. Retrieved 2009-02-25 – via cited by redOrbit.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=San+Jose+Mercury+News&rft.atitle=Stanford+Earns+%24336+Million+Off+Google+Stock&rft.date=2005-12-01&rft.au=Lisa+M.+Krieger&rft_id=http%3A%2F%2Fwww.redorbit.com%2Fnews%2Feducation%2F318480%2Fstanford_earns_336_million_off_google_stock%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRichard_Brandt" class="citation web cs1">Richard Brandt. <a rel="nofollow" class="external text" href="http://www.stanfordalumni.org/news/magazine/2004/novdec/features/startingup.html">"Starting Up. How Google got its groove"</a>. Stanford magazine. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090310190153/http://www.stanfordalumni.org/news/magazine/2004/novdec/features/startingup.html">Archived</a> from the original on 2009-03-10. Retrieved 2009-02-25.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Starting+Up.+How+Google+got+its+groove&rft.pub=Stanford+magazine&rft.au=Richard+Brandt&rft_id=http%3A%2F%2Fwww.stanfordalumni.org%2Fnews%2Fmagazine%2F2004%2Fnovdec%2Ffeatures%2Fstartingup.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-:0-31">^ <a href="#cite_ref-:0_31-0">a</a> <a href="#cite_ref-:0_31-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPageBrin,_SergeyMotwani,_RajeevWinograd,_Terry1999" class="citation report cs1"><a href="/wiki/Larry_Page" title="Larry Page">Page, Lawrence</a>; <a href="/wiki/Sergey_Brin" title="Sergey Brin">Brin, Sergey</a>; <a href="/wiki/Rajeev_Motwani" title="Rajeev Motwani">Motwani, Rajeev</a>; <a href="/wiki/Terry_Winograd" title="Terry Winograd">Winograd, Terry</a> (1999). <a rel="nofollow" class="external text" href="http://dbpubs.stanford.edu:8090/pub/showDoc.Fulltext?lang=en&doc=1999-66&format=pdf&compression=">The PageRank citation ranking: Bringing order to the Web</a> (Report). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060427013310/http://dbpubs.stanford.edu:8090/pub/showDoc.Fulltext?lang=en&doc=1999-66&format=pdf&compression=">Archived</a> from the original on 2006-04-27.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=The+PageRank+citation+ranking%3A+Bringing+order+to+the+Web&rft.date=1999&rft.aulast=Page&rft.aufirst=Lawrence&rft.au=Brin%2C+Sergey&rft.au=Motwani%2C+Rajeev&rft.au=Winograd%2C+Terry&rft_id=http%3A%2F%2Fdbpubs.stanford.edu%3A8090%2Fpub%2FshowDoc.Fulltext%3Flang%3Den%26doc%3D1999-66%26format%3Dpdf%26compression%3D&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988">, published as a technical report on January 29, 1998 <a rel="nofollow" class="external text" href="http://ilpubs.stanford.edu:8090/422/1/1999-66.pdf">PDF</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110818093436/http://ilpubs.stanford.edu:8090/422/1/1999-66.pdf">Archived</a> 2011-08-18 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> <a href="/wiki/Matt_Cutts" title="Matt Cutts">Matt Cutts</a>'s blog: <a rel="nofollow" class="external text" href="http://www.mattcutts.com/blog/seo-for-bloggers/">Straight from Google: What You Need to Know</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100207231048/http://www.mattcutts.com/blog/seo-for-bloggers/">Archived</a> 2010-02-07 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, see page 15 of his slides. </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTaher_HaveliwalaSepandar_Kamvar2003" class="citation journal cs1">Taher Haveliwala & Sepandar Kamvar (March 2003). <a rel="nofollow" class="external text" href="http://www-cs-students.stanford.edu/~taherh/papers/secondeigenvalue.pdf">"The Second Eigenvalue of the Google Matrix"</a> (PDF). Stanford University Technical Report: 7056. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0307056">math/0307056</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003math......7056N">2003math......7056N</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20081217094957/http://www-cs-students.stanford.edu/~taherh/papers/secondeigenvalue.pdf">Archived</a> (PDF) from the original on 2008-12-17.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Stanford+University+Technical+Report&rft.atitle=The+Second+Eigenvalue+of+the+Google+Matrix&rft.pages=7056&rft.date=2003-03&rft_id=info%3Aarxiv%2Fmath%2F0307056&rft_id=info%3Abibcode%2F2003math......7056N&rft.au=Taher+Haveliwala&rft.au=Sepandar+Kamvar&rft_id=http%3A%2F%2Fwww-cs-students.stanford.edu%2F~taherh%2Fpapers%2Fsecondeigenvalue.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGianna_M._Del_CorsoAntonio_GullíFrancesco_Romani2004" class="citation conference cs1">Gianna M. Del Corso; Antonio Gullí; Francesco Romani (2004). "Fast PageRank Computation Via a Sparse Linear System (Extended Abstract)". In Stefano Leonardi (ed.). Algorithms and Models for the Web-Graph: Third International Workshop, WAW 2004, Rome, Italy, October 16, 2004. Proceedings. pp. 118–130. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.58.9060">10.1.1.58.9060</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-30216-2_10">10.1007/978-3-540-30216-2_10</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-23427-2" title="Special:BookSources/978-3-540-23427-2"><bdi>978-3-540-23427-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Fast+PageRank+Computation+Via+a+Sparse+Linear+System+%28Extended+Abstract%29&rft.btitle=Algorithms+and+Models+for+the+Web-Graph%3A+Third+International+Workshop%2C+WAW+2004%2C+Rome%2C+Italy%2C+October+16%2C+2004.+Proceedings&rft.pages=%3Cspan+class%3D%22nowrap%22%3E118-%3C%2Fspan%3E130&rft.date=2004&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.58.9060%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1007%2F978-3-540-30216-2_10&rft.isbn=978-3-540-23427-2&rft.au=Gianna+M.+Del+Corso&rft.au=Antonio+Gull%C3%AD&rft.au=Francesco+Romani&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFArasu,_A.Novak,_J.Tomkins,_A.Tomlin,_J.2002" class="citation conference cs1">Arasu, A.; Novak, J.; Tomkins, A.; Tomlin, J. (2002). "PageRank computation and the structure of the web: Experiments and algorithms". Proceedings of the Eleventh International World Wide Web Conference, Poster Track. Brisbane, Australia. pp. 107–117. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.18.5264">10.1.1.18.5264</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=PageRank+computation+and+the+structure+of+the+web%3A+Experiments+and+algorithms&rft.btitle=Proceedings+of+the+Eleventh+International+World+Wide+Web+Conference%2C+Poster+Track&rft.place=Brisbane%2C+Australia&rft.pages=%3Cspan+class%3D%22nowrap%22%3E107-%3C%2Fspan%3E117&rft.date=2002&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.18.5264%23id-name%3DCiteSeerX&rft.au=Arasu%2C+A.&rft.au=Novak%2C+J.&rft.au=Tomkins%2C+A.&rft.au=Tomlin%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMassimo_Franceschet2010" class="citation arxiv cs1">Massimo Franceschet (2010). "PageRank: Standing on the shoulders of giants". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1002.2858">1002.2858</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.IR">cs.IR</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=PageRank%3A+Standing+on+the+shoulders+of+giants&rft.date=2010&rft_id=info%3Aarxiv%2F1002.2858&rft.au=Massimo+Franceschet&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNicola_Perra_and_Santo_FortunatoFortunato2008" class="citation journal cs1">Nicola Perra and Santo Fortunato; Fortunato (September 2008). "Spectral centrality measures in complex networks". Phys. Rev. E. 78 (3): 36107. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0805.3322">0805.3322</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008PhRvE..78c6107P">2008PhRvE..78c6107P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevE.78.036107">10.1103/PhysRevE.78.036107</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/18851105">18851105</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1755112">1755112</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phys.+Rev.+E&rft.atitle=Spectral+centrality+measures+in+complex+networks&rft.volume=78&rft.issue=3&rft.pages=36107&rft.date=2008-09&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A1755112%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2008PhRvE..78c6107P&rft_id=info%3Aarxiv%2F0805.3322&rft_id=info%3Apmid%2F18851105&rft_id=info%3Adoi%2F10.1103%2FPhysRevE.78.036107&rft.au=Nicola+Perra+and+Santo+Fortunato&rft.au=Fortunato&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFVince_Grolmusz2015" class="citation journal cs1">Vince Grolmusz (2015). "A Note on the PageRank of Undirected Graphs". Information Processing Letters. 115 (6–8): 633–634. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1205.1960">1205.1960</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ipl.2015.02.015">10.1016/j.ipl.2015.02.015</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:9855132">9855132</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Information+Processing+Letters&rft.atitle=A+Note+on+the+PageRank+of+Undirected+Graphs&rft.volume=115&rft.issue=%3Cspan+class%3D%22nowrap%22%3E6%E2%80%93%3C%2Fspan%3E8&rft.pages=%3Cspan+class%3D%22nowrap%22%3E633-%3C%2Fspan%3E634&rft.date=2015&rft_id=info%3Aarxiv%2F1205.1960&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A9855132%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.ipl.2015.02.015&rft.au=Vince+Grolmusz&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPeteris_DaugulisDaugulis2012" class="citation journal cs1">Peteris Daugulis; Daugulis (2012). "A note on a generalization of eigenvector centrality for bipartite graphs and applications". Networks. 59 (2): 261–264. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1610.01544">1610.01544</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fnet.20442">10.1002/net.20442</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1436859">1436859</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Networks&rft.atitle=A+note+on+a+generalization+of+eigenvector+centrality+for+bipartite+graphs+and+applications&rft.volume=59&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E261-%3C%2Fspan%3E264&rft.date=2012&rft_id=info%3Aarxiv%2F1610.01544&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A1436859%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1002%2Fnet.20442&rft.au=Peteris+Daugulis&rft.au=Daugulis&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAtish_Das_SarmaAnisur_Rahaman_MollaGopal_PanduranganEli_Upfal2015" class="citation journal cs1">Atish Das Sarma; Anisur Rahaman Molla; Gopal Pandurangan; Eli Upfal (2015). "Fast Distributed PageRank Computation". Theoretical Computer Science. 561: 113–121. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1208.3071">1208.3071</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.tcs.2014.04.003">10.1016/j.tcs.2014.04.003</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:10284718">10284718</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theoretical+Computer+Science&rft.atitle=Fast+Distributed+PageRank+Computation&rft.volume=561&rft.pages=%3Cspan+class%3D%22nowrap%22%3E113-%3C%2Fspan%3E121&rft.date=2015&rft_id=info%3Aarxiv%2F1208.3071&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A10284718%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.tcs.2014.04.003&rft.au=Atish+Das+Sarma&rft.au=Anisur+Rahaman+Molla&rft.au=Gopal+Pandurangan&rft.au=Eli+Upfal&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-Moskwa-41"><a href="#cite_ref-Moskwa_41-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSusan_Moskwa" class="citation web cs1">Susan Moskwa. <a rel="nofollow" class="external text" href="https://www.google.com/support/forum/p/Webmasters/thread?tid=6a1d6250e26e9e48&hl=en">"PageRank Distribution Removed From WMT"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20091017081100/http://www.google.com/support/forum/p/Webmasters/thread?tid=6a1d6250e26e9e48&hl=en">Archived</a> from the original on October 17, 2009. Retrieved October 16, 2009.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=PageRank+Distribution+Removed+From+WMT&rft.au=Susan+Moskwa&rft_id=https%3A%2F%2Fwww.google.com%2Fsupport%2Fforum%2Fp%2FWebmasters%2Fthread%3Ftid%3D6a1d6250e26e9e48%26hl%3Den&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBartleman2014" class="citation news cs1">Bartleman, Wil (2014-10-12). <a rel="nofollow" class="external text" href="https://managedadmin.com/phoenix-seo-services/google-page-rank-update-coming">"Google Page Rank Update is Not Coming"</a>. Managed Admin. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150402221831/https://managedadmin.com/phoenix-seo-services/google-page-rank-update-coming/">Archived</a> from the original on 2015-04-02. Retrieved 2014-10-12.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Google+Page+Rank+Update+is+Not+Coming&rft.date=2014-10-12&rft.aulast=Bartleman&rft.aufirst=Wil&rft_id=https%3A%2F%2Fmanagedadmin.com%2Fphoenix-seo-services%2Fgoogle-page-rank-update-coming&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSchwartz,_Barry2016" class="citation web cs1"><a href="/wiki/Barry_Schwartz_(technologist)" title="Barry Schwartz (technologist)">Schwartz, Barry</a> (March 8, 2016). <a rel="nofollow" class="external text" href="http://searchengineland.com/google-has-confirmed-they-are-removing-toolbar-pagerank-244230">"Google has confirmed it is removing Toolbar PageRank"</a>. <a href="/wiki/Search_Engine_Land" class="mw-redirect" title="Search Engine Land">Search Engine Land</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160310093405/http://searchengineland.com/google-has-confirmed-they-are-removing-toolbar-pagerank-244230">Archived</a> from the original on March 10, 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Search+Engine+Land&rft.atitle=Google+has+confirmed+it+is+removing+Toolbar+PageRank&rft.date=2016-03-08&rft.au=Schwartz%2C+Barry&rft_id=http%3A%2F%2Fsearchengineland.com%2Fgoogle-has-confirmed-they-are-removing-toolbar-pagerank-244230&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSchwartz2016" class="citation web cs1">Schwartz, Barry (18 April 2016). <a rel="nofollow" class="external text" href="http://searchengineland.com/google-toolbar-pagerank-officially-goes-dark-247553">"Google Toolbar PageRank officially goes dark"</a>. Search Engine Land. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160421224919/http://searchengineland.com/google-toolbar-pagerank-officially-goes-dark-247553">Archived</a> from the original on 2016-04-21.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Search+Engine+Land&rft.atitle=Google+Toolbar+PageRank+officially+goes+dark&rft.date=2016-04-18&rft.aulast=Schwartz&rft.aufirst=Barry&rft_id=http%3A%2F%2Fsearchengineland.com%2Fgoogle-toolbar-pagerank-officially-goes-dark-247553&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSouthern2016" class="citation web cs1">Southern, Matt (2016-04-19). <a rel="nofollow" class="external text" href="https://www.searchenginejournal.com/google-pagerank-official-shuts-doors-public/161874/">"Google PageRank Officially Shuts its Doors to the Public"</a>. Search Engine Journal. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170413154033/https://www.searchenginejournal.com/google-pagerank-official-shuts-doors-public/161874/">Archived</a> from the original on 2017-04-13.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Search+Engine+Journal&rft.atitle=Google+PageRank+Officially+Shuts+its+Doors+to+the+Public&rft.date=2016-04-19&rft.aulast=Southern&rft.aufirst=Matt&rft_id=https%3A%2F%2Fwww.searchenginejournal.com%2Fgoogle-pagerank-official-shuts-doors-public%2F161874%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFishkinJeff_Pollard2007" class="citation web cs1"><a href="/w/index.php?title=Rand_Fishkin&action=edit&redlink=1" class="new" title="Rand Fishkin (page does not exist)">Fishkin, Rand</a>; Jeff Pollard (April 2, 2007). <a rel="nofollow" class="external text" href="http://www.seomoz.org/article/search-ranking-factors">"Search Engine Ranking Factors - Version 2"</a>. seomoz.org. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090507232106/http://www.seomoz.org/article/search-ranking-factors/">Archived</a> from the original on May 7, 2009. Retrieved May 11, 2009.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Search+Engine+Ranking+Factors+-+Version+2&rft.pub=seomoz.org&rft.date=2007-04-02&rft.aulast=Fishkin&rft.aufirst=Rand&rft.au=Jeff+Pollard&rft_id=http%3A%2F%2Fwww.seomoz.org%2Farticle%2Fsearch-ranking-factors&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> Dover, D. Search Engine Optimization Secrets Indianapolis. Wiley. 2011. </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> Viniker, D. The Importance of Keyword Difficulty Screening for SEO. Ed. Schwartz, M. Digital Guidebook Volume 5. News Press. p 160–164. </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://support.google.com/places/bin/answer.py?hl=en&answer=7091">"Ranking of listings: Ranking - Google Places Help"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120526230407/http://support.google.com/places/bin/answer.py?hl=en&answer=7091">Archived</a> from the original on 2012-05-26. Retrieved 2011-05-27.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Ranking+of+listings%3A+Ranking+-+Google+Places+Help&rft_id=https%3A%2F%2Fsupport.google.com%2Fplaces%2Fbin%2Fanswer.py%3Fhl%3Den%26answer%3D7091&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFClark" class="citation news cs1">Clark, Jack. <a rel="nofollow" class="external text" href="https://www.bloomberg.com/news/articles/2015-10-26/google-turning-its-lucrative-web-search-over-to-ai-machines">"Google Turning Its Lucrative Web Search Over to AI Machines"</a>. Bloomberg. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160325232656/http://www.bloomberg.com/news/articles/2015-10-26/google-turning-its-lucrative-web-search-over-to-ai-machines">Archived</a> from the original on 25 March 2016. Retrieved 26 March 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Google+Turning+Its+Lucrative+Web+Search+Over+to+AI+Machines&rft.aulast=Clark&rft.aufirst=Jack&rft_id=https%3A%2F%2Fwww.bloomberg.com%2Fnews%2Farticles%2F2015-10-26%2Fgoogle-turning-its-lucrative-web-search-over-to-ai-machines&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> <a rel="nofollow" class="external text" href="https://www.searchenginewatch.com/2011/07/25/google-directory-has-been-shut-down/">Search Engine Watch: Google Directory Has Been Shut Down July 25, 2011</a> </li> <li id="cite_note-Cutts-0414-52">^ <a href="#cite_ref-Cutts-0414_52-0">a</a> <a href="#cite_ref-Cutts-0414_52-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.mattcutts.com/blog/how-to-report-paid-links/">"How to report paid links"</a>. mattcutts.com/blog. April 14, 2007. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070528054725/http://www.mattcutts.com/blog/how-to-report-paid-links/">Archived</a> from the original on May 28, 2007. Retrieved 2007-05-28.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=How+to+report+paid+links&rft.pub=mattcutts.com%2Fblog&rft.date=2007-04-14&rft_id=http%3A%2F%2Fwww.mattcutts.com%2Fblog%2Fhow-to-report-paid-links%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> <a rel="nofollow" class="external text" href="https://support.google.com/webmasters/answer/66356?hl=en">"Google Link Schemes"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200521035518/https://support.google.com/webmasters/answer/66356?hl=en">Archived</a> 2020-05-21 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> links </li> <li id="cite_note-54"><a href="#cite_ref-54">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://developers.google.com/search/blog/2019/09/evolving-nofollow-new-ways-to-identify">"Evolving"</a>. Google Developers. Retrieved 2022-02-08.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Google+Developers&rft.atitle=Evolving&rft_id=https%3A%2F%2Fdevelopers.google.com%2Fsearch%2Fblog%2F2019%2F09%2Fevolving-nofollow-new-ways-to-identify&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://searchenginewatch.com/article/2334934/So...-You-Think-SEO-Has-Changed">"So...You Think SEO Has Changed"</a>. 19 March 2014. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140331043853/http://searchenginewatch.com/article/2334934/So...-You-Think-SEO-Has-Changed">Archived</a> from the original on 31 March 2014.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=So...You+Think+SEO+Has+Changed&rft.date=2014-03-19&rft_id=http%3A%2F%2Fsearchenginewatch.com%2Farticle%2F2334934%2FSo...-You-Think-SEO-Has-Changed&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-PedMat-56"><a href="#cite_ref-PedMat_56-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMatthew_RichardsonPedro_Domingos,_A.2001" class="citation book cs1">Matthew Richardson & Pedro Domingos, A. (2001). <a rel="nofollow" class="external text" href="http://research.microsoft.com/pubs/66874/qd-pagerank.pdf">The Intelligent Surfer:Probabilistic Combination of Link and Content Information in PageRank</a> (PDF). pp. 1441–1448. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304112752/http://research.microsoft.com/pubs/66874/qd-pagerank.pdf">Archived</a> (PDF) from the original on 2016-03-04.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Intelligent+Surfer%3AProbabilistic+Combination+of+Link+and+Content+Information+in+PageRank&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1441-%3C%2Fspan%3E1448&rft.date=2001&rft.au=Matthew+Richardson&rft.au=Pedro+Domingos%2C+A.&rft_id=http%3A%2F%2Fresearch.microsoft.com%2Fpubs%2F66874%2Fqd-pagerank.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-57"><a href="#cite_ref-57">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGleich2015" class="citation journal cs1">Gleich, David F. (January 2015). "PageRank Beyond the Web". SIAM Review. 57 (3): 321–363. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1407.5107">1407.5107</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F140976649">10.1137/140976649</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:8375649">8375649</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIAM+Review&rft.atitle=PageRank+Beyond+the+Web&rft.volume=57&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E321-%3C%2Fspan%3E363&rft.date=2015-01&rft_id=info%3Aarxiv%2F1407.5107&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A8375649%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1137%2F140976649&rft.aulast=Gleich&rft.aufirst=David+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-Senanayake2015-58"><a href="#cite_ref-Senanayake2015_58-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSenanayakePiraveenanZomaya2015" class="citation journal cs1">Senanayake, Upul; Piraveenan, Mahendra; Zomaya, Albert (2015). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4545754">"The Pagerank-Index: Going beyond Citation Counts in Quantifying Scientific Impact of Researchers"</a>. PLOS ONE. 10 (8): e0134794. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015PLoSO..1034794S">2015PLoSO..1034794S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1371%2Fjournal.pone.0134794">10.1371/journal.pone.0134794</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1932-6203">1932-6203</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4545754">4545754</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26288312">26288312</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=PLOS+ONE&rft.atitle=The+Pagerank-Index%3A+Going+beyond+Citation+Counts+in+Quantifying+Scientific+Impact+of+Researchers&rft.volume=10&rft.issue=8&rft.pages=e0134794&rft.date=2015&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4545754%23id-name%3DPMC&rft_id=info%3Abibcode%2F2015PLoSO..1034794S&rft_id=info%3Apmid%2F26288312&rft_id=info%3Adoi%2F10.1371%2Fjournal.pone.0134794&rft.issn=1932-6203&rft.aulast=Senanayake&rft.aufirst=Upul&rft.au=Piraveenan%2C+Mahendra&rft.au=Zomaya%2C+Albert&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4545754&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-59"><a href="#cite_ref-59">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFG._IvanV._Grolmusz2011" class="citation journal cs1">G. Ivan & V. Grolmusz (2011). <a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbioinformatics%2Fbtq680">"When the Web meets the cell: using personalized PageRank for analyzing protein interaction networks"</a>. Bioinformatics. 27 (3): 405–7. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbioinformatics%2Fbtq680">10.1093/bioinformatics/btq680</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/21149343">21149343</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bioinformatics&rft.atitle=When+the+Web+meets+the+cell%3A+using+personalized+PageRank+for+analyzing+protein+interaction+networks&rft.volume=27&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E405-%3C%2Fspan%3E7&rft.date=2011&rft_id=info%3Adoi%2F10.1093%2Fbioinformatics%2Fbtq680&rft_id=info%3Apmid%2F21149343&rft.au=G.+Ivan&rft.au=V.+Grolmusz&rft_id=https%3A%2F%2Fdoi.org%2F10.1093%252Fbioinformatics%252Fbtq680&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-60"><a href="#cite_ref-60">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFD._Banky_and_G._Ivan_and_V._Grolmusz2013" class="citation journal cs1">D. Banky and G. Ivan and V. Grolmusz (2013). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3558500">"Equal opportunity for low-degree network nodes: a PageRank-based method for protein target identification in metabolic graphs"</a>. PLOS ONE. 8 (1): 405–7. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013PLoSO...854204B">2013PLoSO...854204B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1371%2Fjournal.pone.0054204">10.1371/journal.pone.0054204</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3558500">3558500</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/23382878">23382878</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=PLOS+ONE&rft.atitle=Equal+opportunity+for+low-degree+network+nodes%3A+a+PageRank-based+method+for+protein+target+identification+in+metabolic+graphs&rft.volume=8&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E405-%3C%2Fspan%3E7&rft.date=2013&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3558500%23id-name%3DPMC&rft_id=info%3Apmid%2F23382878&rft_id=info%3Adoi%2F10.1371%2Fjournal.pone.0054204&rft_id=info%3Abibcode%2F2013PLoSO...854204B&rft.au=D.+Banky+and+G.+Ivan+and+V.+Grolmusz&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3558500&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-61"><a href="#cite_ref-61">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBurns2009" class="citation news cs1">Burns, Judith (2009-09-04). <a rel="nofollow" class="external text" href="http://news.bbc.co.uk/2/hi/science/nature/8238462.stm">"Google trick tracks extinctions"</a>. BBC News. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110512060125/http://news.bbc.co.uk/2/hi/science/nature/8238462.stm">Archived</a> from the original on 2011-05-12. Retrieved 2011-05-27.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=BBC+News&rft.atitle=Google+trick+tracks+extinctions&rft.date=2009-09-04&rft.aulast=Burns&rft.aufirst=Judith&rft_id=http%3A%2F%2Fnews.bbc.co.uk%2F2%2Fhi%2Fscience%2Fnature%2F8238462.stm&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-62"><a href="#cite_ref-62">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBenjamin_M._SchmidtMatthew_M._Chingos2007" class="citation journal cs1">Benjamin M. Schmidt & Matthew M. Chingos (2007). <a rel="nofollow" class="external text" href="http://www.people.fas.harvard.edu/~gillum/rankings_paper.pdf">"Ranking Doctoral Programs by Placement: A New Method"</a> (PDF). PS: Political Science and Politics. 40 (July): 523–529. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.582.9402">10.1.1.582.9402</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Fs1049096507070771">10.1017/s1049096507070771</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6012229">6012229</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150213104340/http://www.people.fas.harvard.edu/~gillum/rankings_paper.pdf">Archived</a> (PDF) from the original on 2015-02-13.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=PS%3A+Political+Science+and+Politics&rft.atitle=Ranking+Doctoral+Programs+by+Placement%3A+A+New+Method&rft.volume=40&rft.issue=July&rft.pages=%3Cspan+class%3D%22nowrap%22%3E523-%3C%2Fspan%3E529&rft.date=2007&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.582.9402%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6012229%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2Fs1049096507070771&rft.au=Benjamin+M.+Schmidt&rft.au=Matthew+M.+Chingos&rft_id=http%3A%2F%2Fwww.people.fas.harvard.edu%2F~gillum%2Frankings_paper.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-63"><a href="#cite_ref-63">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJohan_BollenMarko_A._RodriguezHerbert_Van_de_Sompel2006" class="citation conference cs1">Johan Bollen; Marko A. Rodriguez; Herbert Van de Sompel (December 2006). "MESUR: Usage-based metrics of scholarly impact". Proceedings of the 7th ACM/IEEE-CS joint conference on Digital libraries. New York: Association for Computing Machinery. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/cs.GL/0601030">cs.GL/0601030</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006cs........1030B">2006cs........1030B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F1255175.1255273">10.1145/1255175.1255273</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-59593-644-8" title="Special:BookSources/978-1-59593-644-8"><bdi>978-1-59593-644-8</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:3115544">3115544</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=MESUR%3A+Usage-based+metrics+of+scholarly+impact&rft.btitle=Proceedings+of+the+7th+ACM%2FIEEE-CS+joint+conference+on+Digital+libraries&rft.place=New+York&rft.pub=Association+for+Computing+Machinery&rft.date=2006-12&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A3115544%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2006cs........1030B&rft_id=info%3Aarxiv%2Fcs.GL%2F0601030&rft_id=info%3Adoi%2F10.1145%2F1255175.1255273&rft.isbn=978-1-59593-644-8&rft.au=Johan+Bollen&rft.au=Marko+A.+Rodriguez&rft.au=Herbert+Van+de+Sompel&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-64"><a href="#cite_ref-64">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFletcher,_Jack_McKayWennekers,_Thomas2017" class="citation journal cs1">Fletcher, Jack McKay; Wennekers, Thomas (2017). <a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0129065717500137">"From Structure to Activity: Using Centrality Measures to Predict Neuronal Activity"</a>. International Journal of Neural Systems. 28 (2): 1750013. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0129065717500137">10.1142/S0129065717500137</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10026.1%2F9713">10026.1/9713</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/28076982">28076982</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Neural+Systems&rft.atitle=From+Structure+to+Activity%3A+Using+Centrality+Measures+to+Predict+Neuronal+Activity&rft.volume=28&rft.issue=2&rft.pages=1750013&rft.date=2017&rft_id=info%3Ahdl%2F10026.1%2F9713&rft_id=info%3Apmid%2F28076982&rft_id=info%3Adoi%2F10.1142%2FS0129065717500137&rft.au=Fletcher%2C+Jack+McKay&rft.au=Wennekers%2C+Thomas&rft_id=https%3A%2F%2Fdoi.org%2F10.1142%252FS0129065717500137&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-twitterwtf-65"><a href="#cite_ref-twitterwtf_65-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGuptaGoelLinSharma2013" class="citation book cs1">Gupta, Pankaj; Goel, Ashish; Lin, Jimmy; Sharma, Aneesh; Wang, Dong; Zadeh, Reza (2013). <a rel="nofollow" class="external text" href="http://dl.acm.org/citation.cfm?id=2488433">"WTF: The Who to Follow Service at Twitter"</a>. Proceedings of the 22nd International Conference on World Wide Web. ACM. pp. 505–514. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F2488388.2488433">10.1145/2488388.2488433</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4503-2035-1" title="Special:BookSources/978-1-4503-2035-1"><bdi>978-1-4503-2035-1</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:207205045">207205045</a>. Retrieved 11 December 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=WTF%3A+The+Who+to+Follow+Service+at+Twitter&rft.btitle=Proceedings+of+the+22nd+International+Conference+on+World+Wide+Web&rft.pages=%3Cspan+class%3D%22nowrap%22%3E505-%3C%2Fspan%3E514&rft.pub=ACM&rft.date=2013&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A207205045%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F2488388.2488433&rft.isbn=978-1-4503-2035-1&rft.aulast=Gupta&rft.aufirst=Pankaj&rft.au=Goel%2C+Ashish&rft.au=Lin%2C+Jimmy&rft.au=Sharma%2C+Aneesh&rft.au=Wang%2C+Dong&rft.au=Zadeh%2C+Reza&rft_id=http%3A%2F%2Fdl.acm.org%2Fcitation.cfm%3Fid%3D2488433&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-66"><a href="#cite_ref-66">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHa2012" class="citation news cs1">Ha, Anthony (2012-05-08). <a rel="nofollow" class="external text" href="https://techcrunch.com/2012/05/08/swiftype-launch/">"Y Combinator-Backed Swiftype Builds Site Search That Doesn't Suck"</a>. TechCrunch. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140706215017/http://techcrunch.com/2012/05/08/swiftype-launch/">Archived</a> from the original on 2014-07-06. Retrieved 2014-07-08.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=TechCrunch&rft.atitle=Y+Combinator-Backed+Swiftype+Builds+Site+Search+That+Doesn%27t+Suck&rft.date=2012-05-08&rft.aulast=Ha&rft.aufirst=Anthony&rft_id=https%3A%2F%2Ftechcrunch.com%2F2012%2F05%2F08%2Fswiftype-launch%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-67"><a href="#cite_ref-67">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://dbpubs.stanford.edu:8091/diglib/pub/projectdir/google.html">"Working Papers Concerning the Creation of Google"</a>. Google. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20061128230555/http://dbpubs.stanford.edu:8091/diglib/pub/projectdir/google.html">Archived</a> from the original on November 28, 2006. Retrieved November 29, 2006.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Google&rft.atitle=Working+Papers+Concerning+the+Creation+of+Google&rft_id=http%3A%2F%2Fdbpubs.stanford.edu%3A8091%2Fdiglib%2Fpub%2Fprojectdir%2Fgoogle.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-68"><a href="#cite_ref-68">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCho,_J.Garcia-Molina,_H.Page,_L.1998" class="citation conference cs1">Cho, J.; Garcia-Molina, H.; Page, L. (1998). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080603171020/http://dbpubs.stanford.edu:8090/pub/1998-51">"Efficient crawling through URL ordering"</a>. Proceedings of the Seventh Conference on World Wide Web. Archived from <a rel="nofollow" class="external text" href="http://dbpubs.stanford.edu:8090/pub/1998-51">the original</a> on 2008-06-03.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Efficient+crawling+through+URL+ordering&rft.btitle=Proceedings+of+the+Seventh+Conference+on+World+Wide+Web&rft.date=1998&rft.au=Cho%2C+J.&rft.au=Garcia-Molina%2C+H.&rft.au=Page%2C+L.&rft_id=http%3A%2F%2Fdbpubs.stanford.edu%3A8090%2Fpub%2F1998-51&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-69"><a href="#cite_ref-69">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131004234101/http://groups.yahoo.com/group/sd-2/">"Yahoo! Groups"</a>. Groups.yahoo.com. Archived from <a rel="nofollow" class="external text" href="https://groups.yahoo.com/group/sd-2/">the original</a> on 2013-10-04. Retrieved 2013-10-02.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Yahoo%21+Groups&rft.pub=Groups.yahoo.com&rft_id=https%3A%2F%2Fgroups.yahoo.com%2Fgroup%2Fsd-2%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-70"><a href="#cite_ref-70">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation citeseerx cs1">"Autopoietic Information Systems in Modern Organizations". <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.148.9274">10.1.1.148.9274</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=CiteSeerX&rft.atitle=Autopoietic+Information+Systems+in+Modern+Organizations&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.148.9274%23id-name%3DCiteSeerX&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-71"><a href="#cite_ref-71">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFZackLambBall2012" class="citation journal cs1">Zack, Laurie; Lamb, Ron; Ball, Sarah (2012-12-31). <a rel="nofollow" class="external text" href="https://doi.org/10.2140%2Finvolve.2012.5.463">"An application of Google's PageRank to NFL rankings"</a>. Involve: A Journal of Mathematics. 5 (4): 463–471. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2140%2Finvolve.2012.5.463">10.2140/involve.2012.5.463</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1944-4184">1944-4184</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Involve%3A+A+Journal+of+Mathematics&rft.atitle=An+application+of+Google%27s+PageRank+to+NFL+rankings&rft.volume=5&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E463-%3C%2Fspan%3E471&rft.date=2012-12-31&rft_id=info%3Adoi%2F10.2140%2Finvolve.2012.5.463&rft.issn=1944-4184&rft.aulast=Zack&rft.aufirst=Laurie&rft.au=Lamb%2C+Ron&rft.au=Ball%2C+Sarah&rft_id=https%3A%2F%2Fdoi.org%2F10.2140%252Finvolve.2012.5.463&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-72"><a href="#cite_ref-72">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPeñaTouchette2012" class="citation arxiv cs1">Peña, Javier López; Touchette, Hugo (2012-06-28). "A network theory analysis of football strategies". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1206.6904">1206.6904</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.CO">math.CO</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=A+network+theory+analysis+of+football+strategies&rft.date=2012-06-28&rft_id=info%3Aarxiv%2F1206.6904&rft.aulast=Pe%C3%B1a&rft.aufirst=Javier+L%C3%B3pez&rft.au=Touchette%2C+Hugo&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-73"><a href="#cite_ref-73">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBeggsShepherdEmmondsJones2017" class="citation journal cs1">Beggs, Clive B.; Shepherd, Simon J.; Emmonds, Stacey; Jones, Ben (2017-06-02). Zhou, Wei-Xing (ed.). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5456068">"A novel application of PageRank and user preference algorithms for assessing the relative performance of track athletes in competition"</a>. PLOS ONE. 12 (6): e0178458. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017PLoSO..1278458B">2017PLoSO..1278458B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1371%2Fjournal.pone.0178458">10.1371/journal.pone.0178458</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1932-6203">1932-6203</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5456068">5456068</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/28575009">28575009</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=PLOS+ONE&rft.atitle=A+novel+application+of+PageRank+and+user+preference+algorithms+for+assessing+the+relative+performance+of+track+athletes+in+competition&rft.volume=12&rft.issue=6&rft.pages=e0178458&rft.date=2017-06-02&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5456068%23id-name%3DPMC&rft_id=info%3Abibcode%2F2017PLoSO..1278458B&rft_id=info%3Apmid%2F28575009&rft_id=info%3Adoi%2F10.1371%2Fjournal.pone.0178458&rft.issn=1932-6203&rft.aulast=Beggs&rft.aufirst=Clive+B.&rft.au=Shepherd%2C+Simon+J.&rft.au=Emmonds%2C+Stacey&rft.au=Jones%2C+Ben&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5456068&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-74"><a href="#cite_ref-74">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFB._Jiang2006" class="citation journal cs1">B. Jiang (2006). "Ranking spaces for predicting human movement in an urban environment". International Journal of Geographical Information Science. 23 (7): 823–837. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/physics/0612011">physics/0612011</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009IJGIS..23..823J">2009IJGIS..23..823J</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F13658810802022822">10.1080/13658810802022822</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:26880621">26880621</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Geographical+Information+Science&rft.atitle=Ranking+spaces+for+predicting+human+movement+in+an+urban+environment&rft.volume=23&rft.issue=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E823-%3C%2Fspan%3E837&rft.date=2006&rft_id=info%3Aarxiv%2Fphysics%2F0612011&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A26880621%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1080%2F13658810802022822&rft_id=info%3Abibcode%2F2009IJGIS..23..823J&rft.au=B.+Jiang&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-75"><a href="#cite_ref-75">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJiang_B.Zhao_S.Yin_J.2008" class="citation journal cs1 cs1-prop-long-vol">Jiang B.; Zhao S. & Yin J. (2008). "Self-organized natural roads for predicting traffic flow: a sensitivity study". Journal of Statistical Mechanics: Theory and Experiment. P07008 (7): 008. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0804.1630">0804.1630</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008JSMTE..07..008J">2008JSMTE..07..008J</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1742-5468%2F2008%2F07%2FP07008">10.1088/1742-5468/2008/07/P07008</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118605727">118605727</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Statistical+Mechanics%3A+Theory+and+Experiment&rft.atitle=Self-organized+natural+roads+for+predicting+traffic+flow%3A+a+sensitivity+study&rft.volume=P07008&rft.issue=7&rft.pages=008&rft.date=2008&rft_id=info%3Aarxiv%2F0804.1630&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118605727%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F1742-5468%2F2008%2F07%2FP07008&rft_id=info%3Abibcode%2F2008JSMTE..07..008J&rft.au=Jiang+B.&rft.au=Zhao+S.&rft.au=Yin+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-76"><a href="#cite_ref-76">^</a> Roberto Navigli, Mirella Lapata. <a rel="nofollow" class="external text" href="http://www.dsi.uniroma1.it/~navigli/pubs/PAMI_2010_Navigli_Lapata.pdf">"An Experimental Study of Graph Connectivity for Unsupervised Word Sense Disambiguation"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20101214110431/http://www.dsi.uniroma1.it/~navigli/pubs/PAMI_2010_Navigli_Lapata.pdf">Archived</a> 2010-12-14 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI), 32(4), IEEE Press, 2010, pp. 678–692. </li> <li id="cite_note-77"><a href="#cite_ref-77">^</a> M. T. Pilehvar, D. Jurgens and R. Navigli. <a rel="nofollow" class="external text" href="http://wwwusers.di.uniroma1.it/~navigli/pubs/ACL_2013_Pilehvar_Jurgens_Navigli.pdf">Align, Disambiguate and Walk: A Unified Approach for Measuring Semantic Similarity.</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131001004443/http://wwwusers.di.uniroma1.it/~navigli/pubs/ACL_2013_Pilehvar_Jurgens_Navigli.pdf">Archived</a> 2013-10-01 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Proc. of the 51st Annual Meeting of the Association for Computational Linguistics (ACL 2013), Sofia, Bulgaria, August 4–9, 2013, pp. 1341-1351. </li> <li id="cite_note-78"><a href="#cite_ref-78">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAndrea_EsuliFabrizio_Sebastiani" class="citation web cs1">Andrea Esuli & Fabrizio Sebastiani. <a rel="nofollow" class="external text" href="http://nmis.isti.cnr.it/sebastiani/Publications/ACL07.pdf">"PageRanking WordNet synsets: An Application to Opinion-Related Properties"</a> (PDF). In Proceedings of the 35th Meeting of the Association for Computational Linguistics, Prague, CZ, 2007, pp. 424–431. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070628113247/http://nmis.isti.cnr.it/sebastiani/Publications/ACL07.pdf">Archived</a> (PDF) from the original on June 28, 2007. Retrieved June 30, 2007.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=In+Proceedings+of+the+35th+Meeting+of+the+Association+for+Computational+Linguistics%2C+Prague%2C+CZ%2C+2007%2C+pp.+424%E2%80%93431&rft.atitle=PageRanking+WordNet+synsets%3A+An+Application+to+Opinion-Related+Properties&rft.au=Andrea+Esuli&rft.au=Fabrizio+Sebastiani&rft_id=http%3A%2F%2Fnmis.isti.cnr.it%2Fsebastiani%2FPublications%2FACL07.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-79"><a href="#cite_ref-79">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWang_S.Schreckenberg_M.Guhr_T2023" class="citation journal cs1">Wang S.; Schreckenberg M.; Guhr T (2023). <a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1742-5468%2Facf210">"Transitions between quasi-stationary states in traffic systems: Cologne orbital motorways as an example"</a>. Journal of Statistical Mechanics: Theory and Experiment. 2023 (9): 093401. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2302.14596">2302.14596</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2023JSMTE2023i3401W">2023JSMTE2023i3401W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1742-5468%2Facf210">10.1088/1742-5468/acf210</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:257232659">257232659</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Statistical+Mechanics%3A+Theory+and+Experiment&rft.atitle=Transitions+between+quasi-stationary+states+in+traffic+systems%3A+Cologne+orbital+motorways+as+an+example&rft.volume=2023&rft.issue=9&rft.pages=093401&rft.date=2023&rft_id=info%3Aarxiv%2F2302.14596&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A257232659%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F1742-5468%2Facf210&rft_id=info%3Abibcode%2F2023JSMTE2023i3401W&rft.au=Wang+S.&rft.au=Schreckenberg+M.&rft.au=Guhr+T&rft_id=https%3A%2F%2Fdoi.org%2F10.1088%252F1742-5468%252Facf210&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-80"><a href="#cite_ref-80">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://googleblog.blogspot.com/2005/01/preventing-comment-spam.html">"Preventing Comment Spam"</a>. Google. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20050612001742/http://googleblog.blogspot.com/2005/01/preventing-comment-spam.html">Archived</a> from the original on June 12, 2005. Retrieved January 1, 2005.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Google&rft.atitle=Preventing+Comment+Spam&rft_id=http%3A%2F%2Fgoogleblog.blogspot.com%2F2005%2F01%2Fpreventing-comment-spam.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-81"><a href="#cite_ref-81">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.seomoz.org/blog/pagerank-sculpting-parsing-the-value-and-potential-benefits-of-sculpting-pr-with-nofollow">"PageRank Sculpting: Parsing the Value and Potential Benefits of Sculpting PR with Nofollow"</a>. SEOmoz. 14 October 2008. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110514201211/http://www.seomoz.org/blog/pagerank-sculpting-parsing-the-value-and-potential-benefits-of-sculpting-pr-with-nofollow">Archived</a> from the original on 2011-05-14. Retrieved 2011-05-27.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=PageRank+Sculpting%3A+Parsing+the+Value+and+Potential+Benefits+of+Sculpting+PR+with+Nofollow&rft.pub=SEOmoz&rft.date=2008-10-14&rft_id=http%3A%2F%2Fwww.seomoz.org%2Fblog%2Fpagerank-sculpting-parsing-the-value-and-potential-benefits-of-sculpting-pr-with-nofollow&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> <li id="cite_note-82"><a href="#cite_ref-82">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.mattcutts.com/blog/pagerank-sculpting/">"PageRank sculpting"</a>. Mattcutts.com. 2009-06-15. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110511180227/http://www.mattcutts.com/blog/pagerank-sculpting/">Archived</a> from the original on 2011-05-11. Retrieved 2011-05-27.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=PageRank+sculpting&rft.pub=Mattcutts.com&rft.date=2009-06-15&rft_id=http%3A%2F%2Fwww.mattcutts.com%2Fblog%2Fpagerank-sculpting%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3>[<a href="/w/index.php?title=PageRank&action=edit&section=29" title="Edit section: Sources">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 32em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAltmanMoshe_Tennenholtz2005" class="citation conference cs1">Altman, Alon; Moshe Tennenholtz (2005). <a rel="nofollow" class="external text" href="http://www.eecs.harvard.edu/cs286r/courses/fall11/papers/AT%2705.pdf">"Ranking Systems: The PageRank Axioms"</a> (PDF). Proceedings of the 6th ACM conference on Electronic commerce (EC-05). Vancouver, BC. Retrieved 29 September 2014.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Ranking+Systems%3A+The+PageRank+Axioms&rft.btitle=Proceedings+of+the+6th+ACM+conference+on+Electronic+commerce+%28EC-05%29&rft.place=Vancouver%2C+BC&rft.date=2005&rft.aulast=Altman&rft.aufirst=Alon&rft.au=Moshe+Tennenholtz&rft_id=http%3A%2F%2Fwww.eecs.harvard.edu%2Fcs286r%2Fcourses%2Ffall11%2Fpapers%2FAT%252705.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFChengEric_J._Friedman2006" class="citation conference cs1">Cheng, Alice; Eric J. Friedman (2006-06-11). <a rel="nofollow" class="external text" href="http://www.cs.duke.edu/nicl/netecon06/papers/ne06-sybil.pdf">"Manipulability of PageRank under Sybil Strategies"</a> (PDF). Proceedings of the First Workshop on the Economics of Networked Systems (NetEcon06). Ann Arbor, Michigan. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100821011759/http://www.cs.duke.edu/nicl/netecon06/papers/ne06-sybil.pdf">Archived</a> (PDF) from the original on 2010-08-21. Retrieved 2008-01-22.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Manipulability+of+PageRank+under+Sybil+Strategies&rft.btitle=Proceedings+of+the+First+Workshop+on+the+Economics+of+Networked+Systems+%28NetEcon06%29&rft.place=Ann+Arbor%2C+Michigan&rft.date=2006-06-11&rft.aulast=Cheng&rft.aufirst=Alice&rft.au=Eric+J.+Friedman&rft_id=http%3A%2F%2Fwww.cs.duke.edu%2Fnicl%2Fnetecon06%2Fpapers%2Fne06-sybil.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFarahatLoFaroMillerRae2006" class="citation journal cs1">Farahat, Ayman; LoFaro, Thomas; Miller, Joel C.; Rae, Gregory; <a href="/wiki/Lesley_Ward" title="Lesley Ward">Ward, Lesley A.</a> (2006). "Authority Rankings from HITS, PageRank, and SALSA: Existence, Uniqueness, and Effect of Initialization". SIAM Journal on Scientific Computing. 27 (4): 1181–1201. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006SJSC...27.1181F">2006SJSC...27.1181F</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.99.3942">10.1.1.99.3942</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2FS1064827502412875">10.1137/S1064827502412875</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIAM+Journal+on+Scientific+Computing&rft.atitle=Authority+Rankings+from+HITS%2C+PageRank%2C+and+SALSA%3A+Existence%2C+Uniqueness%2C+and+Effect+of+Initialization&rft.volume=27&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1181-%3C%2Fspan%3E1201&rft.date=2006&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.99.3942%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1137%2FS1064827502412875&rft_id=info%3Abibcode%2F2006SJSC...27.1181F&rft.aulast=Farahat&rft.aufirst=Ayman&rft.au=LoFaro%2C+Thomas&rft.au=Miller%2C+Joel+C.&rft.au=Rae%2C+Gregory&rft.au=Ward%2C+Lesley+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHaveliwalaJeh,_GlenKamvar,_Sepandar2003" class="citation conference cs1">Haveliwala, Taher; Jeh, Glen; Kamvar, Sepandar (2003). <a rel="nofollow" class="external text" href="http://www-cs-students.stanford.edu/~taherh/papers/comparison.pdf">"An Analytical Comparison of Approaches to Personalizing PageRank"</a> (PDF). Stanford University Technical Report. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20101216084254/http://www-cs-students.stanford.edu/~taherh/papers/comparison.pdf">Archived</a> (PDF) from the original on 2010-12-16. Retrieved 2008-11-13.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=An+Analytical+Comparison+of+Approaches+to+Personalizing+PageRank&rft.btitle=Stanford+University+Technical+Report&rft.date=2003&rft.aulast=Haveliwala&rft.aufirst=Taher&rft.au=Jeh%2C+Glen&rft.au=Kamvar%2C+Sepandar&rft_id=http%3A%2F%2Fwww-cs-students.stanford.edu%2F~taherh%2Fpapers%2Fcomparison.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLangvilleMeyer,_Carl_D.2003" class="citation journal cs1"><a href="/wiki/Amy_Langville" title="Amy Langville">Langville, Amy N.</a>; Meyer, Carl D. (2003). "Survey: Deeper Inside PageRank". Internet Mathematics. 1 (3).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Internet+Mathematics&rft.atitle=Survey%3A+Deeper+Inside+PageRank&rft.volume=1&rft.issue=3&rft.date=2003&rft.aulast=Langville&rft.aufirst=Amy+N.&rft.au=Meyer%2C+Carl+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLangvilleMeyer,_Carl_D.2006" class="citation book cs1"><a href="/wiki/Amy_Langville" title="Amy Langville">Langville, Amy N.</a>; Meyer, Carl D. (2006). Google's PageRank and Beyond: The Science of Search Engine Rankings. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-12202-1" title="Special:BookSources/978-0-691-12202-1"><bdi>978-0-691-12202-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Google%27s+PageRank+and+Beyond%3A+The+Science+of+Search+Engine+Rankings&rft.pub=Princeton+University+Press&rft.date=2006&rft.isbn=978-0-691-12202-1&rft.aulast=Langville&rft.aufirst=Amy+N.&rft.au=Meyer%2C+Carl+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRichardsonDomingos,_Pedro2002" class="citation conference cs1">Richardson, Matthew; Domingos, Pedro (2002). <a rel="nofollow" class="external text" href="http://www.cs.washington.edu/homes/pedrod/papers/nips01b.pdf">"The intelligent surfer: Probabilistic combination of link and content information in PageRank"</a> (PDF). Proceedings of Advances in Neural Information Processing Systems. Vol. 14. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100628120615/http://www.cs.washington.edu/homes/pedrod/papers/nips01b.pdf">Archived</a> (PDF) from the original on 2010-06-28. Retrieved 2004-09-18.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=The+intelligent+surfer%3A+Probabilistic+combination+of+link+and+content+information+in+PageRank&rft.btitle=Proceedings+of+Advances+in+Neural+Information+Processing+Systems&rft.date=2002&rft.aulast=Richardson&rft.aufirst=Matthew&rft.au=Domingos%2C+Pedro&rft_id=http%3A%2F%2Fwww.cs.washington.edu%2Fhomes%2Fpedrod%2Fpapers%2Fnips01b.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APageRank" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Relevant_patents">Relevant patents</h2>[<a href="/w/index.php?title=PageRank&action=edit&section=30" title="Edit section: Relevant patents">edit</a>]</div> <ul><li><a rel="nofollow" class="external text" href="http://patft.uspto.gov/netacgi/nph-Parser?patentnumber=6,285,999">Original PageRank U.S. Patent—Method for node ranking in a linked database</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140829134755/http://patft.uspto.gov/netacgi/nph-Parser?patentnumber=6,285,999">Archived</a> 2014-08-29 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>—Patent number 6,285,999—September 4, 2001</li> <li><a rel="nofollow" class="external text" href="http://patft1.uspto.gov/netacgi/nph-Parser?Sect1=PTO1&Sect2=HITOFF&d=PALL&p=1&u=%2Fnetahtml%2FPTO%2Fsrchnum.htm&r=1&f=G&l=50&s1=6,799,176.PN.&OS=PN/6,799,176&RS=PN/6,799,176">PageRank U.S. Patent—Method for scoring documents in a linked database</a>—Patent number 6,799,176—September 28, 2004</li> <li><a rel="nofollow" class="external text" href="http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&u=%2Fnetahtml%2FPTO%2Fsearch-adv.htm&r=1&p=1&f=G&l=50&d=PTXT&S1=7,058,628.PN.&OS=pn/7,058,628&RS=PN/7,058,628">PageRank U.S. Patent—Method for node ranking in a linked database</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190828223450/http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&u=%2Fnetahtml%2FPTO%2Fsearch-adv.htm&r=1&p=1&f=G&l=50&d=PTXT&S1=7%2C058%2C628.PN.&OS=pn%2F7%2C058%2C628&RS=PN%2F7%2C058%2C628">Archived</a> 2019-08-28 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>—Patent number 7,058,628—June 6, 2006</li> <li><a rel="nofollow" class="external text" href="http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&u=%2Fnetahtml%2FPTO%2Fsearch-adv.htm&r=1&p=1&f=G&l=50&d=PTXT&S1=7,269,587.PN.&OS=pn/7,269,587&RS=PN/7,269,587">PageRank U.S. Patent—Scoring documents in a linked database</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180331090147/http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2">Archived</a> 2018-03-31 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>—Patent number 7,269,587—September 11, 2007</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=PageRank&action=edit&section=31" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><a href="/wiki/File:Wikiquote-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/34px-Wikiquote-logo.svg.png" decoding="async" width="34" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/51px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/68px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></a></div> <div class="side-box-text plainlist">Wikiquote has quotations related to <a href="https://en.wikiquote.org/wiki/Special:Search/PageRank" class="extiw" title="q:Special:Search/PageRank">PageRank</a>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://www.google.com/insidesearch/howsearchworks/algorithms.html">Algorithms</a> by Google</li> <li><a rel="nofollow" class="external text" href="https://www.google.com/technology/">Our products and services</a> by Google</li> <li><a rel="nofollow" class="external text" href="https://www.ams.org/featurecolumn/archive/pagerank.html">How Google Finds Your Needle in the Web's Haystack</a> by the American Mathematical Society</li></ul> <div style="clear:both;" class=""></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=PageRank&oldid=1281628054">https://en.wikipedia.org/w/index.php?title=PageRank&oldid=1281628054</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Google_Search" title="Category:Google Search">Google Search</a></li><li><a href="/wiki/Category:Search_engine_optimization" title="Category:Search engine optimization">Search engine optimization</a></li><li><a href="/wiki/Category:Reputation_management" title="Category:Reputation management">Reputation management</a></li><li><a href="/wiki/Category:Internet_search_algorithms" title="Category:Internet search algorithms">Internet search algorithms</a></li><li><a href="/wiki/Category:American_inventions" title="Category:American inventions">American inventions</a></li><li><a href="/wiki/Category:Crowdsourcing" title="Category:Crowdsourcing">Crowdsourcing</a></li><li><a href="/wiki/Category:Markov_models" title="Category:Markov models">Markov models</a></li><li><a href="/wiki/Category:Link_analysis" title="Category:Link analysis">Link analysis</a></li><li><a href="/wiki/Category:Graph_algorithms" title="Category:Graph algorithms">Graph algorithms</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><li><a href="/wiki/Category:CS1:_long_volume_value" title="Category:CS1: long volume value">CS1: long volume value</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_June_2013" title="Category:Articles with unsourced statements from June 2013">Articles with unsourced statements from June 2013</a></li><li><a href="/wiki/Category:All_articles_lacking_reliable_references" title="Category:All articles lacking reliable references">All articles lacking reliable references</a></li><li><a href="/wiki/Category:Articles_lacking_reliable_references_from_October_2012" title="Category:Articles lacking reliable references from October 2012">Articles lacking reliable references from October 2012</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_November_2015" title="Category:Articles with unsourced statements from November 2015">Articles with unsourced statements from November 2015</a></li><li><a href="/wiki/Category:Articles_with_example_MATLAB/Octave_code" title="Category:Articles with example MATLAB/Octave code">Articles with example MATLAB/Octave code</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 21 March 2025, at 14:26 (UTC).</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=PageRank&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/static/images/footer/wikimedia-button.svg" width="84" height="29"><img src="/static/images/footer/wikimedia.svg" width="25" height="25" alt="Wikimedia Foundation" lang="en" loading="lazy"></picture></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/w/resources/assets/poweredby_mediawiki.svg" width="88" height="31"><img src="/w/resources/assets/mediawiki_compact.svg" alt="Powered by MediaWiki" lang="en" width="25" height="25" loading="lazy"></picture></a></li> </ul> </footer> </div> </div> </div> <div class="vector-header-container vector-sticky-header-container"> <div id="vector-sticky-header" class="vector-sticky-header"> <div class="vector-sticky-header-start"> <div class="vector-sticky-header-icon-start vector-button-flush-left vector-button-flush-right" aria-hidden="true"> <button class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-sticky-header-search-toggle" tabindex="-1" data-event-name="ui.vector-sticky-search-form.icon"> Search </button> </div> <div role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box"> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail"> <form action="/w/index.php" id="vector-sticky-search-form" class="cdx-search-input cdx-search-input--has-end-button"> <div class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia"> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <div class="vector-sticky-header-context-bar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-sticky-header-toc" class="vector-dropdown mw-portlet mw-portlet-sticky-header-toc vector-sticky-header-toc vector-button-flush-left" > <input type="checkbox" id="vector-sticky-header-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-sticky-header-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-sticky-header-toc-label" for="vector-sticky-header-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-sticky-header-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" >PageRank</div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"> 43 languages </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"> Add topic </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="mw-portlet mw-portlet-dock-bottom emptyPortlet" id="p-dock-bottom"> <ul> </ul> </div> <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.eqiad.main-58c4b96c94-rcjpc","wgBackendResponseTime":238,"wgPageParseReport":{"limitreport":{"cputime":"1.352","walltime":"2.079","ppvisitednodes":{"value":5703,"limit":1000000},"postexpandincludesize":{"value":182959,"limit":2097152},"templateargumentsize":{"value":3460,"limit":2097152},"expansiondepth":{"value":12,"limit":100},"expensivefunctioncount":{"value":5,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":301680,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 1199.357 1 -total"," 62.62% 751.036 1 Template:Reflist"," 24.75% 296.849 30 Template:Cite_web"," 16.08% 192.860 23 Template:Cite_journal"," 9.51% 114.089 1 Template:Short_description"," 5.58% 66.892 2 Template:Pagetype"," 5.24% 62.880 2 Template:Citation_needed"," 5.24% 62.827 3 Template:Fix"," 5.22% 62.588 8 Template:Cite_conference"," 4.91% 58.944 2 Template:NumBlk"]},"scribunto":{"limitreport-timeusage":{"value":"0.729","limit":"10.000"},"limitreport-memusage":{"value":6214142,"limit":52428800}},"cachereport":{"origin":"mw-web.eqiad.main-58c4b96c94-svthz","timestamp":"20250321142624","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"PageRank","url":"https:\/\/en.wikipedia.org\/wiki\/PageRank","sameAs":"http:\/\/www.wikidata.org\/entity\/Q184316","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q184316","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2001-11-05T06:06:13Z","dateModified":"2025-03-21T14:26:13Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/f\/fb\/PageRanks-Example.svg","headline":"algorithm for calculating the authority of a web page based on link structure."}</script> </body> </html>