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</div> </a> <ul id="toc-Random_trees-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conditional_random_graphs" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Conditional_random_graphs"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Conditional random graphs</span> </div> </a> <ul id="toc-Conditional_random_graphs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-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 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Available in 17 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-17" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">17 languages</span> </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%B1%D8%B3%D9%85_%D8%A8%D9%8A%D8%A7%D9%86%D9%8A_%D8%B9%D8%B4%D9%88%D8%A7%D8%A6%D9%8A" title="رسم بياني عشوائي – Arabic" lang="ar" hreflang="ar" data-title="رسم بياني عشوائي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/N%C3%A1hodn%C3%BD_graf" title="Náhodný graf – Czech" lang="cs" hreflang="cs" data-title="Náhodný graf" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Zufallsgraph" title="Zufallsgraph – German" lang="de" hreflang="de" data-title="Zufallsgraph" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Juhuslik_graaf" title="Juhuslik graaf – Estonian" lang="et" hreflang="et" data-title="Juhuslik graaf" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CF%85%CF%87%CE%B1%CE%AF%CE%BF%CF%82_%CE%B3%CF%81%CE%AC%CF%86%CE%BF%CF%82" title="Τυχαίος γράφος – Greek" lang="el" hreflang="el" data-title="Τυχαίος γράφος" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Grafo_aleatorio" title="Grafo aleatorio – Spanish" lang="es" hreflang="es" data-title="Grafo aleatorio" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%AF%D8%B1%D8%A7%D9%81_%D8%AA%D8%B5%D8%A7%D8%AF%D9%81%DB%8C" title="گراف تصادفی – Persian" lang="fa" hreflang="fa" data-title="گراف تصادفی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Graphe_al%C3%A9atoire" title="Graphe aléatoire – French" lang="fr" hreflang="fr" data-title="Graphe aléatoire" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Grafo_aleatorio" title="Grafo aleatorio – Italian" lang="it" hreflang="it" data-title="Grafo aleatorio" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%92%D7%A8%D7%A3_%D7%9E%D7%A7%D7%A8%D7%99" title="גרף מקרי – Hebrew" lang="he" hreflang="he" data-title="גרף מקרי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/V%C3%A9letlen_gr%C3%A1f" title="Véletlen gráf – Hungarian" lang="hu" hreflang="hu" data-title="Véletlen gráf" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Grafo_aleat%C3%B3rio" title="Grafo aleatório – Portuguese" lang="pt" hreflang="pt" data-title="Grafo aleatório" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D1%8B%D0%B9_%D0%B3%D1%80%D0%B0%D1%84" title="Случайный граф – Russian" lang="ru" hreflang="ru" data-title="Случайный граф" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Slumpgraf" title="Slumpgraf – Swedish" lang="sv" hreflang="sv" data-title="Slumpgraf" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D0%B8%D0%BF%D0%B0%D0%B4%D0%BA%D0%BE%D0%B2%D0%B8%D0%B9_%D0%B3%D1%80%D0%B0%D1%84" title="Випадковий граф – Ukrainian" lang="uk" hreflang="uk" data-title="Випадковий граф" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a 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a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle" style="padding-bottom:0.15em;">Part of <a href="/wiki/Category:Network_science" title="Category:Network science">a series</a> on</td></tr><tr><th class="sidebar-title-with-pretitle" style="font-size:175%;"><a href="/wiki/Network_science" title="Network science">Network science</a></th></tr><tr><td class="sidebar-image"><div class="center"><div class="center"> <div style="width: 250px; height: 250px; overflow: hidden;"> <div style="position: relative; top: -0px; left: -0px; width: 250px"><div class="noresize"><span typeof="mw:File"><a href="/wiki/File:Internet_map_1024.jpg" class="mw-file-description"><img alt="Internet_map_1024.jpg" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Internet_map_1024.jpg/250px-Internet_map_1024.jpg" decoding="async" width="250" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Internet_map_1024.jpg/375px-Internet_map_1024.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Internet_map_1024.jpg/500px-Internet_map_1024.jpg 2x" data-file-width="1280" data-file-height="1280" /></a></span></div></div> </div> </div></div></td></tr><tr><th class="sidebar-heading"> <div class="hlist"><ul><li><a href="/wiki/Network_theory" title="Network theory">Theory</a></li></ul></div></th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.5em;"> <ul><li><a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">Graph</a></li> <li><a href="/wiki/Complex_network" title="Complex network">Complex network</a></li> <li><a href="/wiki/Complex_contagion" title="Complex contagion">Contagion</a></li> <li><a href="/wiki/Small-world_network" title="Small-world network">Small-world</a></li> <li><a href="/wiki/Scale-free_network" title="Scale-free network">Scale-free</a></li> <li><a href="/wiki/Community_structure" title="Community structure">Community structure</a></li> <li><a href="/wiki/Percolation_theory" title="Percolation theory">Percolation</a></li> <li><a href="/wiki/Evolving_networks" class="mw-redirect" title="Evolving networks">Evolution</a></li> <li><a href="/wiki/Network_controllability" title="Network controllability">Controllability</a></li> <li><a href="/wiki/Graph_drawing" title="Graph drawing">Graph drawing</a></li> <li><a href="/wiki/Social_capital" title="Social capital">Social capital</a></li> <li><a href="/wiki/Link_analysis" title="Link analysis">Link analysis</a></li> <li><a href="/wiki/Combinatorial_optimization" title="Combinatorial optimization">Optimization</a></li> <li><a href="/wiki/Reciprocity_(network_science)" title="Reciprocity (network science)">Reciprocity</a></li> <li><a href="/wiki/Triadic_closure" title="Triadic closure">Closure</a></li> <li><a href="/wiki/Homophily" title="Homophily">Homophily</a></li> <li><a href="/wiki/Transitive_relation" title="Transitive relation">Transitivity</a></li> <li><a href="/wiki/Preferential_attachment" title="Preferential attachment">Preferential attachment</a></li> <li><a href="/wiki/Balance_theory" title="Balance theory">Balance theory</a></li> <li><a href="/wiki/Network_effect" title="Network effect">Network effect</a></li> <li><a href="/wiki/Social_influence" title="Social influence">Social influence</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Network types</th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.5em;"> <ul><li><a href="/wiki/Computer_network" title="Computer network">Informational (computing)</a></li> <li><a href="/wiki/Telecommunications_network" title="Telecommunications network">Telecommunication</a></li> <li><a href="/wiki/Transport_network" class="mw-redirect" title="Transport network">Transport</a></li> <li><a href="/wiki/Social_network" title="Social network">Social</a></li> <li><a href="/wiki/Scientific_collaboration_network" title="Scientific collaboration network">Scientific collaboration</a></li> <li><a href="/wiki/Biological_network" title="Biological network">Biological</a></li> <li><a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">Artificial neural</a></li> <li><a href="/wiki/Interdependent_networks" title="Interdependent networks">Interdependent</a></li> <li><a href="/wiki/Semantic_network" title="Semantic network">Semantic</a></li> <li><a href="/wiki/Spatial_network" title="Spatial network">Spatial</a></li> <li><a href="/wiki/Dependency_network" title="Dependency network">Dependency</a></li> <li><a href="/wiki/Flow_network" title="Flow network">Flow</a></li> <li><a href="/wiki/Network_on_a_chip" title="Network on a chip">on-Chip</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">Graphs</a></th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.5em;"> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="font-weight:normal;font-style:italic;"> Features</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Clique_(graph_theory)" title="Clique (graph theory)">Clique</a></li> <li><a href="/wiki/Connected_component_(graph_theory)" class="mw-redirect" title="Connected component (graph theory)">Component</a></li> <li><a href="/wiki/Cut_(graph_theory)" title="Cut (graph theory)">Cut</a></li> <li><a href="/wiki/Cycle_(graph_theory)" title="Cycle (graph theory)">Cycle</a></li> <li><a href="/wiki/Graph_(abstract_data_type)" title="Graph (abstract data type)">Data structure</a></li> <li><a href="/wiki/Edge_(graph_theory)" class="mw-redirect" title="Edge (graph theory)">Edge</a></li> <li><a href="/wiki/Loop_(graph_theory)" title="Loop (graph theory)">Loop</a></li> <li><a href="/wiki/Neighbourhood_(graph_theory)" title="Neighbourhood (graph theory)">Neighborhood</a></li> <li><a href="/wiki/Path_(graph_theory)" title="Path (graph theory)">Path</a></li> <li><a href="/wiki/Vertex_(graph_theory)" title="Vertex (graph theory)">Vertex</a></li> <li><span class="nowrap"><a href="/wiki/Adjacency_list" title="Adjacency list">Adjacency list</a> / <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">matrix</a></span></li> <li><span class="nowrap"><a href="/wiki/Incidence_list" class="mw-redirect" title="Incidence list">Incidence list</a> / <a href="/wiki/Incidence_matrix" title="Incidence matrix">matrix</a></span></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-weight:normal;font-style:italic;"> Types</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Bipartite_graph" title="Bipartite graph">Bipartite</a></li> <li><a href="/wiki/Complete_graph" title="Complete graph">Complete</a></li> <li><a href="/wiki/Directed_graph" title="Directed graph">Directed</a></li> <li><a href="/wiki/Hypergraph" title="Hypergraph">Hyper</a></li> <li><a href="/wiki/Labeled_graph" class="mw-redirect" title="Labeled graph">Labeled</a></li> <li><a href="/wiki/Multigraph" title="Multigraph">Multi</a></li> <li><a class="mw-selflink selflink">Random</a></li> <li><a href="/wiki/Weighted_graph" class="mw-redirect" title="Weighted graph">Weighted</a></li></ul></td> </tr></tbody></table></td> </tr><tr><th class="sidebar-heading"> <div class="hlist"><ul><li><a href="/wiki/Metrics_(networking)" title="Metrics (networking)">Metrics</a></li><li><a href="/wiki/List_of_algorithms#Networking" title="List of algorithms">Algorithms</a></li></ul></div></th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.5em;"> <ul><li><a href="/wiki/Centrality" title="Centrality">Centrality</a></li> <li><a href="/wiki/Degree_(graph_theory)" title="Degree (graph theory)">Degree</a></li> <li><a href="/wiki/Network_motif" title="Network motif">Motif</a></li> <li><a href="/wiki/Clustering_coefficient" title="Clustering coefficient">Clustering</a></li> <li><a href="/wiki/Degree_distribution" title="Degree distribution">Degree distribution</a></li> <li><a href="/wiki/Assortativity" title="Assortativity">Assortativity</a></li> <li><a href="/wiki/Distance_(graph_theory)" title="Distance (graph theory)">Distance</a></li> <li><a href="/wiki/Modularity_(networks)" title="Modularity (networks)">Modularity</a></li> <li><a href="/wiki/Efficiency_(network_science)" title="Efficiency (network science)">Efficiency</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Models</th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.5em;"> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="font-weight:normal;font-style:italic;"> Topology</th></tr><tr><td class="sidebar-content"> <ul><li><a class="mw-selflink selflink">Random graph</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model" title="Erdős–Rényi model">Erdős–Rényi</a></li> <li><a href="/wiki/Barab%C3%A1si%E2%80%93Albert_model" title="Barabási–Albert model">Barabási–Albert</a></li> <li><a href="/wiki/Bianconi%E2%80%93Barab%C3%A1si_model" title="Bianconi–Barabási model">Bianconi–Barabási</a></li> <li><a href="/wiki/Fitness_model_(network_theory)" title="Fitness model (network theory)">Fitness model</a></li> <li><a href="/wiki/Watts%E2%80%93Strogatz_model" title="Watts–Strogatz model">Watts–Strogatz</a></li> <li><a href="/wiki/Exponential_random_graph_models" class="mw-redirect" title="Exponential random graph models">Exponential random (ERGM)</a></li> <li><a href="/wiki/Random_geometric_graph" title="Random geometric graph">Random geometric (RGG)</a></li> <li><a href="/wiki/Hyperbolic_geometric_graph" title="Hyperbolic geometric graph">Hyperbolic (HGN)</a></li> <li><a href="/wiki/Hierarchical_network_model" title="Hierarchical network model">Hierarchical</a></li> <li><a href="/wiki/Stochastic_block_model" title="Stochastic block model">Stochastic block</a></li> <li><a href="/wiki/Blockmodeling" title="Blockmodeling">Blockmodeling</a></li> <li><a href="/wiki/Maximum-entropy_random_graph_model" title="Maximum-entropy random graph model">Maximum entropy</a></li> <li><a href="/wiki/Soft_configuration_model" title="Soft configuration model">Soft configuration</a></li> <li><a href="/wiki/Lancichinetti%E2%80%93Fortunato%E2%80%93Radicchi_benchmark" title="Lancichinetti–Fortunato–Radicchi benchmark">LFR Benchmark</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-weight:normal;font-style:italic;"> Dynamics</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Boolean_network" title="Boolean network">Boolean network</a></li> <li><a href="/wiki/Agent-based_model" title="Agent-based model">agent based</a></li> <li><a href="/wiki/Epidemic_model" class="mw-redirect" title="Epidemic model">Epidemic</a>/<a href="/wiki/SIR_model" class="mw-redirect" title="SIR model">SIR</a></li></ul></td> </tr></tbody></table></td> </tr><tr><th class="sidebar-heading"> <div class="hlist"><ul><li>Lists</li><li>Categories</li></ul></div></th></tr><tr><td class="sidebar-content hlist" style="padding-top:0.2em;padding-bottom:0.5em;"> <ul><li><a href="/wiki/List_of_network_theory_topics" title="List of network theory topics">Topics</a></li> <li><a href="/wiki/Social_network_analysis_software" title="Social network analysis software">Software</a></li> <li><a href="/wiki/List_of_network_scientists" title="List of network scientists">Network scientists</a></li></ul> <ul><li><a href="/wiki/Category:Network_theory" title="Category:Network theory">Category:Network theory</a></li> <li><a href="/wiki/Category:Graph_theory" title="Category:Graph theory">Category:Graph theory</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Network_science" title="Template:Network science"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Network_science" title="Template talk:Network science"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Network_science" title="Special:EditPage/Template:Network science"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>random graph</b> is the general term to refer to <a href="/wiki/Probability_distribution" title="Probability distribution">probability distributions</a> over <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a>. Random graphs may be described simply by a probability distribution, or by a <a href="/wiki/Random_process" class="mw-redirect" title="Random process">random process</a> which generates them.<sup id="cite_ref-Random_Graphs_1-0" class="reference"><a href="#cite_note-Random_Graphs-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Introduction_to_Random_graphs_2-0" class="reference"><a href="#cite_note-Introduction_to_Random_graphs-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The theory of random graphs lies at the intersection between <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a> and <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>. From a mathematical perspective, random graphs are used to answer questions about the properties of <i>typical</i> graphs. Its practical applications are found in all areas in which <a href="/wiki/Complex_network" title="Complex network">complex networks</a> need to be modeled – many random graph models are thus known, mirroring the diverse types of complex networks encountered in different areas. In a mathematical context, <i>random graph</i> refers almost exclusively to the <a href="/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model" title="Erdős–Rényi model">Erdős–Rényi random graph model</a>. In other contexts, any graph model may be referred to as a <i>random graph</i>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Models">Models</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_graph&action=edit&section=1" title="Edit section: Models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A random graph is obtained by starting with a set of <i>n</i> isolated vertices and adding successive edges between them at random. The aim of the study in this field is to determine at what stage a particular property of the graph is likely to arise.<sup id="cite_ref-Random_Graphs2_3-0" class="reference"><a href="#cite_note-Random_Graphs2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Different <b>random graph models</b> produce different <a href="/wiki/Probability_distribution" title="Probability distribution">probability distributions</a> on graphs. Most commonly studied is the one proposed by <a href="/wiki/Edgar_Gilbert" title="Edgar Gilbert">Edgar Gilbert</a> but often called the <a href="/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model" title="Erdős–Rényi model">Erdős–Rényi model</a>, denoted <i>G</i>(<i>n</i>,<i>p</i>). In it, every possible edge occurs independently with probability 0 < <i>p</i> < 1. The probability of obtaining <i>any one particular</i> random graph with <i>m</i> edges is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{m}(1-p)^{N-m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{m}(1-p)^{N-m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a651740c801d762a3f4306fedc1d7acf5ab2b46b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:14.328ex; height:3.176ex;" alt="{\displaystyle p^{m}(1-p)^{N-m}}"></span> with the notation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N={\tbinom {n}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N={\tbinom {n}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c60f7fe95d23ef0f6aead7ae0ee99176bcab150f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.278ex; height:3.176ex;" alt="{\displaystyle N={\tbinom {n}{2}}}"></span>.<sup id="cite_ref-Random_Graphs3_4-0" class="reference"><a href="#cite_note-Random_Graphs3-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>A closely related model, also called the Erdős–Rényi model and denoted <i>G</i>(<i>n</i>,<i>M</i>), assigns equal probability to all graphs with exactly <i>M</i> edges. With 0 ≤ <i>M</i> ≤ <i>N</i>, <i>G</i>(<i>n</i>,<i>M</i>) has <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tbinom {N}{M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>N</mi> <mi>M</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tbinom {N}{M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfe7c633b96c7ececeabdc5411e8e063fb61afa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.857ex; height:3.343ex;" alt="{\displaystyle {\tbinom {N}{M}}}"></span> elements and every element occurs with probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/{\tbinom {N}{M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>N</mi> <mi>M</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/{\tbinom {N}{M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9db537a91ea66a3b8e150ced34e4aa07d96d4898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.182ex; height:3.343ex;" alt="{\displaystyle 1/{\tbinom {N}{M}}}"></span>.<sup id="cite_ref-Random_Graphs2_3-1" class="reference"><a href="#cite_note-Random_Graphs2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The <i>G</i>(<i>n</i>,<i>M</i>) model can be viewed as a snapshot at a particular time (<i>M</i>) of the <b>random graph process</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {G}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>G</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {G}}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8340a778e1a962157f13ea507aff67ef7ad4e59f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.045ex; height:3.009ex;" alt="{\displaystyle {\tilde {G}}_{n}}"></span>, a <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic process</a> that starts with <i>n</i> vertices and no edges, and at each step adds one new edge chosen uniformly from the set of missing edges. </p><p>If instead we start with an infinite set of vertices, and again let every possible edge occur independently with probability 0 < <i>p</i> < 1, then we get an object <i>G</i> called an <b>infinite random graph</b>. Except in the trivial cases when <i>p</i> is 0 or 1, such a <i>G</i> <a href="/wiki/Almost_surely" title="Almost surely">almost surely</a> has the following property: </p> <blockquote><p>Given any <i>n</i> + <i>m</i> elements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\ldots ,a_{n},b_{1},\ldots ,b_{m}\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\ldots ,a_{n},b_{1},\ldots ,b_{m}\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91edaec20ab1447185fd3594a2d29992ca51bf22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.475ex; height:2.509ex;" alt="{\displaystyle a_{1},\ldots ,a_{n},b_{1},\ldots ,b_{m}\in V}"></span>, there is a vertex <i>c</i> in <i>V</i> that is adjacent to each of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\ldots ,a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\ldots ,a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/451345cc97e2ed923dd4656fcc400c3f37119cca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.911ex; height:2.009ex;" alt="{\displaystyle a_{1},\ldots ,a_{n}}"></span> and is not adjacent to any of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1},\ldots ,b_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{1},\ldots ,b_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd49a4e6f0dbeba4abaaae3b7987a9eec9eeac0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.903ex; height:2.509ex;" alt="{\displaystyle b_{1},\ldots ,b_{m}}"></span>.</p></blockquote> <p>It turns out that if the vertex set is <a href="/wiki/Countable" class="mw-redirect" title="Countable">countable</a> then there is, <a href="/wiki/Up_to" title="Up to">up to</a> <a href="/wiki/Graph_isomorphism" title="Graph isomorphism">isomorphism</a>, only a single graph with this property, namely the <a href="/wiki/Rado_graph" title="Rado graph">Rado graph</a>. Thus any countably infinite random graph is almost surely the Rado graph, which for this reason is sometimes called simply the <b>random graph</b>. However, the analogous result is not true for uncountable graphs, of which there are many (nonisomorphic) graphs satisfying the above property. </p><p>Another model, which generalizes Gilbert's random graph model, is the <b>random dot-product model</b>. A random dot-product graph associates with each vertex a <a href="/wiki/Real_vector" class="mw-redirect" title="Real vector">real vector</a>. The probability of an edge <i>uv</i> between any vertices <i>u</i> and <i>v</i> is some function of the <a href="/wiki/Dot_product" title="Dot product">dot product</a> <b>u</b> • <b>v</b> of their respective vectors. </p><p>The <a href="/wiki/Network_probability_matrix" title="Network probability matrix">network probability matrix</a> models random graphs through edge probabilities, which represent the probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i,j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i,j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e82f4cb43c1f5cd53898ed7cc80dcf092373f676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:3.193ex; height:2.343ex;" alt="{\displaystyle p_{i,j}}"></span> that a given edge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{i,j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{i,j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9c7a1c781b037551cd83fc9f1939d47f8a3cf4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.018ex; height:2.343ex;" alt="{\displaystyle e_{i,j}}"></span> exists for a specified time period. This model is extensible to directed and undirected; weighted and unweighted; and static or dynamic graphs structure. </p><p>For <i>M</i> ≃ <i>pN</i>, where <i>N</i> is the maximal number of edges possible, the two most widely used models, <i>G</i>(<i>n</i>,<i>M</i>) and <i>G</i>(<i>n</i>,<i>p</i>), are almost interchangeable.<sup id="cite_ref-Handbook_5-0" class="reference"><a href="#cite_note-Handbook-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Random_regular_graph" title="Random regular graph">Random regular graphs</a> form a special case, with properties that may differ from random graphs in general. </p><p>Once we have a model of random graphs, every function on graphs, becomes a <a href="/wiki/Random_variable" title="Random variable">random variable</a>. The study of this model is to determine if, or at least estimate the probability that, a property may occur.<sup id="cite_ref-Random_Graphs3_4-1" class="reference"><a href="#cite_note-Random_Graphs3-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Terminology">Terminology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_graph&action=edit&section=2" title="Edit section: Terminology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The term 'almost every' in the context of random graphs refers to a sequence of spaces and probabilities, such that the <i>error probabilities</i> tend to zero.<sup id="cite_ref-Random_Graphs3_4-2" class="reference"><a href="#cite_note-Random_Graphs3-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_graph&action=edit&section=3" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The theory of random graphs studies typical properties of random graphs, those that hold with high probability for graphs drawn from a particular distribution. For example, we might ask for a given value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> what the probability is that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(n,p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(n,p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad8d6ba8bbe18701bed34c2d5106de6a56e35e08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.234ex; height:2.843ex;" alt="{\displaystyle G(n,p)}"></span> is <a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">connected</a>. In studying such questions, researchers often concentrate on the asymptotic behavior of random graphs—the values that various probabilities converge to as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> grows very large. <a href="/wiki/Percolation_theory" title="Percolation theory">Percolation theory</a> characterizes the connectedness of random graphs, especially infinitely large ones. </p><p>Percolation is related to the robustness of the graph (called also network). Given a random graph of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> nodes and an average degree <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle k\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>k</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle k\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79c8e81fd47c64b42f310aa18c5197183dcbb0d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.021ex; height:2.843ex;" alt="{\displaystyle \langle k\rangle }"></span>. Next we remove randomly a fraction <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9633a8692121eedfa99cace406205e5d1511ef8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.172ex; height:2.509ex;" alt="{\displaystyle 1-p}"></span> of nodes and leave only a fraction <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>. There exists a critical percolation threshold <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{c}={\tfrac {1}{\langle k\rangle }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>k</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{c}={\tfrac {1}{\langle k\rangle }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a99569c6d70801594712783ce52840fb48af5195" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:8.273ex; height:4.176ex;" alt="{\displaystyle p_{c}={\tfrac {1}{\langle k\rangle }}}"></span> below which the network becomes fragmented while above <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/836639c3805ca867b1ff24dc6db7a6b24fc69158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.203ex; height:2.009ex;" alt="{\displaystyle p_{c}}"></span> a giant connected component exists.<sup id="cite_ref-Random_Graphs_1-1" class="reference"><a href="#cite_note-Random_Graphs-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Handbook_5-1" class="reference"><a href="#cite_note-Handbook-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Random_graphs_6-0" class="reference"><a href="#cite_note-Random_graphs-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-On_Random_Graphs_8-0" class="reference"><a href="#cite_note-On_Random_Graphs-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Localized percolation refers to removing a node its neighbors, next nearest neighbors etc. until a fraction of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9633a8692121eedfa99cace406205e5d1511ef8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.172ex; height:2.509ex;" alt="{\displaystyle 1-p}"></span> of nodes from the network is removed. It was shown that for random graph with Poisson distribution of degrees <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{c}={\tfrac {1}{\langle k\rangle }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>k</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{c}={\tfrac {1}{\langle k\rangle }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a99569c6d70801594712783ce52840fb48af5195" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:8.273ex; height:4.176ex;" alt="{\displaystyle p_{c}={\tfrac {1}{\langle k\rangle }}}"></span> exactly as for random removal. </p><p>Random graphs are widely used in the <a href="/wiki/Probabilistic_method" title="Probabilistic method">probabilistic method</a>, where one tries to prove the existence of graphs with certain properties. The existence of a property on a random graph can often imply, via the <a href="/wiki/Szemer%C3%A9di_regularity_lemma" title="Szemerédi regularity lemma">Szemerédi regularity lemma</a>, the existence of that property on almost all graphs. </p><p>In <a href="/wiki/Random_regular_graph" title="Random regular graph">random regular graphs</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(n,r-reg)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>r</mi> <mo>−</mo> <mi>r</mi> <mi>e</mi> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(n,r-reg)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f56909858c0fc1449a150cc2ac7ae18460c840d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.202ex; height:2.843ex;" alt="{\displaystyle G(n,r-reg)}"></span> are the set of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>-regular graphs with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=r(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca5b4b5d48c9035c75618947d9051c2935c9540c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.4ex; height:2.843ex;" alt="{\displaystyle r=r(n)}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> are the natural numbers, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\leq r<n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>≤</mo> <mi>r</mi> <mo><</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\leq r<n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1fe2fc0a252f04bccb9a1fbe5b1d83bafaa87e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.803ex; height:2.343ex;" alt="{\displaystyle 3\leq r<n}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle rn=2m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle rn=2m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52459f4b0df9a3d09de83b51f46159673380e41e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.745ex; height:2.176ex;" alt="{\displaystyle rn=2m}"></span> is even.<sup id="cite_ref-Random_Graphs2_3-2" class="reference"><a href="#cite_note-Random_Graphs2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The degree sequence of a graph <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f43241538481dd8112f3c0ada4237a2c7b2aa6e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.045ex; height:2.343ex;" alt="{\displaystyle G^{n}}"></span> depends only on the number of edges in the sets<sup id="cite_ref-Random_Graphs2_3-3" class="reference"><a href="#cite_note-Random_Graphs2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{n}^{(2)}=\left\{ij\ :\ 1\leq j\leq n,i\neq j\right\}\subset V^{(2)},\qquad i=1,\cdots ,n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mi>i</mi> <mi>j</mi> <mtext> </mtext> <mo>:</mo> <mtext> </mtext> <mn>1</mn> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> <mo>,</mo> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mrow> <mo>}</mo> </mrow> <mo>⊂</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <mspace width="2em" /> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{n}^{(2)}=\left\{ij\ :\ 1\leq j\leq n,i\neq j\right\}\subset V^{(2)},\qquad i=1,\cdots ,n.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0841ba1939842ed3ce26c0f6248e9372ef08b44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.449ex; height:3.509ex;" alt="{\displaystyle V_{n}^{(2)}=\left\{ij\ :\ 1\leq j\leq n,i\neq j\right\}\subset V^{(2)},\qquad i=1,\cdots ,n.}"></span></dd></dl> <p>If edges, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> in a random graph, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab6ce334c0de8fb3417356ce6ed775812acea73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.786ex; height:2.509ex;" alt="{\displaystyle G_{M}}"></span> is large enough to ensure that almost every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab6ce334c0de8fb3417356ce6ed775812acea73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.786ex; height:2.509ex;" alt="{\displaystyle G_{M}}"></span> has minimum degree at least 1, then almost every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab6ce334c0de8fb3417356ce6ed775812acea73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.786ex; height:2.509ex;" alt="{\displaystyle G_{M}}"></span> is connected and, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is even, almost every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab6ce334c0de8fb3417356ce6ed775812acea73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.786ex; height:2.509ex;" alt="{\displaystyle G_{M}}"></span> has a perfect matching. In particular, the moment the last isolated vertex vanishes in almost every random graph, the graph becomes connected.<sup id="cite_ref-Random_Graphs2_3-4" class="reference"><a href="#cite_note-Random_Graphs2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Almost every graph process on an even number of vertices with the edge raising the minimum degree to 1 or a random graph with slightly more than <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n}{4}}\log(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mn>4</mn> </mfrac> </mstyle> </mrow> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {n}{4}}\log(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2f7a78274dd7143981614cbdef0dfffb039321c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.385ex; height:3.176ex;" alt="{\displaystyle {\tfrac {n}{4}}\log(n)}"></span> edges and with probability close to 1 ensures that the graph has a complete matching, with exception of at most one vertex. </p><p>For some constant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span>, almost every labeled graph with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> vertices and at least <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cn\log(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>n</mi> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle cn\log(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9f3b907e2ab170f8551c908f7a4357d1d5f299f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.964ex; height:2.843ex;" alt="{\displaystyle cn\log(n)}"></span> edges is <a href="/wiki/Hamiltonian_cycle" class="mw-redirect" title="Hamiltonian cycle">Hamiltonian</a>. With the probability tending to 1, the particular edge that increases the minimum degree to 2 makes the graph Hamiltonian. </p><p>Properties of random graph may change or remain invariant under graph transformations. <a href="/wiki/Alireza_Mashaghi" title="Alireza Mashaghi">Mashaghi A.</a> et al., for example, demonstrated that a transformation which converts random graphs to their edge-dual graphs (or line graphs) produces an ensemble of graphs with nearly the same degree distribution, but with degree correlations and a significantly higher clustering coefficient.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Colouring">Colouring</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_graph&action=edit&section=4" title="Edit section: Colouring"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a random graph <i>G</i> of order <i>n</i> with the vertex <i>V</i>(<i>G</i>) = {1, ..., <i>n</i>}, by the <a href="/wiki/Greedy_algorithm" title="Greedy algorithm">greedy algorithm</a> on the number of colors, the vertices can be colored with colors 1, 2, ... (vertex 1 is colored 1, vertex 2 is colored 1 if it is not adjacent to vertex 1, otherwise it is colored 2, etc.).<sup id="cite_ref-Random_Graphs2_3-5" class="reference"><a href="#cite_note-Random_Graphs2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The number of proper colorings of random graphs given a number of <i>q</i> colors, called its <a href="/wiki/Chromatic_polynomial" title="Chromatic polynomial">chromatic polynomial</a>, remains unknown so far. The scaling of zeros of the chromatic polynomial of random graphs with parameters <i>n</i> and the number of edges <i>m</i> or the connection probability <i>p</i> has been studied empirically using an algorithm based on symbolic pattern matching.<sup id="cite_ref-Chromatic_Polynomials_of_Random_Graphs_10-0" class="reference"><a href="#cite_note-Chromatic_Polynomials_of_Random_Graphs-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Random_trees">Random trees</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_graph&action=edit&section=5" title="Edit section: Random trees"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Random_tree" title="Random tree">Random tree</a></div> <p>A <a href="/wiki/Random_tree" title="Random tree">random tree</a> is a <a href="/wiki/Tree_(graph_theory)" title="Tree (graph theory)">tree</a> or <a href="/wiki/Arborescence_(graph_theory)" title="Arborescence (graph theory)">arborescence</a> that is formed by a <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic process</a>. In a large range of random graphs of order <i>n</i> and size <i>M</i>(<i>n</i>) the distribution of the number of tree components of order <i>k</i> is asymptotically <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson</a>. Types of random trees include <a href="/wiki/Uniform_spanning_tree" class="mw-redirect" title="Uniform spanning tree">uniform spanning tree</a>, <a href="/wiki/Random_minimum_spanning_tree" title="Random minimum spanning tree">random minimum spanning tree</a>, <a href="/wiki/Random_binary_tree" title="Random binary tree">random binary tree</a>, <a href="/wiki/Treap" title="Treap">treap</a>, <a href="/wiki/Rapidly_exploring_random_tree" title="Rapidly exploring random tree">rapidly exploring random tree</a>, <a href="/wiki/Brownian_tree" title="Brownian tree">Brownian tree</a>, and <a href="/wiki/Random_forest" title="Random forest">random forest</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Conditional_random_graphs">Conditional random graphs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_graph&action=edit&section=6" title="Edit section: Conditional random graphs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider a given random graph model defined on the probability space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d77104a5c3c49cc0634dcf6908db7ad45f738d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},P)}"></span> and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}(G):\Omega \rightarrow R^{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">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">→</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(G):\Omega \rightarrow R^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0deefd1e904627b10cae8d92d98217e5fdd7f3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.008ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(G):\Omega \rightarrow R^{m}}"></span> be a real valued function which assigns to each graph in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> a vector of <i>m</i> properties. For a fixed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} \in R^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>∈</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} \in R^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d28b061fe99de21035280469087e0b2a85f3481f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.765ex; height:2.676ex;" alt="{\displaystyle \mathbf {p} \in R^{m}}"></span>, <i>conditional random graphs</i> are models in which the probability measure <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> assigns zero probability to all graphs such that '<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}(G)\neq \mathbf {p} }"> <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">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(G)\neq \mathbf {p} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/587c77eeb66200b4a32eefe1b599093756886d0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.923ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(G)\neq \mathbf {p} }"></span>. </p><p>Special cases are <i>conditionally uniform random graphs</i>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> assigns equal probability to all the graphs having specified properties. They can be seen as a generalization of the <a href="/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model" title="Erdős–Rényi model">Erdős–Rényi model</a> <i>G</i>(<i>n</i>,<i>M</i>), when the conditioning information is not necessarily the number of edges <i>M</i>, but whatever other arbitrary graph property <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}(G)}"> <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">P</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}(G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79964cec08dc040872b71f5aa356cafb266ca5af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.34ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(G)}"></span>. In this case very few analytical results are available and simulation is required to obtain empirical distributions of average properties. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_graph&action=edit&section=7" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The earliest use of a random graph model was by <a href="/wiki/Helen_Hall_Jennings" title="Helen Hall Jennings">Helen Hall Jennings</a> and <a href="/wiki/Jacob_Moreno" class="mw-redirect" title="Jacob Moreno">Jacob Moreno</a> in 1938 where a "chance sociogram" (a directed Erdős-Rényi model) was considered in studying comparing the fraction of reciprocated links in their network data with the random model.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Another use, under the name "random net", was by <a href="/wiki/Ray_Solomonoff" title="Ray Solomonoff">Ray Solomonoff</a> and <a href="/wiki/Anatol_Rapoport" title="Anatol Rapoport">Anatol Rapoport</a> in 1951, using a model of directed graphs with fixed out-degree and randomly chosen attachments to other vertices.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model" title="Erdős–Rényi model">Erdős–Rényi model</a> of random graphs was first defined by <a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">Paul Erdős</a> and <a href="/wiki/Alfr%C3%A9d_R%C3%A9nyi" title="Alfréd Rényi">Alfréd Rényi</a> in their 1959 paper "On Random Graphs"<sup id="cite_ref-On_Random_Graphs_8-1" class="reference"><a href="#cite_note-On_Random_Graphs-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and independently by Gilbert in his paper "Random graphs".<sup id="cite_ref-Random_graphs_6-1" class="reference"><a href="#cite_note-Random_graphs-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_graph&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Bose%E2%80%93Einstein_condensation:_a_network_theory_approach" class="mw-redirect" title="Bose–Einstein condensation: a network theory approach">Bose–Einstein condensation: a network theory approach</a></li> <li><a href="/wiki/Cavity_method" title="Cavity method">Cavity method</a></li> <li><a href="/wiki/Complex_networks" class="mw-redirect" title="Complex networks">Complex networks</a></li> <li><a href="/wiki/Dual-phase_evolution" title="Dual-phase evolution">Dual-phase evolution</a></li> <li><a href="/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model" title="Erdős–Rényi model">Erdős–Rényi model</a></li> <li><a href="/wiki/Exponential_random_graph_model" class="mw-redirect" title="Exponential random graph model">Exponential random graph model</a></li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></li> <li><a href="/wiki/Interdependent_networks" title="Interdependent networks">Interdependent networks</a></li> <li><a href="/wiki/Network_science" title="Network science">Network science</a></li> <li><a href="/wiki/Percolation" title="Percolation">Percolation</a></li> <li><a href="/wiki/Percolation_theory" title="Percolation theory">Percolation theory</a></li> <li><a href="/wiki/Random_graph_theory_of_gelation" title="Random graph theory of gelation">Random graph theory of gelation</a></li> <li><a href="/wiki/Regular_graph" title="Regular graph">Regular graph</a></li> <li><a href="/wiki/Scale_free_network" class="mw-redirect" title="Scale free network">Scale free network</a></li> <li><a href="/wiki/Semilinear_response" title="Semilinear response">Semilinear response</a></li> <li><a href="/wiki/Stochastic_block_model" title="Stochastic block model">Stochastic block model</a></li> <li><a href="/wiki/Lancichinetti%E2%80%93Fortunato%E2%80%93Radicchi_benchmark" title="Lancichinetti–Fortunato–Radicchi benchmark">Lancichinetti–Fortunato–Radicchi benchmark</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Random_graph&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Random_Graphs-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Random_Graphs_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Random_Graphs_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><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 id="CITEREFBollobás2001" class="citation book cs1">Bollobás, Béla (2001). <i>Random Graphs</i> (2nd ed.). Cambridge University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Random+Graphs&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=2001&rft.aulast=Bollob%C3%A1s&rft.aufirst=B%C3%A9la&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+graph" class="Z3988"></span></span> </li> <li id="cite_note-Introduction_to_Random_graphs-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Introduction_to_Random_graphs_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFriezeKaronski2015" class="citation book cs1">Frieze, Alan; Karonski, Michal (2015). <i>Introduction to Random Graphs</i>. Cambridge University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Random+Graphs&rft.pub=Cambridge+University+Press&rft.date=2015&rft.aulast=Frieze&rft.aufirst=Alan&rft.au=Karonski%2C+Michal&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+graph" class="Z3988"></span></span> </li> <li id="cite_note-Random_Graphs2-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Random_Graphs2_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Random_Graphs2_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Random_Graphs2_3-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Random_Graphs2_3-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Random_Graphs2_3-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Random_Graphs2_3-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/B%C3%A9la_Bollob%C3%A1s" title="Béla Bollobás">Béla Bollobás</a>, <i>Random Graphs</i>, 1985, Academic Press Inc., London Ltd.</span> </li> <li id="cite_note-Random_Graphs3-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Random_Graphs3_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Random_Graphs3_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Random_Graphs3_4-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/B%C3%A9la_Bollob%C3%A1s" title="Béla Bollobás">Béla Bollobás</a>, <i>Probabilistic Combinatorics and Its Applications</i>, 1991, Providence, RI: American Mathematical Society.</span> </li> <li id="cite_note-Handbook-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Handbook_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Handbook_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/B%C3%A9la_Bollob%C3%A1s" title="Béla Bollobás">Bollobas, B.</a> and Riordan, O.M. "Mathematical results on scale-free random graphs" in "Handbook of Graphs and Networks" (S. Bornholdt and H.G. Schuster (eds)), Wiley VCH, Weinheim, 1st ed., 2003</span> </li> <li id="cite_note-Random_graphs-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Random_graphs_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Random_graphs_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGilbert1959" class="citation cs2"><a href="/wiki/Edgar_Gilbert" title="Edgar Gilbert">Gilbert, E. N.</a> (1959), "Random graphs", <i>Annals of Mathematical Statistics</i>, <b>30</b> (4): <span class="nowrap">1141–</span>1144, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177706098">10.1214/aoms/1177706098</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematical+Statistics&rft.atitle=Random+graphs&rft.volume=30&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1141-%3C%2Fspan%3E1144&rft.date=1959&rft_id=info%3Adoi%2F10.1214%2Faoms%2F1177706098&rft.aulast=Gilbert&rft.aufirst=E.+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+graph" class="Z3988"></span>.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNewman2010" class="citation book cs1">Newman, M. E. J. (2010). <i>Networks: An Introduction</i>. Oxford.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Networks%3A+An+Introduction&rft.pub=Oxford&rft.date=2010&rft.aulast=Newman&rft.aufirst=M.+E.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+graph" class="Z3988"></span></span> </li> <li id="cite_note-On_Random_Graphs-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-On_Random_Graphs_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-On_Random_Graphs_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">Erdős, P.</a> <a href="/wiki/Alfr%C3%A9d_R%C3%A9nyi" title="Alfréd Rényi">Rényi, A</a> (1959) "On Random Graphs I" in Publ. Math. Debrecen 6, p. 290–297 <a rel="nofollow" class="external autonumber" href="http://www.renyi.hu/~p_erdos/1959-11.pdf">[1]</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200807021117/https://www.renyi.hu/~p_erdos/1959-11.pdf">Archived</a> 2020-08-07 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamezanpourKarimipourMashaghi2003" class="citation journal cs1">Ramezanpour, A.; Karimipour, V.; Mashaghi, A. (2003). "Generating correlated networks from uncorrelated ones". <i>Phys. Rev. E</i>. <b>67</b> (46107): 046107. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0212469">cond-mat/0212469</a></span>. <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/2003PhRvE..67d6107R">2003PhRvE..67d6107R</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.67.046107">10.1103/PhysRevE.67.046107</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/12786436">12786436</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:33054818">33054818</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=Generating+correlated+networks+from+uncorrelated+ones&rft.volume=67&rft.issue=46107&rft.pages=046107&rft.date=2003&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A33054818%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2003PhRvE..67d6107R&rft_id=info%3Aarxiv%2Fcond-mat%2F0212469&rft_id=info%3Apmid%2F12786436&rft_id=info%3Adoi%2F10.1103%2FPhysRevE.67.046107&rft.aulast=Ramezanpour&rft.aufirst=A.&rft.au=Karimipour%2C+V.&rft.au=Mashaghi%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+graph" class="Z3988"></span></span> </li> <li id="cite_note-Chromatic_Polynomials_of_Random_Graphs-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Chromatic_Polynomials_of_Random_Graphs_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVan_BusselEhrlichFliegnerStolzenberg2010" class="citation journal cs1">Van Bussel, Frank; Ehrlich, Christoph; Fliegner, Denny; Stolzenberg, Sebastian; Timme, Marc (2010). "Chromatic Polynomials of Random Graphs". <i>J. Phys. A: Math. Theor</i>. <b>43</b> (17): 175002. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1709.06209">1709.06209</a></span>. <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/2010JPhA...43q5002V">2010JPhA...43q5002V</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%2F1751-8113%2F43%2F17%2F175002">10.1088/1751-8113/43/17/175002</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:15723612">15723612</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Phys.+A%3A+Math.+Theor.&rft.atitle=Chromatic+Polynomials+of+Random+Graphs&rft.volume=43&rft.issue=17&rft.pages=175002&rft.date=2010&rft_id=info%3Aarxiv%2F1709.06209&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15723612%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F1751-8113%2F43%2F17%2F175002&rft_id=info%3Abibcode%2F2010JPhA...43q5002V&rft.aulast=Van+Bussel&rft.aufirst=Frank&rft.au=Ehrlich%2C+Christoph&rft.au=Fliegner%2C+Denny&rft.au=Stolzenberg%2C+Sebastian&rft.au=Timme%2C+Marc&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+graph" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMorenoJennings1938" class="citation journal cs1">Moreno, Jacob L; Jennings, Helen Hall (Jan 1938). <a rel="nofollow" class="external text" href="https://hal.science/hal-03963403/file/morenojennings1938groupefmr.pdf">"Statistics of Social Configurations"</a> <span class="cs1-format">(PDF)</span>. <i>Sociometry</i>. <b>1</b> (3/4): <span class="nowrap">342–</span>374. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2785588">10.2307/2785588</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2785588">2785588</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sociometry&rft.atitle=Statistics+of+Social+Configurations&rft.volume=1&rft.issue=3%2F4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E342-%3C%2Fspan%3E374&rft.date=1938-01&rft_id=info%3Adoi%2F10.2307%2F2785588&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2785588%23id-name%3DJSTOR&rft.aulast=Moreno&rft.aufirst=Jacob+L&rft.au=Jennings%2C+Helen+Hall&rft_id=https%3A%2F%2Fhal.science%2Fhal-03963403%2Ffile%2Fmorenojennings1938groupefmr.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+graph" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSolomonoffRapoport1951" class="citation journal cs1">Solomonoff, Ray; Rapoport, Anatol (June 1951). "Connectivity of random nets". <i>Bulletin of Mathematical Biophysics</i>. <b>13</b> (2): <span class="nowrap">107–</span>117. <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%2FBF02478357">10.1007/BF02478357</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+Mathematical+Biophysics&rft.atitle=Connectivity+of+random+nets&rft.volume=13&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E107-%3C%2Fspan%3E117&rft.date=1951-06&rft_id=info%3Adoi%2F10.1007%2FBF02478357&rft.aulast=Solomonoff&rft.aufirst=Ray&rft.au=Rapoport%2C+Anatol&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARandom+graph" class="Z3988"></span></span> </li> </ol></div></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style 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.navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Stochastic_processes496" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Stochastic_processes" title="Template:Stochastic processes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Stochastic_processes" title="Template talk:Stochastic processes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Stochastic_processes" title="Special:EditPage/Template:Stochastic processes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Stochastic_processes496" style="font-size:114%;margin:0 4em"><a href="/wiki/Stochastic_process" title="Stochastic process">Stochastic processes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete-time_stochastic_process" class="mw-redirect" title="Discrete-time stochastic process">Discrete time</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li> <li><a href="/wiki/Branching_process" title="Branching process">Branching process</a></li> <li><a href="/wiki/Chinese_restaurant_process" title="Chinese restaurant process">Chinese restaurant process</a></li> <li><a href="/wiki/Galton%E2%80%93Watson_process" title="Galton–Watson process">Galton–Watson process</a></li> <li><a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">Independent and identically distributed random variables</a></li> <li><a href="/wiki/Markov_chain" title="Markov chain">Markov chain</a></li> <li><a href="/wiki/Moran_process" title="Moran process">Moran process</a></li> <li><a href="/wiki/Random_walk" title="Random walk">Random walk</a> <ul><li><a href="/wiki/Loop-erased_random_walk" title="Loop-erased random walk">Loop-erased</a></li> <li><a href="/wiki/Self-avoiding_walk" title="Self-avoiding walk">Self-avoiding</a></li> <li><a href="/wiki/Biased_random_walk_on_a_graph" title="Biased random walk on a graph"> Biased</a></li> <li><a href="/wiki/Maximal_entropy_random_walk" title="Maximal entropy random walk">Maximal entropy</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Continuous-time_stochastic_process" title="Continuous-time stochastic process">Continuous time</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Additive_process" title="Additive process">Additive process</a></li> <li><a href="/wiki/Bessel_process" title="Bessel process">Bessel process</a></li> <li><a href="/wiki/Birth%E2%80%93death_process" title="Birth–death process">Birth–death process</a> <ul><li><a href="/wiki/Birth_process" title="Birth process">pure birth</a></li></ul></li> <li><a href="/wiki/Wiener_process" title="Wiener process">Brownian motion</a> <ul><li><a href="/wiki/Brownian_bridge" title="Brownian bridge">Bridge</a></li> <li><a href="/wiki/Brownian_excursion" title="Brownian excursion">Excursion</a></li> <li><a href="/wiki/Fractional_Brownian_motion" title="Fractional Brownian motion">Fractional</a></li> <li><a href="/wiki/Geometric_Brownian_motion" title="Geometric Brownian motion">Geometric</a></li> <li><a href="/wiki/Brownian_meander" title="Brownian meander">Meander</a></li></ul></li> <li><a href="/wiki/Cauchy_process" title="Cauchy process">Cauchy process</a></li> <li><a href="/wiki/Contact_process_(mathematics)" title="Contact process (mathematics)">Contact process</a></li> <li><a href="/wiki/Continuous-time_random_walk" title="Continuous-time random walk">Continuous-time random walk</a></li> <li><a href="/wiki/Cox_process" title="Cox process">Cox process</a></li> <li><a href="/wiki/Diffusion_process" title="Diffusion process">Diffusion process</a></li> <li><a href="/wiki/Dyson_Brownian_motion" title="Dyson Brownian motion">Dyson Brownian motion</a></li> <li><a href="/wiki/Empirical_process" title="Empirical process">Empirical process</a></li> <li><a href="/wiki/Feller_process" title="Feller process">Feller process</a></li> <li><a href="/wiki/Fleming%E2%80%93Viot_process" title="Fleming–Viot process">Fleming–Viot process</a></li> <li><a href="/wiki/Gamma_process" title="Gamma process">Gamma process</a></li> <li><a href="/wiki/Geometric_process" title="Geometric process">Geometric process</a></li> <li><a href="/wiki/Hawkes_process" title="Hawkes process">Hawkes process</a></li> <li><a href="/wiki/Hunt_process" title="Hunt process">Hunt process</a></li> <li><a href="/wiki/Interacting_particle_system" title="Interacting particle system">Interacting particle systems</a></li> <li><a href="/wiki/It%C3%B4_diffusion" title="Itô diffusion">Itô diffusion</a></li> <li><a href="/wiki/It%C3%B4_process" class="mw-redirect" title="Itô process">Itô process</a></li> <li><a href="/wiki/Jump_diffusion" title="Jump diffusion">Jump diffusion</a></li> <li><a href="/wiki/Jump_process" title="Jump process">Jump process</a></li> <li><a href="/wiki/L%C3%A9vy_process" title="Lévy process">Lévy process</a></li> <li><a href="/wiki/Local_time_(mathematics)" title="Local time (mathematics)">Local time</a></li> <li><a href="/wiki/Markov_additive_process" title="Markov additive process">Markov additive process</a></li> <li><a href="/wiki/McKean%E2%80%93Vlasov_process" title="McKean–Vlasov process">McKean–Vlasov process</a></li> <li><a href="/wiki/Ornstein%E2%80%93Uhlenbeck_process" title="Ornstein–Uhlenbeck process">Ornstein–Uhlenbeck process</a></li> <li><a href="/wiki/Poisson_point_process" title="Poisson point process">Poisson process</a> <ul><li><a href="/wiki/Compound_Poisson_process" title="Compound Poisson process">Compound</a></li> <li><a href="/wiki/Non-homogeneous_Poisson_process" class="mw-redirect" title="Non-homogeneous Poisson process">Non-homogeneous</a></li></ul></li> <li><a href="/wiki/Schramm%E2%80%93Loewner_evolution" title="Schramm–Loewner evolution">Schramm–Loewner evolution</a></li> <li><a href="/wiki/Semimartingale" title="Semimartingale">Semimartingale</a></li> <li><a href="/wiki/Sigma-martingale" title="Sigma-martingale">Sigma-martingale</a></li> <li><a href="/wiki/Stable_process" title="Stable process">Stable process</a></li> <li><a href="/wiki/Superprocess" title="Superprocess">Superprocess</a></li> <li><a href="/wiki/Telegraph_process" title="Telegraph process">Telegraph process</a></li> <li><a href="/wiki/Variance_gamma_process" title="Variance gamma process">Variance gamma process</a></li> <li><a href="/wiki/Wiener_process" title="Wiener process">Wiener process</a></li> <li><a href="/wiki/Wiener_sausage" title="Wiener sausage">Wiener sausage</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Both</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Branching_process" title="Branching process">Branching process</a></li> <li><a href="/wiki/Gaussian_process" title="Gaussian process">Gaussian process</a></li> <li><a href="/wiki/Hidden_Markov_model" title="Hidden Markov model">Hidden Markov model (HMM)</a></li> <li><a href="/wiki/Markov_process" class="mw-redirect" title="Markov process">Markov process</a></li> <li><a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">Martingale</a> <ul><li><a href="/wiki/Martingale_difference_sequence" title="Martingale difference sequence">Differences</a></li> <li><a href="/wiki/Local_martingale" title="Local martingale">Local</a></li> <li><a href="/wiki/Submartingale" class="mw-redirect" title="Submartingale">Sub-</a></li> <li><a href="/wiki/Supermartingale" class="mw-redirect" title="Supermartingale">Super-</a></li></ul></li> <li><a href="/wiki/Random_dynamical_system" title="Random dynamical system">Random dynamical system</a></li> <li><a href="/wiki/Regenerative_process" title="Regenerative process">Regenerative process</a></li> <li><a href="/wiki/Renewal_process" class="mw-redirect" title="Renewal process">Renewal process</a></li> <li><a href="/wiki/Stochastic_chains_with_memory_of_variable_length" title="Stochastic chains with memory of variable length">Stochastic chains with memory of variable length</a></li> <li><a href="/wiki/White_noise" title="White noise">White noise</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fields and other</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dirichlet_process" title="Dirichlet process">Dirichlet process</a></li> <li><a href="/wiki/Gaussian_random_field" title="Gaussian random field">Gaussian random field</a></li> <li><a href="/wiki/Gibbs_measure" title="Gibbs measure">Gibbs measure</a></li> <li><a href="/wiki/Hopfield_model" class="mw-redirect" title="Hopfield model">Hopfield model</a></li> <li><a href="/wiki/Ising_model" title="Ising model">Ising model</a> <ul><li><a href="/wiki/Potts_model" title="Potts model">Potts model</a></li> <li><a href="/wiki/Boolean_network" title="Boolean network">Boolean network</a></li></ul></li> <li><a href="/wiki/Markov_random_field" title="Markov random field">Markov random field</a></li> <li><a href="/wiki/Percolation_theory" title="Percolation theory">Percolation</a></li> <li><a href="/wiki/Pitman%E2%80%93Yor_process" title="Pitman–Yor process">Pitman–Yor process</a></li> <li><a href="/wiki/Point_process" title="Point process">Point process</a> <ul><li><a href="/wiki/Point_process#Cox_point_process" title="Point process">Cox</a></li> <li><a href="/wiki/Poisson_point_process" title="Poisson point process">Poisson</a></li></ul></li> <li><a href="/wiki/Random_field" title="Random field">Random field</a></li> <li><a class="mw-selflink selflink">Random graph</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Time_series" title="Time series">Time series models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH) model</a></li> <li><a href="/wiki/Autoregressive_integrated_moving_average" title="Autoregressive integrated moving average">Autoregressive integrated moving average (ARIMA) model</a></li> <li><a href="/wiki/Autoregressive_model" title="Autoregressive model">Autoregressive (AR) model</a></li> <li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">Autoregressive–moving-average (ARMA) model</a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Generalized autoregressive conditional heteroskedasticity (GARCH) model</a></li> <li><a href="/wiki/Moving-average_model" title="Moving-average model">Moving-average (MA) model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Asset_pricing_model" class="mw-redirect" title="Asset pricing model">Financial models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binomial_options_pricing_model" title="Binomial options pricing model">Binomial options pricing model</a></li> <li><a href="/wiki/Black%E2%80%93Derman%E2%80%93Toy_model" title="Black–Derman–Toy model">Black–Derman–Toy</a></li> <li><a href="/wiki/Black%E2%80%93Karasinski_model" title="Black–Karasinski model">Black–Karasinski</a></li> <li><a href="/wiki/Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes</a></li> <li><a href="/wiki/Chan%E2%80%93Karolyi%E2%80%93Longstaff%E2%80%93Sanders_process" title="Chan–Karolyi–Longstaff–Sanders process">Chan–Karolyi–Longstaff–Sanders (CKLS)</a></li> <li><a href="/wiki/Chen_model" title="Chen model">Chen</a></li> <li><a href="/wiki/Constant_elasticity_of_variance_model" title="Constant elasticity of variance model">Constant elasticity of variance (CEV)</a></li> <li><a href="/wiki/Cox%E2%80%93Ingersoll%E2%80%93Ross_model" title="Cox–Ingersoll–Ross model">Cox–Ingersoll–Ross (CIR)</a></li> <li><a href="/wiki/Garman%E2%80%93Kohlhagen_model" class="mw-redirect" title="Garman–Kohlhagen model">Garman–Kohlhagen</a></li> <li><a href="/wiki/Heath%E2%80%93Jarrow%E2%80%93Morton_framework" title="Heath–Jarrow–Morton framework">Heath–Jarrow–Morton (HJM)</a></li> <li><a href="/wiki/Heston_model" title="Heston model">Heston</a></li> <li><a href="/wiki/Ho%E2%80%93Lee_model" title="Ho–Lee model">Ho–Lee</a></li> <li><a href="/wiki/Hull%E2%80%93White_model" title="Hull–White model">Hull–White</a></li> <li><a href="/wiki/Korn%E2%80%93Kreer%E2%80%93Lenssen_model" title="Korn–Kreer–Lenssen model">Korn-Kreer-Lenssen</a></li> <li><a href="/wiki/LIBOR_market_model" title="LIBOR market model">LIBOR market</a></li> <li><a href="/wiki/Rendleman%E2%80%93Bartter_model" title="Rendleman–Bartter model">Rendleman–Bartter</a></li> <li><a href="/wiki/SABR_volatility_model" title="SABR volatility model">SABR volatility</a></li> <li><a href="/wiki/Vasicek_model" title="Vasicek model">Vašíček</a></li> <li><a href="/wiki/Wilkie_investment_model" title="Wilkie investment model">Wilkie</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/B%C3%BChlmann_model" title="Bühlmann model">Bühlmann</a></li> <li><a href="/wiki/Cram%C3%A9r%E2%80%93Lundberg_model" class="mw-redirect" title="Cramér–Lundberg model">Cramér–Lundberg</a></li> <li><a href="/wiki/Risk_process" class="mw-redirect" title="Risk process">Risk process</a></li> <li><a href="/wiki/Sparre%E2%80%93Anderson_model" class="mw-redirect" title="Sparre–Anderson model">Sparre–Anderson</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Queueing_model" class="mw-redirect" title="Queueing model">Queueing models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bulk_queue" title="Bulk queue">Bulk</a></li> <li><a href="/wiki/Fluid_queue" title="Fluid queue">Fluid</a></li> <li><a href="/wiki/G-network" title="G-network">Generalized queueing network</a></li> <li><a href="/wiki/M/G/1_queue" title="M/G/1 queue">M/G/1</a></li> <li><a href="/wiki/M/M/1_queue" title="M/M/1 queue">M/M/1</a></li> <li><a href="/wiki/M/M/c_queue" title="M/M/c queue">M/M/c</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/C%C3%A0dl%C3%A0g" title="Càdlàg">Càdlàg paths</a></li> <li><a href="/wiki/Continuous_stochastic_process" title="Continuous stochastic process">Continuous</a></li> <li><a href="/wiki/Sample-continuous_process" title="Sample-continuous process">Continuous paths</a></li> <li><a href="/wiki/Ergodicity" title="Ergodicity">Ergodic</a></li> <li><a href="/wiki/Exchangeable_random_variables" title="Exchangeable random variables">Exchangeable</a></li> <li><a href="/wiki/Feller-continuous_process" title="Feller-continuous process">Feller-continuous</a></li> <li><a href="/wiki/Gauss%E2%80%93Markov_process" title="Gauss–Markov process">Gauss–Markov</a></li> <li><a href="/wiki/Markov_property" title="Markov property">Markov</a></li> <li><a href="/wiki/Mixing_(mathematics)" title="Mixing (mathematics)">Mixing</a></li> <li><a href="/wiki/Piecewise-deterministic_Markov_process" title="Piecewise-deterministic Markov process">Piecewise-deterministic</a></li> <li><a href="/wiki/Predictable_process" title="Predictable process">Predictable</a></li> <li><a href="/wiki/Progressively_measurable_process" title="Progressively measurable process">Progressively measurable</a></li> <li><a href="/wiki/Self-similar_process" title="Self-similar process">Self-similar</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationary</a></li> <li><a href="/wiki/Time_reversibility" title="Time reversibility">Time-reversible</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Limit theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li> <li><a href="/wiki/Donsker%27s_theorem" title="Donsker's theorem">Donsker's theorem</a></li> <li><a href="/wiki/Doob%27s_martingale_convergence_theorems" title="Doob's martingale convergence theorems">Doob's martingale convergence theorems</a></li> <li><a href="/wiki/Ergodic_theorem" class="mw-redirect" title="Ergodic theorem">Ergodic theorem</a></li> <li><a href="/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem" title="Fisher–Tippett–Gnedenko theorem">Fisher–Tippett–Gnedenko theorem</a></li> <li><a href="/wiki/Large_deviation_principle" class="mw-redirect" title="Large deviation principle">Large deviation principle</a></li> <li><a href="/wiki/Law_of_large_numbers" title="Law of large numbers">Law of large numbers (weak/strong)</a></li> <li><a href="/wiki/Law_of_the_iterated_logarithm" title="Law of the iterated logarithm">Law of the iterated logarithm</a></li> <li><a href="/wiki/Maximal_ergodic_theorem" title="Maximal ergodic theorem">Maximal ergodic theorem</a></li> <li><a href="/wiki/Sanov%27s_theorem" title="Sanov's theorem">Sanov's theorem</a></li> <li><a href="/wiki/Zero%E2%80%93one_law" title="Zero–one law">Zero–one laws</a> (<a href="/wiki/Blumenthal%27s_zero%E2%80%93one_law" title="Blumenthal's zero–one law">Blumenthal</a>, <a href="/wiki/Borel%E2%80%93Cantelli_lemma" title="Borel–Cantelli lemma">Borel–Cantelli</a>, <a href="/wiki/Engelbert%E2%80%93Schmidt_zero%E2%80%93one_law" title="Engelbert–Schmidt zero–one law">Engelbert–Schmidt</a>, <a href="/wiki/Hewitt%E2%80%93Savage_zero%E2%80%93one_law" title="Hewitt–Savage zero–one law">Hewitt–Savage</a>, <a href="/wiki/Kolmogorov%27s_zero%E2%80%93one_law" title="Kolmogorov's zero–one law"> Kolmogorov</a>, <a href="/wiki/L%C3%A9vy%27s_zero%E2%80%93one_law" class="mw-redirect" title="Lévy's zero–one law">Lévy</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_inequalities#Probability_theory_and_statistics" title="List of inequalities">Inequalities</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Burkholder%E2%80%93Davis%E2%80%93Gundy_inequalities" class="mw-redirect" title="Burkholder–Davis–Gundy inequalities">Burkholder–Davis–Gundy</a></li> <li><a href="/wiki/Doob%27s_martingale_inequality" title="Doob's martingale inequality">Doob's martingale</a></li> <li><a href="/wiki/Doob%27s_upcrossing_inequality" class="mw-redirect" title="Doob's upcrossing inequality">Doob's upcrossing</a></li> <li><a href="/wiki/Kunita%E2%80%93Watanabe_inequality" title="Kunita–Watanabe inequality">Kunita–Watanabe</a></li> <li><a href="/wiki/Marcinkiewicz%E2%80%93Zygmund_inequality" title="Marcinkiewicz–Zygmund inequality">Marcinkiewicz–Zygmund</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tools</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cameron%E2%80%93Martin_formula" class="mw-redirect" title="Cameron–Martin formula">Cameron–Martin formula</a></li> <li><a href="/wiki/Convergence_of_random_variables" title="Convergence of random variables">Convergence of random variables</a></li> <li><a href="/wiki/Dol%C3%A9ans-Dade_exponential" title="Doléans-Dade exponential">Doléans-Dade exponential</a></li> <li><a href="/wiki/Doob_decomposition_theorem" title="Doob decomposition theorem">Doob decomposition theorem</a></li> <li><a href="/wiki/Doob%E2%80%93Meyer_decomposition_theorem" title="Doob–Meyer decomposition theorem">Doob–Meyer decomposition theorem</a></li> <li><a href="/wiki/Doob%27s_optional_stopping_theorem" class="mw-redirect" title="Doob's optional stopping theorem">Doob's optional stopping theorem</a></li> <li><a href="/wiki/Dynkin%27s_formula" title="Dynkin's formula">Dynkin's formula</a></li> <li><a href="/wiki/Feynman%E2%80%93Kac_formula" title="Feynman–Kac formula">Feynman–Kac formula</a></li> <li><a href="/wiki/Filtration_(probability_theory)" title="Filtration (probability theory)">Filtration</a></li> <li><a href="/wiki/Girsanov_theorem" title="Girsanov theorem">Girsanov theorem</a></li> <li><a href="/wiki/Infinitesimal_generator_(stochastic_processes)" title="Infinitesimal generator (stochastic processes)">Infinitesimal generator</a></li> <li><a href="/wiki/It%C3%B4_integral" class="mw-redirect" title="Itô integral">Itô integral</a></li> <li><a href="/wiki/It%C3%B4%27s_lemma" title="Itô's lemma">Itô's lemma</a></li> <li><a href="/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">Karhunen–Loève theorem</a></li> <li><a href="/wiki/Kolmogorov_continuity_theorem" title="Kolmogorov continuity theorem">Kolmogorov continuity theorem</a></li> <li><a href="/wiki/Kolmogorov_extension_theorem" title="Kolmogorov extension theorem">Kolmogorov extension theorem</a></li> <li><a href="/wiki/L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a></li> <li><a href="/wiki/Malliavin_calculus" title="Malliavin calculus">Malliavin calculus</a></li> <li><a href="/wiki/Martingale_representation_theorem" title="Martingale representation theorem">Martingale representation theorem</a></li> <li><a href="/wiki/Optional_stopping_theorem" title="Optional stopping theorem">Optional stopping theorem</a></li> <li><a href="/wiki/Prokhorov%27s_theorem" title="Prokhorov's theorem">Prokhorov's theorem</a></li> <li><a href="/wiki/Quadratic_variation" title="Quadratic variation">Quadratic variation</a></li> <li><a href="/wiki/Reflection_principle_(Wiener_process)" title="Reflection principle (Wiener process)">Reflection principle</a></li> <li><a href="/wiki/Skorokhod_integral" title="Skorokhod integral">Skorokhod integral</a></li> <li><a href="/wiki/Skorokhod%27s_representation_theorem" title="Skorokhod's representation theorem">Skorokhod's representation theorem</a></li> <li><a href="/wiki/Skorokhod_space" class="mw-redirect" title="Skorokhod space">Skorokhod space</a></li> <li><a href="/wiki/Snell_envelope" title="Snell envelope">Snell envelope</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a> <ul><li><a href="/wiki/Tanaka_equation" title="Tanaka equation">Tanaka</a></li></ul></li> <li><a href="/wiki/Stopping_time" title="Stopping time">Stopping time</a></li> <li><a href="/wiki/Stratonovich_integral" title="Stratonovich integral">Stratonovich integral</a></li> <li><a href="/wiki/Uniform_integrability" title="Uniform integrability">Uniform integrability</a></li> <li><a href="/wiki/Usual_hypotheses" class="mw-redirect" title="Usual hypotheses">Usual hypotheses</a></li> <li><a href="/wiki/Wiener_space" class="mw-redirect" title="Wiener space">Wiener space</a> <ul><li><a href="/wiki/Classical_Wiener_space" title="Classical Wiener space">Classical</a></li> <li><a href="/wiki/Abstract_Wiener_space" title="Abstract Wiener space">Abstract</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Disciplines</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial mathematics</a></li> <li><a href="/wiki/Stochastic_control" title="Stochastic control">Control theory</a></li> <li><a href="/wiki/Econometrics" title="Econometrics">Econometrics</a></li> <li><a href="/wiki/Ergodic_theory" title="Ergodic theory">Ergodic theory</a></li> <li><a href="/wiki/Extreme_value_theory" title="Extreme value theory">Extreme value theory (EVT)</a></li> <li><a href="/wiki/Large_deviations_theory" title="Large deviations theory">Large deviations theory</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li> <li><a href="/wiki/Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></li> <li><a href="/wiki/Queueing_theory" title="Queueing theory">Queueing theory</a></li> <li><a href="/wiki/Renewal_theory" title="Renewal theory">Renewal theory</a></li> <li><a href="/wiki/Ruin_theory" title="Ruin theory">Ruin theory</a></li> <li><a href="/wiki/Signal_processing" title="Signal processing">Signal processing</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Stochastic_analysis" class="mw-redirect" title="Stochastic analysis">Stochastic analysis</a></li> <li><a href="/wiki/Time_series_analysis" class="mw-redirect" title="Time series analysis">Time series analysis</a></li> <li><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> 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data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </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"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </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"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </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"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </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"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </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"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </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"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </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"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>17 languages</span> </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"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Add topic</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-5f467697bf-rhxgh","wgBackendResponseTime":125,"wgPageParseReport":{"limitreport":{"cputime":"0.469","walltime":"0.947","ppvisitednodes":{"value":2105,"limit":1000000},"postexpandincludesize":{"value":85927,"limit":2097152},"templateargumentsize":{"value":1698,"limit":2097152},"expansiondepth":{"value":13,"limit":100},"expensivefunctioncount":{"value":3,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":74242,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 446.500 1 -total"," 31.17% 139.190 3 Template:Sidebar"," 27.40% 122.332 1 Template:Reflist"," 25.69% 114.719 1 Template:Network_Science"," 18.34% 81.904 1 Template:Short_description"," 16.31% 72.844 3 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