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(PDF) Percolation in Random Graphs: A Finite Approach
<!DOCTYPE html> <html > <head> <meta charset="utf-8"> <meta rel="search" type="application/opensearchdescription+xml" href="/open_search.xml" title="Academia.edu"> <meta content="width=device-width, initial-scale=1" name="viewport"> <meta name="google-site-verification" content="bKJMBZA7E43xhDOopFZkssMMkBRjvYERV-NaN4R6mrs"> <meta name="csrf-param" content="authenticity_token" /> <meta name="csrf-token" content="m62JtpL-iGbiflA5J6e6968akrhTlfIOriDtpEv8gbP_vGZcxbdQy1JoMyErTAdNUxSrGudkbwONVERwZkyeEw" /> <meta name="citation_title" content="Percolation in Random Graphs: A Finite Approach" /> <meta name="citation_author" content="Lyle Muller" /> <meta name="twitter:card" content="summary" /> <meta name="twitter:url" content="https://www.academia.edu/21181248/Percolation_in_Random_Graphs_A_Finite_Approach" /> <meta name="twitter:title" content="Percolation in Random Graphs: A Finite Approach" /> <meta name="twitter:description" content="We propose an approach to calculate the critical percolation threshold for finite-sized Erdos-Renyi digraphs using minimal Hamiltonian cycles. We obtain an analytically exact result, valid non-asymptotically for all graph sizes, which scales in" /> <meta name="twitter:image" content="https://gravatar.com/avatar/e17a752ffd1ecbeb9658caf3992db8d3?s=200" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/21181248/Percolation_in_Random_Graphs_A_Finite_Approach" /> <meta property="og:title" content="Percolation in Random Graphs: A Finite Approach" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="We propose an approach to calculate the critical percolation threshold for finite-sized Erdos-Renyi digraphs using minimal Hamiltonian cycles. We obtain an analytically exact result, valid non-asymptotically for all graph sizes, which scales in" /> <meta property="article:author" content="https://salk.academia.edu/LyleMuller" /> <meta name="description" content="We propose an approach to calculate the critical percolation threshold for finite-sized Erdos-Renyi digraphs using minimal Hamiltonian cycles. We obtain an analytically exact result, valid non-asymptotically for all graph sizes, which scales in" /> <title>(PDF) Percolation in Random Graphs: A Finite Approach</title> <link rel="canonical" href="https://www.academia.edu/21181248/Percolation_in_Random_Graphs_A_Finite_Approach" /> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "single_work", 'action': "show", 'controller_action': 'single_work#show', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two cases. // ab_test_bucket should be of the form <ab_test_name>:<bucket> 'ab_test_bucket': null, }) </script> <script> var $controller_name = 'single_work'; var $action_name = "show"; var $rails_env = 'production'; var $app_rev = '1e60a92a442ff83025cbe4f252857ee7c49c0bbe'; var $domain = 'academia.edu'; var $app_host = "academia.edu"; var $asset_host = "academia-assets.com"; var $start_time = new Date().getTime(); var $recaptcha_key = "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB"; var $recaptcha_invisible_key = "6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj"; var $disableClientRecordHit = false; </script> <script> window.require = { config: function() { return function() {} } } </script> <script> window.Aedu = window.Aedu || {}; window.Aedu.hit_data = null; window.Aedu.serverRenderTime = new Date(1740564596000); window.Aedu.timeDifference = new Date().getTime() - 1740564596000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"We propose an approach to calculate the critical percolation threshold for finite-sized Erdos-Renyi digraphs using minimal Hamiltonian cycles. We obtain an analytically exact result, valid non-asymptotically for all graph sizes, which scales in accordance with results obtained for infinite random graphs using the emergence of a giant connected component as marking the percolation transition. 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{"work":{"id":21181248,"created_at":"2016-01-29T13:45:04.779-08:00","from_world_paper_id":147517914,"updated_at":"2021-03-23T00:24:37.426-07:00","_data":{"abstract":"We propose an approach to calculate the critical percolation threshold for finite-sized Erdos-Renyi digraphs using minimal Hamiltonian cycles. We obtain an analytically exact result, valid non-asymptotically for all graph sizes, which scales in accordance with results obtained for infinite random graphs using the emergence of a giant connected component as marking the percolation transition. Our approach is general and can be applied to all graph models for which an algebraic formulation of the adjacency matrix is available."},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Percolation in Random Graphs: A Finite Approach","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [34187500]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "control"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":41750968,"attachmentType":"pdf"}"><img alt="First page of “Percolation in Random Graphs: A Finite Approach”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/41750968/mini_magick20190218-6983-fvc66l.png?1550539689" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">Percolation in Random Graphs: A Finite Approach</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="34187500" href="https://salk.academia.edu/LyleMuller"><img alt="Profile image of Lyle Muller" class="ds-work-card--author-avatar" src="https://gravatar.com/avatar/e17a752ffd1ecbeb9658caf3992db8d3?s=65" />Lyle Muller</a></div><div class="ds-work-card--detail"><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">4 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 21181248; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=21181248"; const getWorkViews = async (workId) => { const response = await fetch(worksViewsPath); if (!response.ok) { throw new Error('Failed to load work views'); } const data = await response.json(); return data.views[workId]; }; // Get the view count for the work - we send this immediately rather than waiting for // the DOM to load, so it can be available as soon as possible (but without holding up // the backend or other resource requests, because it's a bit expensive and not critical). const viewCount = await getWorkViews(workId); const updateViewCount = (viewCount) => { try { const viewCountNumber = parseInt(viewCount, 10); if (viewCountNumber === 0) { // Remove the whole views element if there are zero views. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); return; } const commaizedViewCount = viewCountNumber.toLocaleString(); const viewCountBody = document.getElementById('work-metadata-view-count'); if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">We propose an approach to calculate the critical percolation threshold for finite-sized Erdos-Renyi digraphs using minimal Hamiltonian cycles. We obtain an analytically exact result, valid non-asymptotically for all graph sizes, which scales in accordance with results obtained for infinite random graphs using the emergence of a giant connected component as marking the percolation transition. Our approach is general and can be applied to all graph models for which an algebraic formulation of the adjacency matrix is available.</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":41750968,"attachmentType":"pdf","workUrl":"https://www.academia.edu/21181248/Percolation_in_Random_Graphs_A_Finite_Approach"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":41750968,"attachmentType":"pdf","workUrl":"https://www.academia.edu/21181248/Percolation_in_Random_Graphs_A_Finite_Approach"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div><div class="ds-signup-banner-trigger-container"><div class="ds-signup-banner-trigger ds-signup-banner-trigger-control"></div></div><div class="ds-signup-banner ds-signup-banner-control"><div id="ds-signup-banner-close-button"><button class="ds2-5-button ds2-5-button--secondary ds2-5-button--inverse"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">close</span></button></div><div class="ds-signup-banner-ctas" data-impression-entity-id="21181248" data-impression-entity-type="2" data-impression-source="signup-banner"><img src="//a.academia-assets.com/images/academia-logo-capital-white.svg" /><h4 class="ds2-5-heading-serif-sm">Sign up for access to the world's latest research</h4><button class="ds2-5-button ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{"location":"signup-banner"}">Sign up for free<span class="material-symbols-outlined" style="font-size: 20px" translate="no">arrow_forward</span></button></div><div class="ds-signup-banner-divider"></div><div class="ds-signup-banner-reasons"><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Get notified about relevant papers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Save papers to use in your research</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Join the discussion with peers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Track your impact</span></div></div></div><script>(() => { // Set up signup banner show/hide behavior: // 1. 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Dubé</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2015</p><p class="ds-related-work--abstract ds2-5-body-sm">We present a comprehensive and versatile theoretical framework to study site and bond percolation on clustered and correlated random graphs. Our contribution can be summarized in three main points. (i) We introduce a set of iterative equations that solve the exact distribution of the size and composition of components in finite size quenched or random multitype graphs. (ii) We define a very general random graph ensemble that encompasses most of the models published to this day, and also that permits to model structural properties not yet included in a theoretical framework. Site and bond percolation on this ensemble is solved exactly in the infinite size limit using probability generating functions [i.e., the percolation threshold, the size and the composition of the giant (extensive) and small components]. Several examples and applications are also provided. (iii) Our approach can be adapted to model interdependent graphs-whose most striking feature is the emergence of an extensive component via a discontinuous phase transition-in an equally general fashion. We show how a graph can successively undergo a continuous then a discontinuous phase transition, and preliminary results suggest that clustering increases the amplitude of the discontinuity at the transition.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"General and exact approach to percolation on random graphs","attachmentId":98573654,"attachmentType":"pdf","work_url":"https://www.academia.edu/96766298/General_and_exact_approach_to_percolation_on_random_graphs","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/96766298/General_and_exact_approach_to_percolation_on_random_graphs"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="1" data-entity-id="51237760" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/51237760/Percolation_on_Sparse_Random_Graphs_with_Given_Degree_Sequence">Percolation on Sparse Random Graphs with Given Degree Sequence</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32988049" href="https://bham.academia.edu/NikolaosFountoulakis">Nikolaos Fountoulakis</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Internet Mathematics, 2007</p><p class="ds-related-work--abstract ds2-5-body-sm">We study the two most common types of percolation process on a sparse random graph with a given degree sequence. Namely, we examine first a bond percolation process where the edges of the graph are retained with probability p and afterwards we focus on site percolation where the vertices are retained with probability p. We establish critical values for p above which a giant component emerges in both cases. Moreover, we show that in fact these coincide. As a special case, our results apply to power law random graphs. We obtain rigorous proofs for formulas derived by several physicists for such graphs.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Percolation on Sparse Random Graphs with Given Degree Sequence","attachmentId":69044685,"attachmentType":"pdf","work_url":"https://www.academia.edu/51237760/Percolation_on_Sparse_Random_Graphs_with_Given_Degree_Sequence","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/51237760/Percolation_on_Sparse_Random_Graphs_with_Given_Degree_Sequence"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="2" data-entity-id="16867090" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/16867090/Random_sequential_renormalization_and_agglomerative_percolation_in_networks_Application_to_Erd%C3%B6s_R%C3%A9nyi_and_scale_free_graphs">Random sequential renormalization and agglomerative percolation in networks: Application to Erdös-Rényi and scale-free graphs</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="26329343" href="https://independent.academia.edu/PeterGrassberger">Peter Grassberger</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2011</p><p class="ds-related-work--abstract ds2-5-body-sm">We study the statistical behavior under random sequential renormalization (RSR) of several network models including Erdös-Rényi (ER) graphs, scale-free networks, and an annealed model related to ER graphs. In RSR the network is locally coarse grained by choosing at each renormalization step a node at random and joining it to all its neighbors. Compared to previous (quasi-)parallel renormalization methods [Song et al., Nature (London) 433, 392 (2005)], RSR allows a more fine-grained analysis of the renormalization group (RG) flow and unravels new features that were not discussed in the previous analyses. In particular, we find that all networks exhibit a second-order transition in their RG flow. This phase transition is associated with the emergence of a giant hub and can be viewed as a new variant of percolation, called agglomerative percolation. We claim that this transition exists also in previous graph renormalization schemes and explains some of the scaling behavior seen there. For critical trees it happens as N/N(0) → 0 in the limit of large systems (where N(0) is the initial size of the graph and N its size at a given RSR step). In contrast, it happens at finite N/N(0) in sparse ER graphs and in the annealed model, while it happens for N/N(0) → 1 on scale-free networks. Critical exponents seem to depend on the type of the graph but not on the average degree and obey usual scaling relations for percolation phenomena. For the annealed model they agree with the exponents obtained from a mean-field theory. At late times, the networks exhibit a starlike structure in agreement with the results of Radicchi et al. [Phys. Rev. Lett. 101, 148701 (2008)]. While degree distributions are of main interest when regarding the scheme as network renormalization, mass distributions (which are more relevant when considering &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;quot;supernodes&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;quot; as clusters) are much easier to study using the fast Newman-Ziff algorithm for percolation, allowing us to obtain very high statistics.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Random sequential renormalization and agglomerative percolation in networks: Application to Erdös-Rényi and scale-free graphs","attachmentId":42381478,"attachmentType":"pdf","work_url":"https://www.academia.edu/16867090/Random_sequential_renormalization_and_agglomerative_percolation_in_networks_Application_to_Erd%C3%B6s_R%C3%A9nyi_and_scale_free_graphs","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/16867090/Random_sequential_renormalization_and_agglomerative_percolation_in_networks_Application_to_Erd%C3%B6s_R%C3%A9nyi_and_scale_free_graphs"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="3" data-entity-id="55758945" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/55758945/Explosive_percolation_in_graphs">Explosive percolation in graphs</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="38312703" href="https://independent.academia.edu/FortunatoSanto">Santo Fortunato</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Physics: Conference Series, 2011</p><p class="ds-related-work--abstract ds2-5-body-sm">Percolation is perhaps the simplest example of a process exhibiting a phase transition and one of the most studied phenomena in statistical physics. The percolation transition is continuous if sites/bonds are occupied independently with the same probability. However, alternative rules for the occupation of sites/bonds might affect the order of the transition. A recent set of rules proposed by Achlioptas et al. [Science 323, 1453 (2009)], characterized by competitive link addition, was claimed to lead to a discontinuous connectedness transition, named "explosive percolation". In this work we survey a numerical study of the explosive percolation transition on various types of graphs, from lattices to scale-free networks, and show the consistency of these results with recent analytical work showing that the transition is actually continuous.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Explosive percolation in graphs","attachmentId":71477861,"attachmentType":"pdf","work_url":"https://www.academia.edu/55758945/Explosive_percolation_in_graphs","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/55758945/Explosive_percolation_in_graphs"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="64235495" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/64235495/Percolation_on_random_graphs_with_a_fixed_degree_sequence">Percolation on random graphs with a fixed degree sequence</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32988049" href="https://bham.academia.edu/NikolaosFountoulakis">Nikolaos Fountoulakis</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2016</p><p class="ds-related-work--abstract ds2-5-body-sm">We consider bond percolation on random graphs with given degrees and bounded average degree. In particular, we consider the order of the largest component after the random deletion of the edges of such a random graph. We give a rough characterisation of those degree distributions for which bond percolation with high probability leaves a component of linear order, known usually as a giant component. We show that essentially the critical condition has to do with the tail of the degree distribution. Our proof makes use of recent technique introduced by Joos et al. [FOCS 2016, pp. 695--703], which is based on the switching method and avoids the use of the classic configuration model as well as the hypothesis of having a limiting object. Thus our results hold for sparse degree sequences without the usual restrictions that accompany the configuration model.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Percolation on random graphs with a fixed degree sequence","attachmentId":76362996,"attachmentType":"pdf","work_url":"https://www.academia.edu/64235495/Percolation_on_random_graphs_with_a_fixed_degree_sequence","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/64235495/Percolation_on_random_graphs_with_a_fixed_degree_sequence"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="22808355" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/22808355/First_passage_percolation_on_random_graphs_with_finite_mean_degrees">First passage percolation on random graphs with finite mean degrees</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="44457585" href="https://independent.academia.edu/GerardHooghiemstra">Gerard Hooghiemstra</a></div><p class="ds-related-work--metadata ds2-5-body-xs">The Annals of Applied Probability, 2010</p><p class="ds-related-work--abstract ds2-5-body-sm">We study first passage percolation on the configuration model. Assuming that each edge has an independent exponentially distributed edge weight, we derive explicit distributional asymptotics for the minimum weight between two randomly chosen connected vertices in the network, as well as for the number of edges on the least weight path, the so-called hopcount.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"First passage percolation on random graphs with finite mean degrees","attachmentId":43357734,"attachmentType":"pdf","work_url":"https://www.academia.edu/22808355/First_passage_percolation_on_random_graphs_with_finite_mean_degrees","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/22808355/First_passage_percolation_on_random_graphs_with_finite_mean_degrees"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="51779498" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/51779498/Inducing_effect_on_the_percolation_transition_in_complex_networks">Inducing effect on the percolation transition in complex networks</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="11694809" href="https://itpcas.academia.edu/HaiJunZhou">Hai-Jun Zhou</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Nature Communications, 2013</p><p class="ds-related-work--abstract ds2-5-body-sm">Percolation theory concerns the emergence of connected clusters that percolate through a networked system. Previous studies ignored the effect that a node outside the percolating cluster may actively induce its inside neighbours to exit the percolating cluster. Here we study this inducing effect on the classical site percolation and K-core percolation, showing that the inducing effect always causes a discontinuous percolation transition. We precisely predict the percolation threshold and core size for uncorrelated random networks with arbitrary degree distributions. For low-dimensional lattices the percolation threshold fluctuates considerably over realizations, yet we can still predict the core size once the percolation occurs. The core sizes of real-world networks can also be well predicted using degree distribution as the only input. Our work therefore provides a theoretical framework for quantitatively understanding discontinuous breakdown phenomena in various complex systems.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Inducing effect on the percolation transition in complex networks","attachmentId":69351199,"attachmentType":"pdf","work_url":"https://www.academia.edu/51779498/Inducing_effect_on_the_percolation_transition_in_complex_networks","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/51779498/Inducing_effect_on_the_percolation_transition_in_complex_networks"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="18707778" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/18707778/The_Critical_Point_of_k_Clique_Percolation_in_the_Erd%C5%91s_R%C3%A9nyi_Graph">The Critical Point of k-Clique Percolation in the Erdős–Rényi Graph</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="38775154" href="https://elte-hu.academia.edu/Tam%C3%A1sVicsek">Tamás Vicsek</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Statistical Physics, 2007</p><p class="ds-related-work--abstract ds2-5-body-sm">Motivated by the success of a k-clique percolation method for the identification of overlapping communities in large real networks, here we study the k-clique percolation problem in the Erdős–Rnyi graph. When the probability p of two nodes being connected is above a certain threshold p c (k), the complete subgraphs of size k (the k-cliques) are organized into a giant</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The Critical Point of k-Clique Percolation in the Erdős–Rényi Graph","attachmentId":40941979,"attachmentType":"pdf","work_url":"https://www.academia.edu/18707778/The_Critical_Point_of_k_Clique_Percolation_in_the_Erd%C5%91s_R%C3%A9nyi_Graph","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/18707778/The_Critical_Point_of_k_Clique_Percolation_in_the_Erd%C5%91s_R%C3%A9nyi_Graph"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="112360847" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/112360847/Effect_of_volume_growth_on_the_percolation_threshold_in_random_directed_acyclic_graphs_with_a_given_degree_distribution">Effect of volume growth on the percolation threshold in random directed acyclic graphs with a given degree distribution</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="72172644" href="https://independent.academia.edu/PietIedema">Piet Iedema</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical review, 2020</p><p class="ds-related-work--abstract ds2-5-body-sm">In every network, a distance between a pair of nodes can be defined as the length of the shortest path connecting these nodes, and therefore one may speak of a ball, its volume, and how it grows as a function of the radius. Spatial networks tend to feature peculiar volume scaling functions, as well as other topological features, including clustering, degree-degree correlation, clique complexes, and heterogeneity. Here we investigate a nongeometric random graph with a given degree distribution and an additional constraint on the volume scaling function. We show that such structures fall into the category of m-colored random graphs and study the percolation transition by using this theory. We prove that for a given degree distribution the percolation threshold for weakly connected components is not affected by the volume growth function. Additionally, we show that the size of the giant component and the cyclomatic number are not affected by volume scaling. These findings may explain the surprisingly good performance of network models that neglect volume scaling. Even though this paper focuses on the implications of the volume growth, the model is generic and might lead to insights in the field of random directed acyclic graphs and their applications.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Effect of volume growth on the percolation threshold in random directed acyclic graphs with a given degree distribution","attachmentId":109615706,"attachmentType":"pdf","work_url":"https://www.academia.edu/112360847/Effect_of_volume_growth_on_the_percolation_threshold_in_random_directed_acyclic_graphs_with_a_given_degree_distribution","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/112360847/Effect_of_volume_growth_on_the_percolation_threshold_in_random_directed_acyclic_graphs_with_a_given_degree_distribution"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="96774882" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/96774882/Heterogeneous_micro_structure_of_percolation_in_sparse_networks">Heterogeneous micro-structure of percolation in sparse networks</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="53713911" href="https://independent.academia.edu/ReimerKuehn">Reimer Kuehn</a></div><p class="ds-related-work--metadata ds2-5-body-xs">EPL (Europhysics Letters), 2017</p><p class="ds-related-work--abstract ds2-5-body-sm">We examine the heterogeneous responses of individual nodes in sparse networks to the random removal of a fraction of edges. Using the message-passing formulation of percolation, we discover considerable variation across the network in the probability of a particular node to remain part of the giant component, and in the expected size of small clusters containing that node. In the vicinity of the percolation threshold, weakly non-linear analysis reveals that node-to-node heterogeneity is captured by the recently introduced notion of non-backtracking centrality. We supplement these results for fixed finite networks by a population dynamics approach to analyse random graph models in the infinite system size limit, also providing closed-form approximations for the large mean degree limit of Erdős-Rényi random graphs. Interpreted in terms of the application of percolation to realworld processes, our results shed light on the heterogeneous exposure of different nodes to cascading failures, epidemic spread, and information flow.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Heterogeneous micro-structure of percolation in sparse networks","attachmentId":98579825,"attachmentType":"pdf","work_url":"https://www.academia.edu/96774882/Heterogeneous_micro_structure_of_percolation_in_sparse_networks","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/96774882/Heterogeneous_micro_structure_of_percolation_in_sparse_networks"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":41750968,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":41750968,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_41750968" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. 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href="https://independent.academia.edu/PeterGrassberger">Peter Grassberger</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review Letters, 2011</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Percolation Transitions Are Not Always Sharpened by Making Networks Interdependent","attachmentId":47091326,"attachmentType":"pdf","work_url":"https://www.academia.edu/10831942/Percolation_Transitions_Are_Not_Always_Sharpened_by_Making_Networks_Interdependent","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" 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href="https://independent.academia.edu/SRedner">S. Redner</a><span>, </span><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="26206077" href="https://independent.academia.edu/VishalSood2">Vishal Sood</a><span>, </span><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="36832619" href="https://independent.academia.edu/DBenavraham">Daniel ben-Avraham</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Physics A: Mathematical and General, 2005</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"First-passage properties of the Erdos–Renyi random graph","attachmentId":47456145,"attachmentType":"pdf","work_url":"https://www.academia.edu/27202684/First_passage_properties_of_the_Erdos_Renyi_random_graph","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" 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