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{"work":{"id":11267437,"created_at":"2015-03-04T09:09:48.277-08:00","from_world_paper_id":132697400,"updated_at":"2024-11-16T20:24:22.962-08:00","_data":{"grobid_abstract":"We consider a class of random, weighted networks, obtained through a redefinition of patterns in an Hopfield-like model and, by performing percolation processes, we get information about topology and resilience properties of the networks themselves. Given the weighted nature of the graphs, different kinds of bond percolation can be studied: stochastic (deleting links randomly) and deterministic (deleting links based on rank weights), each mimicking a different physical process. The evolution of the network is accordingly different, as evidenced by the behavior of the largest component size and of the distribution of cluster sizes. In particular, we can derive that weak ties are crucial in order to maintain the graph connected and that, when they are the most prone to failure, the giant component typically shrinks without abruptly breaking apart; these results have been recently evidenced in several kinds of social networks.","publication_date":"2011,,","publication_name":"Physical Review E","grobid_abstract_attachment_id":"46781892"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Percolation on correlated random networks","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [27211760]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "full_page_mobile_sutd_modal"; 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":46781892,"attachmentType":"pdf"}"><img alt="First page of “Percolation on correlated random networks”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/46781892/mini_magick20190208-24708-2mdftu.png?1549683733" /><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 on correlated random networks</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="27211760" href="https://uniroma1.academia.edu/ElenaAgliari"><img alt="Profile image of Elena Agliari" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Elena Agliari</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2011, Physical Review E</p><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">11 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 = 11267437; 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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 consider a class of random, weighted networks, obtained through a redefinition of patterns in an Hopfield-like model and, by performing percolation processes, we get information about topology and resilience properties of the networks themselves. Given the weighted nature of the graphs, different kinds of bond percolation can be studied: stochastic (deleting links randomly) and deterministic (deleting links based on rank weights), each mimicking a different physical process. The evolution of the network is accordingly different, as evidenced by the behavior of the largest component size and of the distribution of cluster sizes. In particular, we can derive that weak ties are crucial in order to maintain the graph connected and that, when they are the most prone to failure, the giant component typically shrinks without abruptly breaking apart; these results have been recently evidenced in several kinds of social networks.</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":46781892,"attachmentType":"pdf","workUrl":"https://www.academia.edu/11267437/Percolation_on_correlated_random_networks"}">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":46781892,"attachmentType":"pdf","workUrl":"https://www.academia.edu/11267437/Percolation_on_correlated_random_networks"}"><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="11267437" 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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Their percolation transition is usually of a first order compared to the second order transition found in classical networks. We analytically analyze the effect of different distributions of dependencies links on the robustness of networks. For a random Erdös-Rényi (ER) network with average degree k that is divided into dependency clusters of size s, the fraction of nodes that belong to the giant component, P∞, is given by P∞ = p s-1 [1 -exp (-kpP∞)] s where 1 -p is the initial fraction of removed nodes. Our general result coincides with the known Erdös-Rényi equation for random networks for s = 1 and with the result of Parshani et al (PNAS, in press, 2011) for s = 2. For networks with Poissonian distribution of dependency links we find that P∞ is given by P∞ = f k,p (P∞)e ( s -1)(pf k,p (P∞)-1) where f k,p (P∞) ≡ 1 -exp (-kpP∞) and s is the mean value of the size of dependency clusters. For networks with Gaussian distribution of dependency links we show how the average and width of the distribution affect the robustness of the networks.</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 in networks composed of connectivity and dependency links","attachmentId":85401968,"attachmentType":"pdf","work_url":"https://www.academia.edu/78309384/Percolation_in_networks_composed_of_connectivity_and_dependency_links","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/78309384/Percolation_in_networks_composed_of_connectivity_and_dependency_links"><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="96766284" 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/96766284/Bond_percolation_on_a_class_of_correlated_and_clustered_random_graphs">Bond percolation on a class of correlated and clustered random 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="143525165" href="https://ulaval.academia.edu/LouisJDub%C3%A9">Louis J. Dubé</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Physics A: Mathematical and Theoretical, 2012</p><p class="ds-related-work--abstract ds2-5-body-sm">We introduce a formalism for computing bond percolation properties of a class of correlated and clustered random graphs. This class of graphs is a generalization of the Configuration Model where nodes of different types are connected via different types of hyperedges, edges that can link more than 2 nodes. We argue that the multitype approach coupled with the use of clustered hyperedges can reproduce a wide spectrum of complex patterns, and thus enhances our capability to model real complex networks. As an illustration of this claim, we use our formalism to highlight unusual behaviors of the size and composition of the components (small and giant) in a synthetic, albeit realistic, social network.</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":"Bond percolation on a class of correlated and clustered random graphs","attachmentId":98573650,"attachmentType":"pdf","work_url":"https://www.academia.edu/96766284/Bond_percolation_on_a_class_of_correlated_and_clustered_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/96766284/Bond_percolation_on_a_class_of_correlated_and_clustered_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="2" data-entity-id="78309382" 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/78309382/The_Combined_Effect_of_Connectivity_and_Dependency_Links_on_Percolation_of_Networks">The Combined Effect of Connectivity and Dependency Links on Percolation of 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="222516340" href="https://biu.academia.edu/AmirBashan">Amir Bashan</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Statistical Physics, 2011</p><p class="ds-related-work--abstract ds2-5-body-sm">Percolation theory is extensively studied in statistical physics and mathematics with applications in diverse fields. However, the research is focused on systems with only one type of links, connectivity links. We review a recently developed mathematical framework for analyzing percolation properties of realistic scenarios of networks having links of two types, connectivity and dependency links. This formalism was applied to study Erdös-Rényi (ER) networks that include also dependency links. For an ER network with average degree k that is composed of dependency clusters of size s, the fraction of nodes that belong to the giant component, P ∞ , is given by s where 1 -p is the initial fraction of randomly removed nodes. Here, we apply the formalism to the study of randomregular (RR) networks and find a formula for the size of the giant component in the percolation process: P ∞ = p s-1 (1 -r k ) s where r is the solution of r = p s (r k-1 -1)(1 -r k ) + 1. These general results coincide, for s = 1, with the known equations for percolation in ER and RR networks respectively without dependency links. In contrast to s = 1, where the percolation transition is second order, for s > 1 it is of first order. Comparing the percolation behavior of ER and RR networks we find a remarkable difference regarding their resilience. We show, analytically and numerically, that in ER networks with low connectivity degree or large dependency clusters, removal of even a finite number (zero fraction) of the network nodes will trigger a cascade of failures that fragments the whole network. Specifically, for any given s there exists a critical degree value, k min , such that an ER network with k ≤ k min is unstable and collapse when removing even a single node. This result is in</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 Combined Effect of Connectivity and Dependency Links on Percolation of Networks","attachmentId":85401964,"attachmentType":"pdf","work_url":"https://www.academia.edu/78309382/The_Combined_Effect_of_Connectivity_and_Dependency_Links_on_Percolation_of_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/78309382/The_Combined_Effect_of_Connectivity_and_Dependency_Links_on_Percolation_of_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="3" data-entity-id="96766298" 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/96766298/General_and_exact_approach_to_percolation_on_random_graphs">General and exact approach to percolation on random 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="143525165" href="https://ulaval.academia.edu/LouisJDub%C3%A9">Louis J. 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="4" data-entity-id="2881960" 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/2881960/The_Unreasonable_Effectiveness_of_Tree_Based_Theory_for_Bond_Percolation_on_Networks_with_Clustering">The Unreasonable Effectiveness of Tree-Based Theory for Bond Percolation on Networks with Clustering</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="2700276" href="https://oxford.academia.edu/MasonPorter">Mason Porter</a></div><p class="ds-related-work--abstract ds2-5-body-sm">Abstract. We demonstrate that a tree-based theory for bond percolation yields extremely accurate results for several networks with high levels of clustering. We find that such a theory works well as long as the mean intervertex distance ℓ is sufficiently small—ie, as long as it is close to the value of ℓ in a random network with negligible clustering and the same degree-degree correlations. We confirm this hypothesis numerically using real-world networks from various domains and on several classes of synthetic clustered networks.</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 Unreasonable Effectiveness of Tree-Based Theory for Bond Percolation on Networks with Clustering","attachmentId":30816616,"attachmentType":"pdf","work_url":"https://www.academia.edu/2881960/The_Unreasonable_Effectiveness_of_Tree_Based_Theory_for_Bond_Percolation_on_Networks_with_Clustering","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/2881960/The_Unreasonable_Effectiveness_of_Tree_Based_Theory_for_Bond_Percolation_on_Networks_with_Clustering"><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="102130287" 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/102130287/The_dynamic_nature_of_percolation_on_networks_with_triadic_interactions">The dynamic nature of percolation on networks with triadic interactions</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="35256933" href="https://qmul.academia.edu/GinestraBianconi">Ginestra Bianconi</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Nature Communications</p><p class="ds-related-work--abstract ds2-5-body-sm">Percolation establishes the connectivity of complex networks and is one of the most fundamental critical phenomena for the study of complex systems. On simple networks, percolation displays a second-order phase transition; on multiplex networks, the percolation transition can become discontinuous. However, little is known about percolation in networks with higher-order interactions. Here, we show that percolation can be turned into a fully fledged dynamical process when higher-order interactions are taken into account. By introducing signed triadic interactions, in which a node can regulate the interactions between two other nodes, we define triadic percolation. We uncover that in this paradigmatic model the connectivity of the network changes in time and that the order parameter undergoes a period doubling and a route to chaos. We provide a general theory for triadic percolation which accurately predicts the full phase diagram on random graphs as confirmed by extensive numerical si...</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 dynamic nature of percolation on networks with triadic interactions","attachmentId":102475567,"attachmentType":"pdf","work_url":"https://www.academia.edu/102130287/The_dynamic_nature_of_percolation_on_networks_with_triadic_interactions","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/102130287/The_dynamic_nature_of_percolation_on_networks_with_triadic_interactions"><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="74780890" 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/74780890/Percolation_of_interdependent_network_of_networks">Percolation of interdependent network of 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="36769513" href="https://independent.academia.edu/JianxiGao">Jianxi Gao</a></div><p class="ds-related-work--abstract ds2-5-body-sm">Complex networks appear in almost every aspect of science and technology. Previous work in network theory has focused primarily on analyzing single networks that do not interact with other networks, despite the fact that many real-world networks interact with and depend on each other. Very recently an analytical framework for studying the percolation properties of interacting networks has been introduced. Here we review the analytical framework and the results for percolation laws for a Network Of Networks (NONs) formed by n interdependent random networks. The percolation properties of a network of networks differ greatly from those of single isolated networks. In particular, because the constituent networks of a NON are connected by node dependencies, a NON is subject to cascading failure. When there is strong interdependent coupling between networks, the percolation transition is discontinuous (first-order) phase transition, unlike the wellknown continuous second-order transition in single isolated networks. Moreover, although networks with broader degree distributions, e.g., scale-free networks, are more robust when analyzed as single networks, they become more vulnerable in a NON. We also review the effect of space embedding on network vulnerability. It is shown that for spatially embedded networks any finite fraction of dependency nodes will lead to abrupt 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":"Percolation of interdependent network of networks","attachmentId":82806881,"attachmentType":"pdf","work_url":"https://www.academia.edu/74780890/Percolation_of_interdependent_network_of_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/74780890/Percolation_of_interdependent_network_of_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="5447164" 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/5447164/Dynamic_Networks_and_Directed_Percolation">Dynamic Networks and Directed Percolation</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="7630487" href="https://bu.academia.edu/EugeneStanley">Eugene Stanley</a></div><p class="ds-related-work--abstract ds2-5-body-sm">We introduce a model for dynamic networks, where the links or the strengths of the links change over time. We solve the model by mapping dynamic networks to the problem of directed percolation, where the direction corresponds to the time evolution of the network. We show that the dynamic network undergoes a percolation phase transition at a critical concentration pc, that decreases with the rate r at which the network links are changed. The behavior near criticality is universal and independent of r. We find that for dynamic random networks fundamental laws are changed. i) The size of the giant component at criticality scales with the network size N for all values of r, rather than as N 2/3 in static networks. ii) In the presence of a broad distribution of disorder, the optimal path length between two nodes in a dynamic network scales as N 1/2 , compared to N 1/3 in a static network.</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":"Dynamic Networks and Directed Percolation","attachmentId":32570429,"attachmentType":"pdf","work_url":"https://www.academia.edu/5447164/Dynamic_Networks_and_Directed_Percolation","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/5447164/Dynamic_Networks_and_Directed_Percolation"><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="16145828" 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/16145828/Percolation_in_networks_of_networks_with_random_matching_of_nodes_in_different_layers">Percolation in networks of networks with random matching of nodes in different layers</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="35256933" href="https://qmul.academia.edu/GinestraBianconi">Ginestra Bianconi</a></div><p class="ds-related-work--abstract ds2-5-body-sm">We consider robustness and percolation properties of the networks of networks, in which random nodes in different individual networks (layers) can be interdependent. We explore the emergence of the giant mutually connected component, generalizing the percolation cluster in a single network to interdependent networks, and observe the strong effect of loops of interdependencies. In particular, we find that the giant mutual component does not emerge in a loop formed by any number of layers. In contrast, we observe multiple hybrid transitions in networks of networks formed by infinite number of randomly connected layers, corresponding to the percolation of layers with different number of interdependencies. In particular we find that layers with many interdependencies are more fragile than layers with less interdependencies. These hybrid transitions, combining a discontinuity and a singularity, are responsible for joining a finite fraction of nodes in different layers to the giant mutually connected component. In the case of partial interdependence, when only a fraction of interlinks between layers provide interdependence, some of these transitions can become 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":"Percolation in networks of networks with random matching of nodes in different layers","attachmentId":42669742,"attachmentType":"pdf","work_url":"https://www.academia.edu/16145828/Percolation_in_networks_of_networks_with_random_matching_of_nodes_in_different_layers","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/16145828/Percolation_in_networks_of_networks_with_random_matching_of_nodes_in_different_layers"><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":46781892,"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":46781892,"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_46781892" 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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class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2013</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 of interdependent networks with intersimilarity","attachmentId":66993288,"attachmentType":"pdf","work_url":"https://www.academia.edu/48316126/Percolation_of_interdependent_networks_with_intersimilarity","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" href="https://www.academia.edu/48316126/Percolation_of_interdependent_networks_with_intersimilarity"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" 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data-author-id="132586" href="https://unizar.academia.edu/yamirmoreno">yamir moreno</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2003</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":"Resilience to damage of graphs with degree correlations","attachmentId":4860094,"attachmentType":"pdf","work_url":"https://www.academia.edu/811003/Resilience_to_damage_of_graphs_with_degree_correlations","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" href="https://www.academia.edu/811003/Resilience_to_damage_of_graphs_with_degree_correlations"><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-related-work-sidebar-card" data-collection-position="7" data-entity-id="18707846" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/18707846/Clique_Percolation_in_Random_Networks">Clique Percolation in Random Networks</a><div class="ds-related-work--metadata"><a class="js-related-work-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">Physical Review Letters, 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":"Clique 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