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{"work":{"id":74007336,"created_at":"2022-03-18T05:18:39.731-07:00","from_world_paper_id":198832962,"updated_at":"2024-11-24T11:17:46.780-08:00","_data":{"grobid_abstract":"We study the stability of network communication after removal of q = 1 − p links under the assumption that communication is effective only if the shortest path between nodes i and j after removal is shorter than aℓij(a ≥ 1) where ℓij is the shortest path before removal. For a large class of networks, we find a new percolation transition atpc = (κo − 1) (1−a)/a , where κo ≡ k 2 / k and k is the node degree. Abovepc, order N nodes can communicate within the limited path length aℓij, while belowpc, N δ (δ \u003c 1) nodes can communicate. Our analytical results are supported by simulations. We expect our results to influence network design, routing algorithms, and immunization strategies, where short paths are most relevant.","publication_date":"2007,,","grobid_abstract_attachment_id":"83817562"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Limited Percolation on Complex Networks","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [36407694]; 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":83817562,"attachmentType":"pdf"}"><img alt="First page of “Limited Percolation on Complex Networks”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/83817562/mini_magick20220412-23672-h9x6ak.png?1649753469" /><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">Limited Percolation on Complex 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="36407694" href="https://biu.academia.edu/SHavlin"><img alt="Profile image of S. Havlin" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />S. Havlin</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2007</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">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 = 74007336; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=74007336"; 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 study the stability of network communication after removal of q = 1 − p links under the assumption that communication is effective only if the shortest path between nodes i and j after removal is shorter than aℓij(a ≥ 1) where ℓij is the shortest path before removal. For a large class of networks, we find a new percolation transition atpc = (κo − 1) (1−a)/a , where κo ≡ k 2 / k and k is the node degree. Abovepc, order N nodes can communicate within the limited path length aℓij, while belowpc, N δ (δ < 1) nodes can communicate. Our analytical results are supported by simulations. We expect our results to influence network design, routing algorithms, and immunization strategies, where short paths are most relevant.</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":83817562,"attachmentType":"pdf","workUrl":"https://www.academia.edu/74007336/Limited_Percolation_on_Complex_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":83817562,"attachmentType":"pdf","workUrl":"https://www.academia.edu/74007336/Limited_Percolation_on_Complex_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-premium-marketing"></div></div><div class="ds-signup-banner ds-signup-banner-premium-marketing"><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="premium-banner-content" data-impression-entity-id="74007336" data-impression-entity-type="2" data-impression-source="premium-banner-desktop"><div class="left"><img src="//a.academia-assets.com/images/academia-logo-capital-white.svg" /><span>Get access to the world's latest research</span></div><div class="right"><div class="card free"><div class="header">Free</div><div class="feature-list"><div class="feature"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">check</span><span>Download one paper at a time</span></div><div class="feature"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">check</span><span>Save papers to bookmarks</span></div><div class="feature"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">check</span><span>Basic search</span></div></div><button class="ds2-5-button ds2-5-button--secondary ds2-5-button--small ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{"location":"premium-banner-desktop-free"}">Sign up for free</button></div><div class="card premium"><div class="pill">Recommended</div><div class="header premium">Premium</div><div class="feature-list"><div class="feature"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">check</span><span>Get highly curated PDF packages</span></div><div class="feature"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">check</span><span>Track your impact with Mentions</span></div><div class="feature"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">check</span><span>Access advanced search filters</span></div><div class="feature"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">check</span><span>Support Academia’s mission</span></div><div class="feature"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">check</span><span>Create your personal website</span></div></div><button class="ds2-5-button ds2-5-button--small ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{"location":"premium-banner-desktop-upgrade","submitText":"Try Premium for $1"}">Try Premium for $1</button></div></div></div></div><script>(() => { // Set up signup banner show/hide behavior: // 1. 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For a large class of networks, we find a new percolation transition atp c = (κ o − 1) (1−a)/a , where κ o ≡ k 2 / k and k is the node degree. Belowp c , only a fraction N δ of the network nodes can communicate, where δ ≡ a(1 − | log p|/ log (κ o − 1)) < 1, while abovep c , order N nodes can communicate within the limited path length aℓ ij . Our analytical results are supported by simulations on Erdős-Rényi and scale-free network models. We expect our results to influence the design of networks, routing algorithms, and immunization strategies, where short paths are most relevant.</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":"Limited Path Percolation in Complex Networks","attachmentId":43493206,"attachmentType":"pdf","work_url":"https://www.academia.edu/15179266/Limited_Path_Percolation_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/15179266/Limited_Path_Percolation_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="1" data-entity-id="17156043" 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/17156043/Unusual_percolation_in_simple_small_world_networks">Unusual percolation in simple small-world 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="36747772" href="https://independent.academia.edu/MehranKardar">Mehran Kardar</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2009</p><p class="ds-related-work--abstract ds2-5-body-sm">We present an exact solution of percolation in a generalized class of Watts-Strogatz graphs defined on a 1-dimensional underlying lattice. We find a non-classical critical point in the limit of the number of long-range bonds in the system going to zero, with a discontinuity in the percolation probability and a divergence in the mean finite-cluster size. We show that the critical behavior falls into one of three regimes depending on the proportion of occupied long-range to unoccupied nearest-neighbor bonds, with each regime being characterized by different critical exponents. The three regimes can be united by a single scaling function around the critical point. These results can be used to identify the number of long-range links necessary to secure connectivity in a communication or transportation chain. As an example, we can resolve the communication problem in a game of "telephone".</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":"Unusual percolation in simple small-world networks","attachmentId":42300610,"attachmentType":"pdf","work_url":"https://www.academia.edu/17156043/Unusual_percolation_in_simple_small_world_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/17156043/Unusual_percolation_in_simple_small_world_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="2" data-entity-id="78309384" 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/78309384/Percolation_in_networks_composed_of_connectivity_and_dependency_links">Percolation in networks composed of connectivity and dependency links</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">Physical Review E, 2011</p><p class="ds-related-work--abstract ds2-5-body-sm">Networks composed from both connectivity and dependency links were found to be more vulnerable compared to classical networks with only connectivity links. 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="3" 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="4" data-entity-id="28599095" 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/28599095/L_hop_percolation_on_networks_with_arbitrary_degree_distributions_and_its_applications">L-hop percolation on networks with arbitrary degree distributions and its applications</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="53241330" href="https://uccs.academia.edu/ShouhuaiXu">Shouhuai Xu</a></div><p class="ds-related-work--abstract ds2-5-body-sm">Site percolation has been used to help understand analytically the robustness of complex networks in the presence of random node deletion (or failure). In this paper we move a further step beyond random node deletion by considering that a node can be deleted because it is chosen or because it is within some L-hop distance of a chosen node. Using the generating functions approach, we present analytic results on the percolation threshold as well as the mean size, and size distribution, of nongiant components of complex networks under such operations. The introduction of parameter L is both conceptually interesting because it accommodates a sort of nonindependent node deletion, which is often difficult to tackle analytically, and practically interesting because it offers useful insights for cybersecurity (such as botnet defense).</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":"L-hop percolation on networks with arbitrary degree distributions and its applications","attachmentId":48961012,"attachmentType":"pdf","work_url":"https://www.academia.edu/28599095/L_hop_percolation_on_networks_with_arbitrary_degree_distributions_and_its_applications","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/28599095/L_hop_percolation_on_networks_with_arbitrary_degree_distributions_and_its_applications"><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="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="6398618" 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/6398618/Impact_of_Single_Links_in_Competitive_Percolation_How_complex_networks_grow_under_competition">Impact of Single Links in Competitive Percolation -- How complex networks grow under competition</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="10050633" href="https://univ-paris1.academia.edu/AnnaLevina">Anna Levina</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2011</p><p class="ds-related-work--abstract ds2-5-body-sm">How a complex network is connected crucially impacts its dynamics and function. Percolation, the transition to extensive connectedness upon gradual addition of links, was long believed to be continuous but recent numerical evidence on "explosive percolation" suggests that it might as well be discontinuous if links compete for addition. Here we analyze the microscopic mechanisms underlying discontinuous percolation processes and reveal a strong impact of single link additions. We show that in generic competitive percolation processes, including those displaying explosive percolation, single links do not induce a discontinuous gap in the largest cluster size in the thermodynamic limit. Nevertheless, our results highlight that for large finite systems single links may still induce observable gaps because gap sizes scale weakly algebraically with system size. Several essentially macroscopic clusters coexist immediately before the transition, thus announcing discontinuous percolation. These results explain how single links may drastically change macroscopic connectivity in networks where links add competitively.</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":"Impact of Single Links in Competitive Percolation -- How complex networks grow under competition","attachmentId":33208068,"attachmentType":"pdf","work_url":"https://www.academia.edu/6398618/Impact_of_Single_Links_in_Competitive_Percolation_How_complex_networks_grow_under_competition","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/6398618/Impact_of_Single_Links_in_Competitive_Percolation_How_complex_networks_grow_under_competition"><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="51779514" 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/51779514/Core_Percolation_on_Complex_Networks">Core Percolation on 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">Physical Review Letters, 2012</p><p class="ds-related-work--abstract ds2-5-body-sm">As a fundamental structural transition in complex networks, core percolation is related to a wide range of important problems. Yet, previous theoretical studies of core percolation have been focusing on the classical Erdős-Rényi random networks with Poisson degree distribution, which are quite unlike many real-world networks with scale-free or fat-tailed degree distributions. Here we show that core percolation can be analytically studied for complex networks with arbitrary degree distributions. We derive the condition for core percolation and find that purely scale-free networks have no core for any degree exponents. We show that for undirected networks if core percolation occurs then it is always continuous while for directed networks it becomes discontinuous when the in-and out-degree distributions are different. We also apply our theory to real-world directed networks and find, surprisingly, that they often have much larger core sizes as compared to random models. These findings would help us better understand the interesting interplay between the structural and dynamical properties of complex networks. Network science has emerged as a prominent field in complex system research, which provides us a novel perspective to better understand complexity[1-3]. In the last decade considerable advances about structural and dynamical properties of complex networks have been made[4-6]. Among them, structural transitions in networks were extensively studied due to their big impacts on numerous dynamical processes on networks. Particularly interesting are the emergence of a giant connected component[7-10] , k-core percolation[11-13], k-clique percolation[14, 15], and explosive percolation[16-18]. These structural transitions affect many properties of networks, e.g. robustness and resilience to breakdowns[9, 19, 20], cascading failure in interdependent networks[21-24],</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":"Core Percolation on Complex Networks","attachmentId":69351164,"attachmentType":"pdf","work_url":"https://www.academia.edu/51779514/Core_Percolation_on_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/51779514/Core_Percolation_on_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="9" data-entity-id="60625238" 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/60625238/Weighted_Percolation_on_Directed_Networks">Weighted Percolation on Directed 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="32706411" href="https://independent.academia.edu/OttEdward">Edward Ott</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review Letters, 2008</p><p class="ds-related-work--abstract ds2-5-body-sm">We present an analysis of the percolation transition for general node removal strategies valid for locally tree-like directed networks. On the basis of heuristic arguments we predict that, if the probability of removing node i is pi, the network disintegrates if pi is such that the largest eigenvalue of the matrix with entries Aij(1 − pi) is less than 1, where A is the adjacency matrix of the network. The knowledge or applicability of a Markov network model is not required by our theory, thus making it applicable to situations not covered by previous works. We test our predicted percolation criterion against numerical results for different networks and node removal strategies.</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":"Weighted Percolation on Directed Networks","attachmentId":73985008,"attachmentType":"pdf","work_url":"https://www.academia.edu/60625238/Weighted_Percolation_on_Directed_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/60625238/Weighted_Percolation_on_Directed_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":83817562,"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":83817562,"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_83817562" 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. You can download the paper by clicking the button above.</p></div></div></div></div><div class="ds-sidebar--container js-work-sidebar"><div class="ds-related-content--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="0" data-entity-id="16145825" 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/16145825/Percolation_on_interdependent_networks_with_a_fraction_of_antagonistic_interactions">Percolation on interdependent networks with a fraction of antagonistic interactions</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="35256933" href="https://qmul.academia.edu/GinestraBianconi">Ginestra Bianconi</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline 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js-related-work-sidebar-card" data-collection-position="1" data-entity-id="6146974" 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/6146974/Percolation_of_a_general_network_of_networks">Percolation of a general network of 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="7630487" href="https://bu.academia.edu/EugeneStanley">Eugene Stanley</a></div><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 a general network of networks","attachmentId":33040751,"attachmentType":"pdf","work_url":"https://www.academia.edu/6146974/Percolation_of_a_general_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-related-work-grid-card-view-pdf" href="https://www.academia.edu/6146974/Percolation_of_a_general_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-related-work-sidebar-card" data-collection-position="2" data-entity-id="10831942" 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/10831942/Percolation_Transitions_Are_Not_Always_Sharpened_by_Making_Networks_Interdependent">Percolation Transitions Are Not Always Sharpened by Making Networks Interdependent</a><div class="ds-related-work--metadata"><a 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href="https://www.academia.edu/10831942/Percolation_Transitions_Are_Not_Always_Sharpened_by_Making_Networks_Interdependent"><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="3" data-entity-id="113634538" 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/113634538/Scaling_of_percolation_transitions_on_Erd%C3%B6s_R%C3%A9nyi_networks_under_centrality_based_attacks">Scaling of percolation transitions on Erdös-Rényi networks under centrality-based attacks</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="98865348" href="https://independent.academia.edu/IgnacioPerotti">Ignacio Perotti</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2020</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":"Scaling of percolation transitions on Erdös-Rényi networks under centrality-based attacks","attachmentId":110544382,"attachmentType":"pdf","work_url":"https://www.academia.edu/113634538/Scaling_of_percolation_transitions_on_Erd%C3%B6s_R%C3%A9nyi_networks_under_centrality_based_attacks","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/113634538/Scaling_of_percolation_transitions_on_Erd%C3%B6s_R%C3%A9nyi_networks_under_centrality_based_attacks"><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="4" data-entity-id="79567352" 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/79567352/Bootstrap_percolation_and_the_geometry_of_complex_networks">Bootstrap percolation and the geometry of complex 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="32988049" href="https://bham.academia.edu/NikolaosFountoulakis">Nikolaos Fountoulakis</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2014</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" 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class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/16145772/Percolation_on_interacting_antagonistic_networks">Percolation on interacting, antagonistic 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="35256933" href="https://qmul.academia.edu/GinestraBianconi">Ginestra Bianconi</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Statistical Mechanics: Theory and Experiment, 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 on interacting, antagonistic networks","attachmentId":42669763,"attachmentType":"pdf","work_url":"https://www.academia.edu/16145772/Percolation_on_interacting_antagonistic_networks","alternativeTracking":true}"><span 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class="js-related-work-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><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-related-work-grid-card-view-pdf" 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