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{"work":{"id":85476136,"created_at":"2022-08-23T17:05:35.408-07:00","from_world_paper_id":213502080,"updated_at":"2024-11-25T03:26:29.974-08:00","_data":{"publisher":"Springer Nature","grobid_abstract":"We virtually dissect complex networks in order to understand their internal structure, just as doctors do with the bodies of animals. Our novel method classifies network links into four categories: bone, fat, cartilage, and muscle, based on network connectivity. We derive an efficient percolation strategy from this new viewpoint of network anatomy, which enables abrupt-like percolation transition through removal of a small amount of cartilage links, which play a crucial role in network connectivity. Furthermore, we find nontrivial scaling laws in the relationships between four types of links in each cluster and evaluate power exponents, which characterize network structures as seen in the real largescale network of trading business firms and in the Erdős-Rényi network. Finally, we observe changes in the transition point for random bond percolation process, demonstrating that the addition of muscle links enhances network robustness, while fat links are irrelevant. These findings aid in controlling the percolation transition for an arbitrary network. Different networks, such as human and business relationship networks, and power networks, are everywhere in our world 1-4. Most complex networks in social systems can be categorized as having scale-free and small-world properties 5, 6. Many methods quantifying such inhomogeneous networks have been proposed from various fields including biology, information science and physics 4, 6-9. It is important to understand how a network can be made robust under attack, including methods for intentional removal of nodes and links because such networks form the basis of the society and economy 10-14. Thus, it is necessary to determine what elements contribute to reinforcing network connectivity, and to find practical ways to enhance robustness of the system. Percolation theory has been studied in the fields of mathematics and physics to clarify macroscopic connectivity from a microscopic viewpoint 4, 15-20. Specifically, the percolation transition properties of complex networks have been attracting the attention of many scientists since the proposal of the scale-free network model 5. It has been reported that a scale-free network is fragile against targeted attacks, but robust against random failures 10. Furthermore, recent percolation models have been extended to explain a discontinuous percolation transition (DPT) 2, 18, 19, 21, 22. It has been suggested that a real power network carries the risk of massive blackouts due to cascading failures in a multi-layered network 2 , and discontinuity in the explosive percolation model has attracted great interest in recent years due to its simple yet diverse characteristics 23-25. Classification of nodes and links, such as community extraction, is also an important field of study, and comprehensive graphical expressions have become available for this application 26-30. In the theory of percolation transition for square lattices, there are studies classifying \"backbone links\" based on significant contribution to overall connectivity 31-33 , however, such classifications have not been yet introduced to percolation study in complex networks. In this study, we further generalize the anatomical concept of a \"backbone\" by introducing a novel \"network anatomy\", which virtually dissects any given complex network and classifies its links based on their contribution to network connectivity. In the Method section, all network links are classified into four categories: bone, fat, cartilage, and muscle links, as an analogy for the anatomy of animal bodies. In the Result 1 section, we show that a percolation strategy assembled from these link categories enables the abrupt-like transition in a large-scale real network, as well as artificial networks. Nontrivial scaling laws are observed among the four classified link types and scaling exponents that characterize a network are shown in the Result 2 section. In the Result 3 section, we observe shifts of the percolation transition point caused by doping fat and muscle links to clarify the functional","publication_date":"2017,,","publication_name":"Scientific Reports","grobid_abstract_attachment_id":"90163458"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Network Anatomy Controlling Abrupt-like Percolation Transition","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [34060367]; 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":90163458,"attachmentType":"pdf"}"><img alt="First page of “Network Anatomy Controlling Abrupt-like Percolation Transition”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/90163458/mini_magick20220824-1-1we38dk.png?1661299710" /><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">Network Anatomy Controlling Abrupt-like Percolation Transition</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="34060367" href="https://independent.academia.edu/HidekiTakayasu"><img alt="Profile image of Hideki Takayasu" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Hideki Takayasu</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2017, Scientific Reports</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">8 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 = 85476136; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=85476136"; 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 virtually dissect complex networks in order to understand their internal structure, just as doctors do with the bodies of animals. Our novel method classifies network links into four categories: bone, fat, cartilage, and muscle, based on network connectivity. We derive an efficient percolation strategy from this new viewpoint of network anatomy, which enables abrupt-like percolation transition through removal of a small amount of cartilage links, which play a crucial role in network connectivity. Furthermore, we find nontrivial scaling laws in the relationships between four types of links in each cluster and evaluate power exponents, which characterize network structures as seen in the real largescale network of trading business firms and in the Erdős-Rényi network. Finally, we observe changes in the transition point for random bond percolation process, demonstrating that the addition of muscle links enhances network robustness, while fat links are irrelevant. These findings aid in controlling the percolation transition for an arbitrary network. Different networks, such as human and business relationship networks, and power networks, are everywhere in our world 1-4. Most complex networks in social systems can be categorized as having scale-free and small-world properties 5, 6. Many methods quantifying such inhomogeneous networks have been proposed from various fields including biology, information science and physics 4, 6-9. It is important to understand how a network can be made robust under attack, including methods for intentional removal of nodes and links because such networks form the basis of the society and economy 10-14. Thus, it is necessary to determine what elements contribute to reinforcing network connectivity, and to find practical ways to enhance robustness of the system. Percolation theory has been studied in the fields of mathematics and physics to clarify macroscopic connectivity from a microscopic viewpoint 4, 15-20. Specifically, the percolation transition properties of complex networks have been attracting the attention of many scientists since the proposal of the scale-free network model 5. It has been reported that a scale-free network is fragile against targeted attacks, but robust against random failures 10. Furthermore, recent percolation models have been extended to explain a discontinuous percolation transition (DPT) 2, 18, 19, 21, 22. It has been suggested that a real power network carries the risk of massive blackouts due to cascading failures in a multi-layered network 2 , and discontinuity in the explosive percolation model has attracted great interest in recent years due to its simple yet diverse characteristics 23-25. Classification of nodes and links, such as community extraction, is also an important field of study, and comprehensive graphical expressions have become available for this application 26-30. In the theory of percolation transition for square lattices, there are studies classifying "backbone links" based on significant contribution to overall connectivity 31-33 , however, such classifications have not been yet introduced to percolation study in complex networks. In this study, we further generalize the anatomical concept of a "backbone" by introducing a novel "network anatomy", which virtually dissects any given complex network and classifies its links based on their contribution to network connectivity. In the Method section, all network links are classified into four categories: bone, fat, cartilage, and muscle links, as an analogy for the anatomy of animal bodies. In the Result 1 section, we show that a percolation strategy assembled from these link categories enables the abrupt-like transition in a large-scale real network, as well as artificial networks. Nontrivial scaling laws are observed among the four classified link types and scaling exponents that characterize a network are shown in the Result 2 section. In the Result 3 section, we observe shifts of the percolation transition point caused by doping fat and muscle links to clarify the functional</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":90163458,"attachmentType":"pdf","workUrl":"https://www.academia.edu/85476136/Network_Anatomy_Controlling_Abrupt_like_Percolation_Transition"}">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":90163458,"attachmentType":"pdf","workUrl":"https://www.academia.edu/85476136/Network_Anatomy_Controlling_Abrupt_like_Percolation_Transition"}"><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="85476136" 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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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="1" 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="2" 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="3" 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="4" 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="5" 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="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="50591935" 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/50591935/Evolution_of_networks">Evolution 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="107117040" href="https://independent.academia.edu/JoseMendes58">Jose Mendes</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Advances in Physics, 2002</p><p class="ds-related-work--abstract ds2-5-body-sm">We review the recent fast progress in statistical physics of evolving networks. Interest has focused mainly on the structural properties of random complex networks in communications, biology, social sciences and economics. A number of giant artificial networks of such a kind came into existence recently. This opens a wide field for the study of their topology, evolution, and complex processes occurring in them. Such networks possess a rich set of scaling properties. A number of them are scale-free and show striking resilience against random breakdowns. In spite of large sizes of these networks, the distances between most their vertices are short-a feature known as the "smallworld" effect. We discuss how growing networks self-organize into scale-free structures and the role of the mechanism of preferential linking. We consider the topological and structural properties of evolving networks, and percolation in these networks. We present a number of models demonstrating the main features of evolving networks and discuss current approaches for their simulation and analytical study. Applications of the general results to particular networks in Nature are discussed. We demonstrate the generic connections of the network growth processes with the general problems of non-equilibrium physics, econophysics, evolutionary biology, etc. CONTENTS 42 IX L. Eigenvalue spectrum of the adjacency matrix 42 IX M. Scale-free trees 43 X. Non-scale-free networks with preferential linking 43 XI. Percolation on networks 44 XI A. Theory of percolation on undirected equilibrium networks 44 XI B. Percolation on directed equilibrium networks 48 XI C. Failures and attacks 49 XI D. Resilience against random breakdowns 50 XI E. Intentional damage 52 XI F. Disease spread within networks 54 XI G. Anomalous percolation on growing networks 55 XII. Growth of networks and self-organized criticality 57 XII A. Linking with sand-pile problems 57 XII B. Preferential linking and the Simon model 57 XII C. Multiplicative stochastic models and the generalized Lotka-Volterra equation 58 XIII. Concluding remarks 58 Acknowledgements 59 References 59</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":"Evolution of networks","attachmentId":68515792,"attachmentType":"pdf","work_url":"https://www.academia.edu/50591935/Evolution_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/50591935/Evolution_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="8" data-entity-id="15179266" 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/15179266/Limited_Path_Percolation_in_Complex_Networks">Limited Path Percolation 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="34232414" href="https://independent.academia.edu/EduardoColgateLopez">Eduardo Colgate Lopez</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review Letters, 2007</p><p class="ds-related-work--abstract ds2-5-body-sm">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 aℓ ij (a ≥ 1) where ℓ ij is the shortest path before removal. 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="9" data-entity-id="127257743" 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/127257743/Abrupt_transition_in_the_structural_formation_of_interconnected_networks">Abrupt transition in the structural formation of interconnected 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="138515774" href="https://independent.academia.edu/alexarenas29">alex arenas</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Nature Physics, 2013</p><p class="ds-related-work--abstract ds2-5-body-sm">Our world is linked by a complex mesh of networks through which information, people and goods flow. These networks are interdependent on each other, and present structural and dynamical features 1-6 different from those observed in isolated networks . Although examples of such dissimilar properties are becoming more abundant-such as in diffusion, robustness and competition-it is not yet clear where these differences are rooted. Here we show that the process of building independent networks into an interconnected network of networks undergoes a structurally sharp transition as the interconnections are formed. Depending on the relative importance of inter-and intra-layer connections, we find that the entire interdependent system can be tuned between two regimes: in one regime, the various layers are structurally decoupled and they act as independent entities; in the other regime, network layers are indistinguishable and the whole system behaves as a singlelevel network. We analytically show that the transition between the two regimes is discontinuous even for finite-size networks. Thus, any real-world interconnected system is potentially at risk of abrupt changes in its structure, which may manifest new dynamical properties. Interacting, interdependent or multiplex networks are different ways of naming the same class of complex systems where networks are not considered as isolated entities but interacting with each other. In multiplex, the nodes at each network are instances of the same entity; thus, the networks are representing simply different categorical relationships between entities, and usually categories are represented by layers. Interdependent networks is a more general framework where nodes can be different at each network. Many, if not all, real networks are coupled with other real networks. Examples can be found in several domains: social networks (for example, Facebook, Twitter and so on) are coupled because they share the same actors 10 ; multimodal transportation networks are composed of different layers (for example, bus, subway and so on) that share the same locations 11 ; the functioning of communication and power grid systems depends one on the other 1 . So far, all phenomena that have been studied on interdependent networks, including percolation 1,3 , epidemics 4 and linear dynamical systems 5 , have provided results that differ much from those valid in the case of isolated complex networks. Sometimes the difference is radical: for example, whereas isolated scale-free networks are robust against failures of their nodes or edges 12 , scale-free interdependent networks are instead very fragile 1,3 . Given such observations, two fundamentally important theoretical questions are in order: Why do dynamical and critical phenomena running on interdependent network models differ</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":"Abrupt transition in the structural formation of interconnected networks","attachmentId":121017230,"attachmentType":"pdf","work_url":"https://www.academia.edu/127257743/Abrupt_transition_in_the_structural_formation_of_interconnected_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/127257743/Abrupt_transition_in_the_structural_formation_of_interconnected_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":90163458,"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":90163458,"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_90163458" 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="4819113" 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/4819113/Explosive_Percolation_in_Scale_Free_Networks">Explosive Percolation in Scale-Free 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="6230751" href="https://independent.academia.edu/FilippoRadicchi">Filippo Radicchi</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review Letters, 2009</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 Scale-Free Networks","attachmentId":32112139,"attachmentType":"pdf","work_url":"https://www.academia.edu/4819113/Explosive_Percolation_in_Scale_Free_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/4819113/Explosive_Percolation_in_Scale_Free_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="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 class="js-related-work-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 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" 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="94284448" 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/94284448/Networks_formed_from_interdependent_networks">Networks formed from interdependent 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="36769513" href="https://independent.academia.edu/JianxiGao">Jianxi Gao</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Nature Physics, 2011</p><div class="ds-related-work--ctas"><button 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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/15452926/Explosive_percolation_on_real_world_networks">Explosive percolation on real-world 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="34595670" href="https://vmwork.academia.edu/KimmoKaski">Kimmo Kaski</a><span>, </span><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="22576039" href="https://ceu.academia.edu/JanosKertesz">Janos Kertesz</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv preprint arXiv:1010.3171, 2010</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 on real-world 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href="https://www.academia.edu/16913417/Percolation_in_directed_scale_free_networks">Percolation in directed scale-free 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="36832619" href="https://independent.academia.edu/DBenavraham">Daniel ben-Avraham</a><span>, </span><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="36407694" href="https://biu.academia.edu/SHavlin">S. Havlin</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2002</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 directed scale-free networks","attachmentId":39254526,"attachmentType":"pdf","work_url":"https://www.academia.edu/16913417/Percolation_in_directed_scale_free_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/16913417/Percolation_in_directed_scale_free_networks"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" 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class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32507196" href="https://ksu.academia.edu/CaterinaScoglio">Caterina Scoglio</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2015</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":"Exact coupling threshold for structural transition reveals diversified behaviors in interconnected networks","attachmentId":47355615,"attachmentType":"pdf","work_url":"https://www.academia.edu/27104505/Exact_coupling_threshold_for_structural_transition_reveals_diversified_behaviors_in_interconnected_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 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js-related-work-sidebar-card" data-collection-position="12" data-entity-id="6099857" 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/6099857/Double_percolation_phase_transition_in_clustered_complex_networks">Double percolation phase transition in clustered 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="3076056" href="https://barcelona.academia.edu/PolDeSimon">Pol De Simon</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":"Double percolation phase transition in clustered complex 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href="https://www.academia.edu/16145820/Multiple_percolation_transitions_in_a_configuration_model_of_a_network_of_networks">Multiple percolation transitions in a configuration model of a 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="35256933" href="https://qmul.academia.edu/GinestraBianconi">Ginestra Bianconi</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2014</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":"Multiple percolation transitions in a configuration model of a network of networks","attachmentId":42669728,"attachmentType":"pdf","work_url":"https://www.academia.edu/16145820/Multiple_percolation_transitions_in_a_configuration_model_of_a_network_of_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="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><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-related-work-grid-card-view-pdf" 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