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<!DOCTYPE html> <html > <head> <meta charset="utf-8"> <meta rel="search" type="application/opensearchdescription+xml" href="/open_search.xml" title="Academia.edu"> <meta content="width=device-width, initial-scale=1" name="viewport"> <meta name="google-site-verification" content="bKJMBZA7E43xhDOopFZkssMMkBRjvYERV-NaN4R6mrs"> <meta name="csrf-param" content="authenticity_token" /> <meta name="csrf-token" content="zz7CChivIhAnNUqOWn758ficruQr_KKw2FgS_PEqy1RAyj-l-nRO_jQ9w4dYTlizjo0Tr76O_pCnO9nT9OCsxg" /> <meta name="citation_title" content="Lifespan method as a tool to study criticality in absorbing-state phase transitions" /> <meta name="citation_publication_date" content="2015/01/01" /> <meta name="citation_journal_title" content="Physical review. E, Statistical, nonlinear, and soft matter physics" /> <meta name="citation_author" content="Angelica Mata" /> <meta name="twitter:card" content="summary" /> <meta name="twitter:url" content="https://www.academia.edu/30169125/Lifespan_method_as_a_tool_to_study_criticality_in_absorbing_state_phase_transitions" /> <meta name="twitter:title" content="Lifespan method as a tool to study criticality in absorbing-state phase transitions" /> <meta name="twitter:description" content="In a recent work, a new numerical method (the lifespan method) has been introduced to study the critical properties of epidemic processes on complex networks [M. Boguñá, C. Castellano, and R. Pastor-Satorras, Phys. Rev. Lett. 111, 068701" /> <meta name="twitter:image" content="https://0.academia-photos.com/15839788/6002415/15628407/s200_angelica.mata.jpg" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/30169125/Lifespan_method_as_a_tool_to_study_criticality_in_absorbing_state_phase_transitions" /> <meta property="og:title" content="Lifespan method as a tool to study criticality in absorbing-state phase transitions" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="In a recent work, a new numerical method (the lifespan method) has been introduced to study the critical properties of epidemic processes on complex networks [M. Boguñá, C. Castellano, and R. Pastor-Satorras, Phys. Rev. Lett. 111, 068701" /> <meta property="article:author" content="https://dwcc.academia.edu/AngelicaMata" /> <meta name="description" content="In a recent work, a new numerical method (the lifespan method) has been introduced to study the critical properties of epidemic processes on complex networks [M. Boguñá, C. Castellano, and R. Pastor-Satorras, Phys. Rev. Lett. 111, 068701" /> <title>(PDF) Lifespan method as a tool to study criticality in absorbing-state phase transitions</title> <link rel="canonical" href="https://www.academia.edu/30169126/The_lifespan_method_as_a_tool_to_study_criticality_in_absorbing_state_phase_transitions" /> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "single_work", 'action': "show", 'controller_action': 'single_work#show', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two cases. // ab_test_bucket should be of the form <ab_test_name>:<bucket> 'ab_test_bucket': null, }) </script> <script> var $controller_name = 'single_work'; var $action_name = "show"; var $rails_env = 'production'; var $app_rev = '1e60a92a442ff83025cbe4f252857ee7c49c0bbe'; var $domain = 'academia.edu'; var $app_host = "academia.edu"; var $asset_host = "academia-assets.com"; var $start_time = new Date().getTime(); var $recaptcha_key = "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB"; var $recaptcha_invisible_key = "6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj"; var $disableClientRecordHit = false; </script> <script> window.require = { config: function() { return function() {} } } </script> <script> window.Aedu = window.Aedu || {}; window.Aedu.hit_data = null; window.Aedu.serverRenderTime = new Date(1740572651000); window.Aedu.timeDifference = new Date().getTime() - 1740572651000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"In a recent work, a new numerical method (the lifespan method) has been introduced to study the critical properties of epidemic processes on complex networks [M. Boguñá, C. Castellano, and R. Pastor-Satorras, Phys. Rev. Lett. 111, 068701 (2013)PRLTAO0031-900710.1103/PhysRevLett.111.068701]. Here, we present a detailed analysis of the viability of this method for the study of the critical properties of generic absorbing-state phase transitions in lattices. Focusing on the well-understood case of the contact process, we develop a finite-size scaling theory to measure the critical point and its associated critical exponents. We show the validity of the method by studying numerically the contact process on a one-dimensional lattice and comparing the findings of the lifespan method with the standard quasistationary method. We find that the lifespan method gives results that are perfectly compatible with those of quasistationary simulations and with analytical results. Our observations co...","author":[{"@context":"https://schema.org","@type":"Person","name":"Angelica Mata","url":"https://dwcc.academia.edu/AngelicaMata"}],"contributor":[],"dateCreated":"2016-11-29","dateModified":"2016-11-29","datePublished":"2015-01-01","headline":"Lifespan method as a tool to study criticality in absorbing-state phase transitions","image":"https://attachments.academia-assets.com/50626382/thumbnails/1.jpg","inLanguage":"en","keywords":["Engineering","Mathematical Sciences","Physical sciences"],"publication":"Physical review. E, Statistical, nonlinear, and soft matter physics","publisher":{"@context":"https://schema.org","@type":"Organization","name":null},"sourceOrganization":[{"@context":"https://schema.org","@type":"EducationalOrganization","name":"dwcc"}],"thumbnailUrl":"https://attachments.academia-assets.com/50626382/thumbnails/1.jpg","url":"https://www.academia.edu/30169125/Lifespan_method_as_a_tool_to_study_criticality_in_absorbing_state_phase_transitions"}</script><style type="text/css">@media(max-width: 567px){:root{--token-mode: Rebrand;--dropshadow: 0 2px 4px 0 #22223340;--primary-brand: #0645b1;--error-dark: #b60000;--success-dark: #05b01c;--inactive-fill: #ebebee;--hover: #0c3b8d;--pressed: #082f75;--button-primary-fill-inactive: #ebebee;--button-primary-fill: #0645b1;--button-primary-text: #ffffff;--button-primary-fill-hover: #0c3b8d;--button-primary-fill-press: #082f75;--button-primary-icon: #ffffff;--button-primary-fill-inverse: #ffffff;--button-primary-text-inverse: 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false; window.loswp.work = {"work":{"id":30169125,"created_at":"2016-11-29T21:23:48.082-08:00","from_world_paper_id":159460110,"updated_at":"2021-01-18T10:04:12.657-08:00","_data":{"abstract":"In a recent work, a new numerical method (the lifespan method) has been introduced to study the critical properties of epidemic processes on complex networks [M. Boguñá, C. Castellano, and R. Pastor-Satorras, Phys. Rev. Lett. 111, 068701 (2013)PRLTAO0031-900710.1103/PhysRevLett.111.068701]. Here, we present a detailed analysis of the viability of this method for the study of the critical properties of generic absorbing-state phase transitions in lattices. Focusing on the well-understood case of the contact process, we develop a finite-size scaling theory to measure the critical point and its associated critical exponents. We show the validity of the method by studying numerically the contact process on a one-dimensional lattice and comparing the findings of the lifespan method with the standard quasistationary method. We find that the lifespan method gives results that are perfectly compatible with those of quasistationary simulations and with analytical results. Our observations co...","publication_date":"2015,,","publication_name":"Physical review. E, Statistical, nonlinear, and soft matter physics"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Lifespan method as a tool to study criticality in absorbing-state phase transitions","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [15839788]; 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":50626382,"attachmentType":"pdf"}"><img alt="First page of “Lifespan method as a tool to study criticality in absorbing-state phase transitions”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/50626382/mini_magick20190128-12903-yzbrf6.png?1548704980" /><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">Lifespan method as a tool to study criticality in absorbing-state phase transitions</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="15839788" href="https://dwcc.academia.edu/AngelicaMata"><img alt="Profile image of Angelica Mata" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/15839788/6002415/15628407/s65_angelica.mata.jpg" />Angelica Mata</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2015, Physical review. E, Statistical, nonlinear, and soft matter physics</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">9 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 = 30169125; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=30169125"; 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">In a recent work, a new numerical method (the lifespan method) has been introduced to study the critical properties of epidemic processes on complex networks [M. Boguñá, C. Castellano, and R. Pastor-Satorras, Phys. Rev. Lett. 111, 068701 (2013)PRLTAO0031-900710.1103/PhysRevLett.111.068701]. Here, we present a detailed analysis of the viability of this method for the study of the critical properties of generic absorbing-state phase transitions in lattices. Focusing on the well-understood case of the contact process, we develop a finite-size scaling theory to measure the critical point and its associated critical exponents. We show the validity of the method by studying numerically the contact process on a one-dimensional lattice and comparing the findings of the lifespan method with the standard quasistationary method. We find that the lifespan method gives results that are perfectly compatible with those of quasistationary simulations and with analytical results. Our observations co...</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":50626382,"attachmentType":"pdf","workUrl":"https://www.academia.edu/30169125/Lifespan_method_as_a_tool_to_study_criticality_in_absorbing_state_phase_transitions"}">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":50626382,"attachmentType":"pdf","workUrl":"https://www.academia.edu/30169125/Lifespan_method_as_a_tool_to_study_criticality_in_absorbing_state_phase_transitions"}"><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="30169125" 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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Rev. Lett. 111, 068701 (2013)]. Here, we present a detailed analysis of the viability of this method for the study of the critical properties of generic absorbing-state phase transitions in lattices. Focusing on the well understood case of the contact process, we develop a finite-size scaling theory to measure the critical point and its associated critical exponents. We show the validity of the method by studying numerically the contact process on a one-dimensional lattice and comparing the findings of the lifespan method with the standard quasi-stationary method. We find that the lifespan method gives results that are perfectly compatible with those of quasi-stationary simulations and with analytical results. Our observations confirm that the lifespan method is a fully legitimate tool for the study of the critical properties of absorbing phase-transitions in general substrates, either random or regular.</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 lifespan method as a tool to study criticality in absorbing-state phase transitions","attachmentId":50626381,"attachmentType":"pdf","work_url":"https://www.academia.edu/30169126/The_lifespan_method_as_a_tool_to_study_criticality_in_absorbing_state_phase_transitions","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/30169126/The_lifespan_method_as_a_tool_to_study_criticality_in_absorbing_state_phase_transitions"><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="32101260" 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/32101260/Phase_transitions_with_infinitely_many_absorbing_states_in_complex_networks">Phase transitions with infinitely many absorbing states 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="33676389" href="https://ufv-br.academia.edu/SilvioFerreira">Silvio Ferreira</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2013</p><p class="ds-related-work--abstract ds2-5-body-sm">We instigate the properties of the threshold contact process (TCP), a process showing an absorbing-state phase transition with infinitely many absorbing states, on random complex networks. The finite size scaling exponents characterizing the transition are obtained in a heterogeneous mean field (HMF) approximation and compared with extensive simulations, particularly in the case of heterogeneous scale-free networks. We observe that the TCP exhibits the same critical properties as the contact process (CP), which undergoes an absorbing-state phase transition to a single absorbing state. The accordance among the critical exponents of different models and networks leads to conjecture that the critical behavior of the contact process in a HMF theory is a universal feature of absorbing state phase transitions in complex networks, depending only on the locality of the interactions and independent of the number of absorbing states. The conditions for the applicability of the conjecture are discussed considering a parallel with the susceptible-infected-susceptible epidemic spreading model, which in fact belongs to a different universality class in complex 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":"Phase transitions with infinitely many absorbing states in complex networks","attachmentId":52350237,"attachmentType":"pdf","work_url":"https://www.academia.edu/32101260/Phase_transitions_with_infinitely_many_absorbing_states_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/32101260/Phase_transitions_with_infinitely_many_absorbing_states_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="2" data-entity-id="32258135" 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/32258135/Absorbing_state_phase_transitions_with_infinitely_many_absorbing_states_in_complex_networks">Absorbing state phase transitions with infinitely many absorbing states 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="62573188" href="https://ifmuz.academia.edu/RenanSander">Renan Sander</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2012</p><p class="ds-related-work--abstract ds2-5-body-sm">We instigate the properties of the threshold contact process (TCP), a process showing an absorbing-state phase transition with infinitely many absorbing states, on random complex networks. The finite size scaling exponents characterizing the transition are obtained in a heterogeneous mean field (HMF) approximation and compared with extensive simulations, particularly in the case of heterogeneous scale-free networks. We observe that the TCP exhibits the same critical properties as the contact process (CP), which undergoes an absorbing-state phase transition to a single absorbing state. The accordance among the critical exponents of different models and networks leads to conjecture that the critical behavior of the contact process in a HMF theory is a universal feature of absorbing state phase transitions in complex networks, depending only on the locality of the interactions and independent of the number of absorbing states. The conditions for the applicability of the conjecture are discussed considering a parallel with the susceptible-infected-susceptible epidemic spreading model, which in fact belongs to a different universality class in complex 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":"Absorbing state phase transitions with infinitely many absorbing states in complex networks","attachmentId":52479209,"attachmentType":"pdf","work_url":"https://www.academia.edu/32258135/Absorbing_state_phase_transitions_with_infinitely_many_absorbing_states_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/32258135/Absorbing_state_phase_transitions_with_infinitely_many_absorbing_states_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="3" data-entity-id="86077409" 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/86077409/Phase_transitions_in_the_quadratic_contact_process_on_complex_networks">Phase transitions in the quadratic contact process 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="133790086" href="https://independent.academia.edu/ChrisVarghese6">Chris Varghese</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2013</p><p class="ds-related-work--abstract ds2-5-body-sm">The quadratic contact process (QCP) is a natural extension of the well studied linear contact process where infected (1) individuals infect susceptible (0) neighbors at rate λ and infected individuals recover (1 −→ 0) at rate 1. In the QCP, a combination of two 1's is required to effect a 0 −→ 1 change. We extend the study of the QCP, which so far has been limited to lattices, to complex networks. We define two versions of the QCP-vertex centered (VQCP) and edge centered (EQCP) with birth events 1 − 0 − 1 −→ 1 − 1 − 1 and 1 − 1 − 0 −→ 1 − 1 − 1 respectively, where '−' represents an edge. We investigate the effects of network topology by considering the QCP on random regular, Erdős-Rényi and power law random graphs. We perform mean field calculations as well as simulations to find the steady state fraction of occupied vertices as a function of the birth rate. We find that on the random regular and Erdős-Rényi graphs, there is a discontinuous phase transition with a region of bistability, whereas on the heavy tailed power law graph, the transition is continuous. The critical birth rate is found to be positive in the former but zero in the latter.</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":"Phase transitions in the quadratic contact process on complex networks","attachmentId":90610847,"attachmentType":"pdf","work_url":"https://www.academia.edu/86077409/Phase_transitions_in_the_quadratic_contact_process_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/86077409/Phase_transitions_in_the_quadratic_contact_process_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="79741222" 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/79741222/Self_organized_criticality_as_an_absorbing_state_phase_transition">Self-organized criticality as an absorbing-state phase transition</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="51513548" href="https://independent.academia.edu/ronalddickman">ronald dickman</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 1998</p><p class="ds-related-work--abstract ds2-5-body-sm">We explore the connection between self-organized criticality and phase transitions in models with absorbing states. Sandpile models are found to exhibit criticality only when a pair of relevant parameters-dissipation ǫ and driving field hare set to their critical values. The critical values of ǫ and h are both equal to zero. The first result is due to the absence of saturation (no bound on energy) in the sandpile model, while the second result is common to other absorbing-state transitions. The original definition of the sandpile model places it at the point (ǫ = 0, h = 0 +): it is critical by definition. We argue power-law avalanche distributions are a general feature of models with infinitely many absorbing configurations, when they are subject to slow driving at the critical point. Our assertions are supported by simulations of the sandpile at ǫ = h = 0 and fixed energy density ζ (no drive, periodic boundaries), and of the slowly-driven pair contact process. We formulate a field theory for the sandpile model, in which the order parameter is coupled to a conserved energy density, which plays the role of an effective creation rate.</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":"Self-organized criticality as an absorbing-state phase transition","attachmentId":86353584,"attachmentType":"pdf","work_url":"https://www.academia.edu/79741222/Self_organized_criticality_as_an_absorbing_state_phase_transition","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/79741222/Self_organized_criticality_as_an_absorbing_state_phase_transition"><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="120358925" 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/120358925/Hybrid_phase_transition_into_an_absorbing_state_Percolation_and_avalanches">Hybrid phase transition into an absorbing state: Percolation and avalanches</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="22576039" href="https://ceu.academia.edu/JanosKertesz">Janos Kertesz</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2016</p><p class="ds-related-work--abstract ds2-5-body-sm">Interdependent networks are more fragile under random attacks than simplex networks, because interlayer dependencies lead to cascading failures and finally to a sudden collapse. This is a hybrid phase transition (HPT), meaning that at the transition point the order parameter has a jump but there are also critical phenomena related to it. Here we study these phenomena on the Erdős-Rényi and the two dimensional interdependent networks and show that the hybrid percolation transition exhibits two kinds of critical behaviors: divergence of the fluctuations of the order parameter and power-law size distribution of finite avalanches at a transition point. At the transition point global or "infinite" avalanches occur while the finite ones have a power law size distribution; thus the avalanche statistics also has the nature of a HPT. The exponent βm of the order parameter is 1/2 under general conditions, while the value of the exponent γm characterizing the fluctuations of the order parameter depends on the system. The critical behavior of the finite avalanches can be described by another set of exponents, βa and γa. These two critical behaviors are coupled by a scaling law: 1 − βm = γa.</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":"Hybrid phase transition into an absorbing state: Percolation and avalanches","attachmentId":115535555,"attachmentType":"pdf","work_url":"https://www.academia.edu/120358925/Hybrid_phase_transition_into_an_absorbing_state_Percolation_and_avalanches","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/120358925/Hybrid_phase_transition_into_an_absorbing_state_Percolation_and_avalanches"><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="59162574" 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/59162574/Critical_behavior_of_a_one_dimensional_diffusive_epidemic_process">Critical behavior of a one-dimensional diffusive epidemic process</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="65671377" href="https://independent.academia.edu/GezaOdor">Geza Odor</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2001</p><p class="ds-related-work--abstract ds2-5-body-sm">The phase transition of the one-dimensional diffusive pair contact process is investigated by N cluster mean-field approximations and high precision simulations. The Nϭ3,4 cluster approximations exhibit smooth transition line to absorbing state by varying the diffusion rate D with ␤ 2 ϭ2 mean-field order parameter exponent of the pair density. This contradicts with former Nϭ2 results, where two different mean-field behavior was found along the transition line. Extensive dynamical simulations on Lϭ10 5 lattices give estimates for the order parameter exponents of the particles for 0.05рDр0.7. These data may support former two distinct class findings. However, the gap between low-and high-D exponents is narrower than previously estimated and the possibility for interpreting numerical data as a single class behavior with exponents ␣ ϭ0.21(1), ␤ϭ0.40(1) assuming logarithmic corrections is shown. Finite-size scaling results are also presented.</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":"Critical behavior of a one-dimensional diffusive epidemic process","attachmentId":73228074,"attachmentType":"pdf","work_url":"https://www.academia.edu/59162574/Critical_behavior_of_a_one_dimensional_diffusive_epidemic_process","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/59162574/Critical_behavior_of_a_one_dimensional_diffusive_epidemic_process"><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="81981876" 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/81981876/Critical_behavior_in_reaction_diffusion_systems_exhibiting_absorbing_phase_transitions">Critical behavior in reaction-diffusion systems exhibiting absorbing phase transitions</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="65671377" href="https://independent.academia.edu/GezaOdor">Geza Odor</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Brazilian Journal of Physics, 2003</p><p class="ds-related-work--abstract ds2-5-body-sm">Phase transitions of reaction-diffusion systems with a site occupation restriction, particle creation requiring n > 2 parents, and in which explicit diffusion of single particles (A) is possible, are reviewed. Arguments based on mean-field approximation and simulations are given which support novel kind of nonequilibrium criticality. These are in contradiction with the implications of a suggested phenomenological, multiplicative noise Langevin equation approach and with some recent numerical analyses. Simulation results for one-and twodimensional binary spreading model, 2A → 4A, 4A → 2A, reveal a new type of mean-field criticality characterized by the critical exponents α = 1/3 and β = 1/2, as suggested in a recent preprint [cond-mat/0210615].</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":"Critical behavior in reaction-diffusion systems exhibiting absorbing phase transitions","attachmentId":87835796,"attachmentType":"pdf","work_url":"https://www.academia.edu/81981876/Critical_behavior_in_reaction_diffusion_systems_exhibiting_absorbing_phase_transitions","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/81981876/Critical_behavior_in_reaction_diffusion_systems_exhibiting_absorbing_phase_transitions"><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="88895925" 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/88895925/Universality_class_of_absorbing_transitions_with_continuously_varying_critical_exponents">Universality class of absorbing transitions with continuously varying critical exponents</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="65884543" href="https://independent.academia.edu/ParkHyunggyu">Hyunggyu Park</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2004</p><p class="ds-related-work--abstract ds2-5-body-sm">The well-established universality classes of absorbing critical phenomena are directed percolation (DP) and directed Ising (DI) classes. Recently, the pair contact process with diffusion (PCPD) has been investigated extensively and claimed to exhibit a new type of critical phenomena distinct from both DP and DI classes. Noticing that the PCPD possesses a long-term memory effect, we introduce a generalized version of the PCPD (GPCPD) with a parameter controlling the memory effect. The GPCPD connects the DP fixed point to the PCPD point continuously. Monte Carlo simulations show that the GPCPD displays novel type critical phenomena which are characterized by continuously varying critical exponents. The same critical behaviors are also observed in models where two species of particles are coupled cyclically. We suggest that the long-term memory may serve as a marginal perturbation to the ordinary DP fixed point.</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":"Universality class of absorbing transitions with continuously varying critical exponents","attachmentId":92792919,"attachmentType":"pdf","work_url":"https://www.academia.edu/88895925/Universality_class_of_absorbing_transitions_with_continuously_varying_critical_exponents","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/88895925/Universality_class_of_absorbing_transitions_with_continuously_varying_critical_exponents"><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="14719460" 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/14719460/Critical_behavior_of_the_contact_process_in_a_multiscale_network">Critical behavior of the contact process in a multiscale network</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="33676389" href="https://ufv-br.academia.edu/SilvioFerreira">Silvio Ferreira</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2007</p><p class="ds-related-work--abstract ds2-5-body-sm">Inspired by dengue and yellow fever epidemics, we investigated the contact process (CP) in a multiscale network constituted by one-dimensional chains connected through a Barabási-Albert scale-free network. In addition to the CP dynamics inside the chains, the exchange of individuals between connected chains (travels) occurs at a constant rate. A finite epidemic threshold and an epidemic mean lifetime diverging exponentially in the subcritical phase, concomitantly with a power law divergence of the outbreak's duration, were found. A generalized scaling function involving both regular and SF components was proposed for the quasistationary analysis and the associated critical exponents determined, demonstrating that the CP on this hybrid network and nonvanishing travel rates establishes a new universality class.</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":"Critical behavior of the contact process in a multiscale network","attachmentId":43944202,"attachmentType":"pdf","work_url":"https://www.academia.edu/14719460/Critical_behavior_of_the_contact_process_in_a_multiscale_network","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/14719460/Critical_behavior_of_the_contact_process_in_a_multiscale_network"><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":50626382,"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":50626382,"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_50626382" 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="122151662" 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/122151662/Phase_Diagram_of_the_Contact_Process_on_Barabasi_Albert_Networks">Phase Diagram of the Contact Process on Barabasi-Albert 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="259186772" href="https://independent.academia.edu/AntoniodeMacedoFilho">Antonio de Macedo Filho</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":"Phase Diagram of the Contact Process on Barabasi-Albert Networks","attachmentId":116874208,"attachmentType":"pdf","work_url":"https://www.academia.edu/122151662/Phase_Diagram_of_the_Contact_Process_on_Barabasi_Albert_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/122151662/Phase_Diagram_of_the_Contact_Process_on_Barabasi_Albert_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="25456029" 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/25456029/Critical_Phenomena_and_Diffusion_In_Complex_Systems">Critical Phenomena and Diffusion In Complex Systems</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="48958377" href="https://independent.academia.edu/AlexanderDubkov">Alexander Dubkov</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Arxiv preprint arXiv:0810.1486, 2008</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":"Critical Phenomena and Diffusion In Complex Systems","attachmentId":45773810,"attachmentType":"pdf","work_url":"https://www.academia.edu/25456029/Critical_Phenomena_and_Diffusion_In_Complex_Systems","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/25456029/Critical_Phenomena_and_Diffusion_In_Complex_Systems"><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="79000382" 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/79000382/Criticality_of_a_contact_process_with_coupled_diffusive_and_nondiffusive_fields">Criticality of a contact process with coupled diffusive and nondiffusive fields</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="22811599" href="https://ufal.academia.edu/IramGl%C3%A9ria">Iram Gléria</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2007</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":"Criticality of a contact process with coupled diffusive and nondiffusive fields","attachmentId":85867731,"attachmentType":"pdf","work_url":"https://www.academia.edu/79000382/Criticality_of_a_contact_process_with_coupled_diffusive_and_nondiffusive_fields","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/79000382/Criticality_of_a_contact_process_with_coupled_diffusive_and_nondiffusive_fields"><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="21773440" 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/21773440/Finite_size_scaling_and_universality_at_non_equilibrium_phase_transitions_revisited">Finite-size scaling and universality at non-equilibrium phase transitions revisited</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="43055775" href="https://jinr.academia.edu/JordanBrankov">Jordan Brankov</a><span>, </span><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="42986670" href="https://independent.academia.edu/NadezhdaBunzarova">Nadezhda Bunzarova</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":"Finite-size scaling and universality at non-equilibrium phase transitions revisited","attachmentId":42541279,"attachmentType":"pdf","work_url":"https://www.academia.edu/21773440/Finite_size_scaling_and_universality_at_non_equilibrium_phase_transitions_revisited","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span 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