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{"work":{"id":86077409,"created_at":"2022-09-03T05:32:17.985-07:00","from_world_paper_id":214278478,"updated_at":"2024-11-25T13:22:13.270-08:00","_data":{"publisher":"American Physical Society (APS)","grobid_abstract":"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.","publication_date":"2013,,","publication_name":"Physical Review E","grobid_abstract_attachment_id":"90610847"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Phase transitions in the quadratic contact process on complex networks","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [133790086]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "control"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":90610847,"attachmentType":"pdf"}"><img alt="First page of “Phase transitions in the quadratic contact process on complex networks”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/90610847/mini_magick20220903-1-1fe0847.png?1662210584" /><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">Phase transitions in the quadratic contact process on complex networks</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="133790086" href="https://independent.academia.edu/ChrisVarghese6"><img alt="Profile image of Chris Varghese" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Chris Varghese</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2013, Physical Review E</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">19 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 = 86077409; 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if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">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. 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Studying the role of quenched disorder, both intrinsic to nodes and topological, is a key challenge. With this in mind, here we analyze the contact process, i.e. the simplest model for propagation phenomena, with node-dependent infection rates (i.e. intrinsic quenched disorder) on complex networks. We find Griffiths phases and other rare region effects, leading rather generically to anomalously slow (algebraic, logarithmic, etc. ) relaxation, on Erdős-Rényi networks. We predict similar effects to exist for other topologies as long as a non-vanishing percolation threshold exists. More strikingly, we find that Griffiths phases can also emerge -even with constant epidemic ratesas a consequence of mere topological heterogeneity. In particular, we find Griffiths phases in finite dimensional networks as, for instance, a family of generalized small-world networks. These results have a broad spectrum of implications for propagation phenomena and other dynamical processes on networks, and are relevant for the analysis of both models and empirical data.</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":"Rare-region effects in the contact process on networks","attachmentId":47068545,"attachmentType":"pdf","work_url":"https://www.academia.edu/10862899/Rare_region_effects_in_the_contact_process_on_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/10862899/Rare_region_effects_in_the_contact_process_on_networks"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="1" data-entity-id="59161708" 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/59161708/Slow_dynamics_of_the_contact_process_on_complex_networks">Slow dynamics of the 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="65671377" href="https://independent.academia.edu/GezaOdor">Geza Odor</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2013</p><p class="ds-related-work--abstract ds2-5-body-sm">The Contact Process has been studied on complex networks exhibiting different kinds of quenched disorder. Numerical evidence is found for Griffiths phases and other rare region effects, in Erdős Rényi networks, leading rather generically to anomalously slow (algebraic, logarithmic,...) relaxation. More surprisingly, it turns out that Griffiths phases can also emerge in the absence of quenched disorder, as a consequence of sole topological heterogeneity in networks with finite topological dimension. In case of scale-free networks, exhibiting infinite topological dimension, slow dynamics can be observed on tree-like structures and a superimposed weight pattern. In the infinite size limit the correlated subspaces of vertices seem to cause a smeared phase transition. These results have a broad spectrum of implications for propagation phenomena and other dynamical process on networks and are relevant for the analysis of both models and empirical data.</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":"Slow dynamics of the contact process on complex networks","attachmentId":73227678,"attachmentType":"pdf","work_url":"https://www.academia.edu/59161708/Slow_dynamics_of_the_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/59161708/Slow_dynamics_of_the_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="2" data-entity-id="30169127" 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/30169127/Multiple_phase_transitions_of_the_susceptible_infected_susceptible_epidemic_model_on_complex_networks">Multiple phase transitions of the susceptible-infected-susceptible epidemic model 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="15839788" href="https://dwcc.academia.edu/AngelicaMata">Angelica Mata</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2014</p><p class="ds-related-work--abstract ds2-5-body-sm">The epidemic threshold of the susceptible-infected-susceptible (SIS) dynamics on random networks having a power law degree distribution with exponent $\gamma>3$ has been investigated using different mean-field approaches, which predict different outcomes. We performed extensive simulations in the quasistationary state for a comparison with these mean-field theories. We observed concomitant multiple transitions in individual networks presenting large gaps in the degree distribution and the obtained multiple epidemic thresholds are well described by different mean-field theories. We observed that the transitions involving thresholds which vanishes at the thermodynamic limit involve localized states, in which a vanishing fraction of the network effectively contribute to epidemic activity, whereas an endemic state, with a finite density of infected vertices, occurs at a finite threshold. The multiple transitions are related to the activations of distinct sub-domains of the network, which are not directly connected.</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 phase transitions of the susceptible-infected-susceptible epidemic model on complex networks","attachmentId":50626361,"attachmentType":"pdf","work_url":"https://www.academia.edu/30169127/Multiple_phase_transitions_of_the_susceptible_infected_susceptible_epidemic_model_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/30169127/Multiple_phase_transitions_of_the_susceptible_infected_susceptible_epidemic_model_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="3" data-entity-id="30169123" 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/30169123/Multiple_transitions_of_the_susceptible_infected_susceptible_epidemic_model_on_complex_networks">Multiple transitions of the susceptible-infected-susceptible epidemic model 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="15839788" href="https://dwcc.academia.edu/AngelicaMata">Angelica Mata</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":"Multiple transitions of the susceptible-infected-susceptible epidemic model on complex networks","attachmentId":50626383,"attachmentType":"pdf","work_url":"https://www.academia.edu/30169123/Multiple_transitions_of_the_susceptible_infected_susceptible_epidemic_model_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/30169123/Multiple_transitions_of_the_susceptible_infected_susceptible_epidemic_model_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="77264222" 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/77264222/Random_Graph_Dynamics">Random Graph Dynamics</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="42322807" href="https://duke.academia.edu/RDurrett">R. Durrett</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2006</p><p class="ds-related-work--abstract ds2-5-body-sm">Erdös-Renyi Random Graphs The first random graph model was introduced by Erdös and Rényi in the late 1950's. To define the model, we begin with the set of vertices V = {1, 2,. .. n}. For 1 ≤ x < y ≤ n let η x,y be independent = 1 with probability p = λ/n and 0 otherwise. Let η y ,x = η x,y. If η x,y = 1 there is an edge from x to y. A large Erdös-Renyi random graph has a degree distribution that is Poisson with mean λ. However in many technological and social networks, the degree distribution p k follows a power law: p k ∼ Ck −α. Rick Durrett (Cornell) Random Graph Dynamics 2 / Figure: Sweden sex partners follow power law Rick Durrett (Cornell) Random Graph Dynamics 3 / 109 Fixed Degree Distributions Molloy and Reed (1995) were the first to construct graphs with specified degree distributions. We will use the approach of Newman, Strogatz, and Watts (2001, 2002) to define the model. Let d 1 ,. .. d n be independent and have P(d i = k) = p k. Since we want d i to be the degree of vertex i, we condition on E n = {d 1 + • • • + d n is even}. If the probability P(E 1) ∈ (0, 1) then P(E n) → 1/2 as n → ∞ so the conditioning will have little effect on the finite dimensional distributions. Rick Durrett (Cornell) Random Graph Dynamics 4 / Attach d i half-edges to vertex i and then pair the half-edges at random. This can produce parallel edges or self-loops, but if Ed 2 i < ∞ then with probability bounded away from 0 we get an ordinary graph. Rick Durrett (Cornell) Random Graph Dynamics 6 / Contact Process Consider the contact process on a power-law random graph. In this model infected individuals become healthy at rate 1 (and are again susceptible to the disease) susceptible individuals become infected at a rate λ times the number of infected neighbors. Pastor-Satorras and Vespigniani (2001a, 2001b, 2002) have made an extensive study of this model using mean-field methods (See Section 4.8.). Rick Durrett (Cornell) Random Graph Dynamics 7 / 109 Contact Process Conjectures Let λ c be the critical value for prolonged persistence. If λ > λ c there will be a quasi-stationary distribution with density ρ(λ) ∼ C (λ − λ c) β If α ≤ 3 then λ c = 0. If 3 < α < 4, λ c > 0 but the critical exponent β > 1. If α > 4 then λ c > 0 and β = 1. Problem. Berger, Borgs, Chayes, Saberi (2005) prove persistence for time exp(cn 1/2) for any λ > 0 when α = 3. Does it last for exp(cn)? Rick Durrett (Cornell) Random Graph Dynamics 8 / Voter models Vertex voter model. Each vertex x changes at rate 1. Pick a neighbor at random and set ξ(x) = ξ(y). Genealogical process jumps at rate 1. Stationary distribution π(x) = cd(x). Edge voter model. Each edge becomes active at rate 1. Flip a coin to give it an orientation (x, y) then set ξ t (x) = ξ t (y). Genealogical process of a site is a random walk that jumps to a randomly chosen neighbor at rate d(x). Stationary distribution is uniform. Rick Durrett (Cornell) Random Graph Dynamics 9 / 109 Rick Durrett (Cornell) Random Graph Dynamics 11 / 109 Rick Durrett (Cornell) Random Graph Dynamics 17 / 109 Rick Durrett (Cornell) Random Graph Dynamics 23 / 109 Rick Durrett (Cornell) Random Graph Dynamics 29 / 109 Rick Durrett (Cornell) Random Graph Dynamics 65 / 109 Rick Durrett (Cornell) Random Graph Dynamics 83 / 109 Rick Durrett (Cornell) Random Graph Dynamics 89 / 109 Rick Durrett (Cornell) Random Graph Dynamics 95 / 109</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Random Graph Dynamics","attachmentId":84692867,"attachmentType":"pdf","work_url":"https://www.academia.edu/77264222/Random_Graph_Dynamics","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/77264222/Random_Graph_Dynamics"><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="811007" 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/811007/Epidemic_incidence_in_correlated_complex_networks">Epidemic incidence in correlated 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="132586" href="https://unizar.academia.edu/yamirmoreno">yamir moreno</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E, 2003</p><p class="ds-related-work--abstract ds2-5-body-sm">We introduce a numerical method to solve epidemic models on the underlying topology of complex networks. The approach exploits the mean-field like rate equations describing the system and allows to work with very large system sizes, where Monte Carlo simulations are useless due to memory needs. We then study the SIR epidemiological model on assortative networks, providing numerical evidence of the absence of epidemic thresholds. Besides, the time profiles of the populations are analyzed. Finally, we stress that the present method would allow to solve arbitrary epidemic-like models provided that they can be described by mean-field rate equations. 89.75.Fb, 05.70.Jk, 05.40.a A few years ago, Watts and Strogatz [1] introduced a model able to produce networks with properties of both regular lattices and random graphs with small diameter. Their model soon led to a burst of activity in the field [2, 3], further spurred by Barabasi and collaborators who found that many seemingly diverse systems share several topological properties such as a power law behavior in their connectivity distributions when represented as networks . These complex networks are formed by a set of many elements (or nodes) that are linked together through edges (or links) if they interact directly. Empirical evidence supports that in notable networks, such as metabolic or communication webs, the probability P (k) that any node has k links to other nodes is distributed accordingly to a power law P (k) ∼ k −γ [5, 6, 7], with γ ≤ 3 in most cases.</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":"Epidemic incidence in correlated complex networks","attachmentId":4860097,"attachmentType":"pdf","work_url":"https://www.academia.edu/811007/Epidemic_incidence_in_correlated_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/811007/Epidemic_incidence_in_correlated_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="6" data-entity-id="25564501" 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/25564501/Griffiths_effects_of_the_susceptible_infected_susceptible_epidemic_model_on_random_power_law_networks">Griffiths effects of the susceptible-infected-susceptible epidemic model on random power-law 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="14784369" href="https://ufv.academia.edu/WesleyCota">Wesley Cota</a><span>, </span><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--abstract ds2-5-body-sm">We provide numerical evidence for slow dynamics of the susceptible-infected-susceptible model evolving on finite-size random networks with power-law degree distributions. Extensive simulations were done by averaging the activity density over many realizations of networks. We investigated the effects of outliers in both highly fluctuating (natural cutoff) and non-fluctuating (hard cutoff) most connected vertices. Logarithmic and power-law decays in time were found for natural and hard cutoffs, respectively. This happens in extended regions of the control parameter space λ1 < λ < λ2, suggesting Griffiths effects, induced by the topological inhomogeneities. Optimal fluctuation theory considering sample-to-sample fluctuations of the pseudo thresholds is presented to explain the observed slow dynamics. A quasistationary analysis shows that response functions remain bounded at λ2. We argue these to be signals of a smeared transition. However, in the thermodynamic limit the Griffiths effects loose their relevancy and have a conventional critical point at λc = 0. Since many real networks are composed by heterogeneous and weakly connected modules, the slow dynamics found in our analysis of independent and finite networks can play an important role for the deeper understanding of such 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":"Griffiths effects of the susceptible-infected-susceptible epidemic model on random power-law networks","attachmentId":45894573,"attachmentType":"pdf","work_url":"https://www.academia.edu/25564501/Griffiths_effects_of_the_susceptible_infected_susceptible_epidemic_model_on_random_power_law_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/25564501/Griffiths_effects_of_the_susceptible_infected_susceptible_epidemic_model_on_random_power_law_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="93455322" 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/93455322/The_contact_process_on_scale_free_networks_evolving_by_vertex_updating">The contact process on scale-free networks evolving by vertex updating</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="115658820" href="https://independent.academia.edu/EmmanuelJacob19">Emmanuel Jacob</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Royal Society open science, 2017</p><p class="ds-related-work--abstract ds2-5-body-sm">We study the contact process on a class of evolving scale-free networks, where each node updates its connections at independent random times. We give a rigorous mathematical proof that there is a transition between a phase where for all infection rates the infection survives for a long time, at least exponential in the network size, and a phase where for sufficiently small infection rates extinction occurs quickly, at most polynomially in the network size. The phase transition occurs when the power-law exponent crosses the value four. This behaviour is in contrast with that of the contact process on the corresponding static model, where there is no phase transition, as well as that of a classical mean-field approximation, which has a phase transition at power-law exponent three. The new observation behind our result is that temporal variability of networks can simultaneously increase the rate at which the infection spreads in the network, and decrease the time at which the infection...</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 contact process on scale-free networks evolving by vertex updating","attachmentId":96189281,"attachmentType":"pdf","work_url":"https://www.academia.edu/93455322/The_contact_process_on_scale_free_networks_evolving_by_vertex_updating","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/93455322/The_contact_process_on_scale_free_networks_evolving_by_vertex_updating"><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="67216007" 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/67216007/Explicit_bounds_for_critical_infection_rates_and_expected_extinction_times_of_the_contact_process_on_finite_random_graphs">Explicit bounds for critical infection rates and expected extinction times of the contact process on finite random graphs</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="41342035" href="https://radboud.academia.edu/EricCator">Eric Cator</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv: Probability, 2018</p><p class="ds-related-work--abstract ds2-5-body-sm">We introduce a method to prove metastability of the contact process on Erdős-Renyi graphs and on configuration model graphs. The method relies on uniformly bounding the total infection rate from below, over all sets with a fixed number of nodes. Once this bound is established, a simple comparison with a well chosen birth-and-death process will show the exponential growth of the extinction time. Our paper complements recent results on the metastability of the contact process: under a certain minimal edge density condition, we give explicit lower bounds on the infection rate needed to get metastability, and we have explicit exponentially growing lower bounds on the expected extinction time.</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":"Explicit bounds for critical infection rates and expected extinction times of the contact process on finite random graphs","attachmentId":78116253,"attachmentType":"pdf","work_url":"https://www.academia.edu/67216007/Explicit_bounds_for_critical_infection_rates_and_expected_extinction_times_of_the_contact_process_on_finite_random_graphs","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/67216007/Explicit_bounds_for_critical_infection_rates_and_expected_extinction_times_of_the_contact_process_on_finite_random_graphs"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="100176777" 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/100176777/Critical_avalanches_of_susceptible_infected_susceptible_dynamics_in_finite_networks">Critical avalanches of susceptible-infected-susceptible dynamics in finite 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="187493871" href="https://independent.academia.edu/Notarmuzi">Daniele Notarmuzi</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Physical Review E</p><p class="ds-related-work--abstract ds2-5-body-sm">We investigate the avalanche temporal statistics of the Susceptible-Infected-Susceptible (SIS) model when the dynamics is critical and takes place on finite random networks. By considering numerical simulations on annealed topologies we show that the survival probability always exhibits three distinct dynamical regimes. Size-dependent crossover timescales separating them scale differently for homogeneous and for heterogeneous networks. The phenomenology can be qualitatively understood based on known features of the SIS dynamics on networks. A fully quantitative approach based on Langevin theory is shown to perfectly reproduce the results for homogeneous networks, while failing in the heterogeneous case. The analysis is extended to quenched random networks, which behave in agreement with the annealed case for strongly homogeneous and strongly heterogeneous 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":"Critical avalanches of susceptible-infected-susceptible dynamics in finite networks","attachmentId":101072964,"attachmentType":"pdf","work_url":"https://www.academia.edu/100176777/Critical_avalanches_of_susceptible_infected_susceptible_dynamics_in_finite_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/100176777/Critical_avalanches_of_susceptible_infected_susceptible_dynamics_in_finite_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":90610847,"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":90610847,"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_90610847" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. 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