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      Equivalence and its invalidation between non-Markovian and Markovian spreading dynamics on complex networks

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          Abstract

          Epidemic spreading processes in the real world depend on human behaviors and, consequently, are typically non-Markovian in that the key events underlying the spreading dynamics cannot be described as a Poisson random process and the corresponding event time is not exponentially distributed. In contrast to Markovian type of spreading dynamics for which mathematical theories have been well developed, we lack a comprehensive framework to analyze and fully understand non-Markovian spreading processes. Here we develop a mean-field theory to address this challenge, and demonstrate that the theory enables accurate prediction of both the transient phase and the steady states of non-Markovian susceptible-infected-susceptible spreading dynamics on synthetic and empirical networks. We further find that the existence of equivalence between non-Markovian and Markovian spreading depends on a specific edge activation mechanism. In particular, when temporal correlations are absent on active edges, the equivalence can be expected; otherwise, an exact equivalence no longer holds.

          Abstract

          When modelling epidemic spreading on complex networks, one useful simplification is to assume that the dynamics are Markovian, i.e. memoryless. Here the authors present a more general non-Markovian approach which is able to accurately reproduce the transient and stationary regime on different substrates.

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          Most cited references49

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          Epidemic Spreading in Scale-Free Networks

          The Internet has a very complex connectivity recently modeled by the class of scale-free networks. This feature, which appears to be very efficient for a communications network, favors at the same time the spreading of computer viruses. We analyze real data from computer virus infections and find the average lifetime and persistence of viral strains on the Internet. We define a dynamical model for the spreading of infections on scale-free networks, finding the absence of an epidemic threshold and its associated critical behavior. This new epidemiological framework rationalizes data of computer viruses and could help in the understanding of other spreading phenomena on communication and social networks.
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            A universal model for mobility and migration patterns.

            Introduced in its contemporary form in 1946 (ref. 1), but with roots that go back to the eighteenth century, the gravity law is the prevailing framework with which to predict population movement, cargo shipping volume and inter-city phone calls, as well as bilateral trade flows between nations. Despite its widespread use, it relies on adjustable parameters that vary from region to region and suffers from known analytic inconsistencies. Here we introduce a stochastic process capturing local mobility decisions that helps us analytically derive commuting and mobility fluxes that require as input only information on the population distribution. The resulting radiation model predicts mobility patterns in good agreement with mobility and transport patterns observed in a wide range of phenomena, from long-term migration patterns to communication volume between different regions. Given its parameter-free nature, the model can be applied in areas where we lack previous mobility measurements, significantly improving the predictive accuracy of most of the phenomena affected by mobility and transport processes.
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              Epidemic dynamics and endemic states in complex networks.

              We study by analytical methods and large scale simulations a dynamical model for the spreading of epidemics in complex networks. In networks with exponentially bounded connectivity we recover the usual epidemic behavior with a threshold defining a critical point below that the infection prevalence is null. On the contrary, on a wide range of scale-free networks we observe the absence of an epidemic threshold and its associated critical behavior. This implies that scale-free networks are prone to the spreading and the persistence of infections whatever spreading rate the epidemic agents might possess. These results can help understanding computer virus epidemics and other spreading phenomena on communication and social networks.
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                Author and article information

                Contributors
                tangminghan007@gmail.com
                Journal
                Nat Commun
                Nat Commun
                Nature Communications
                Nature Publishing Group UK (London )
                2041-1723
                23 August 2019
                23 August 2019
                2019
                : 10
                : 3748
                Affiliations
                [1 ]ISNI 0000 0004 0369 6365, GRID grid.22069.3f, School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, , East China Normal University, ; Shanghai, 200241 China
                [2 ]ISNI 0000 0004 0369 4060, GRID grid.54549.39, Web Sciences Center, , University of Electronic Science and Technology of China, ; Chengdu, 611731 China
                [3 ]ISNI 0000 0004 0369 4060, GRID grid.54549.39, Big Data Research Center, , University of Electronic Science and Technology of China, ; Chengdu, 611731 China
                [4 ]ISNI 0000 0004 0369 6365, GRID grid.22069.3f, Shanghai Key Laboratory of Multidimensional Information Processing, , East China Normal University, ; Shanghai, 200241 China
                [5 ]ISNI 0000 0001 2151 2636, GRID grid.215654.1, School of Electrical, Computer and Energy Engineering, , Arizona State University, ; Tempe, AZ 85287 USA
                Author information
                http://orcid.org/0000-0001-7296-9193
                http://orcid.org/0000-0002-3017-7435
                http://orcid.org/0000-0002-0723-733X
                Article
                11763
                10.1038/s41467-019-11763-z
                6707263
                31444336
                937f8fd0-b991-4d78-ba33-a6e3e8f8a28e
                © The Author(s) 2019

                Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

                History
                : 9 January 2019
                : 30 July 2019
                Funding
                Funded by: FundRef https://doi.org/10.13039/501100001809, National Natural Science Foundation of China (National Science Foundation of China);
                Award ID: 11575041
                Award Recipient :
                Funded by: Natural Science Foundation of Shanghai under Grant No. 18ZR1412200, and the Science and Technology Commission of Shanghai Municipality under Grant No. 18dz2271000.
                Funded by: Vannevar Bush Faculty Fellowship program sponsored by the Basic Research Office of the Assistant Secretary of Defense for Research and Engineering and funded by the Office of Naval Research through Grant No. N00014-16-1-2828
                Categories
                Article
                Custom metadata
                © The Author(s) 2019

                Uncategorized
                complex networks,nonlinear phenomena,phase transitions and critical phenomena

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