Cumulative Merging Percolation and the Epidemic Transition of the Susceptible-Infected-Susceptible Model in Networks

We consider cumulative merging percolation (CMP), a long-range percolation process describing the iterative merging of clusters in networks, depending on their mass and mutual distance. For a specific class of CMP processes, which represents a generalization of degree-ordered percolation, we derive a scaling solution on uncorrelated complex networks, unveiling the existence of diverse mechanisms leading to the formation of a percolating cluster. The scaling solution accurately reproduces universal properties of the transition. This finding is used to infer the critical properties of the susceptible-infected-susceptible model for epidemics in infinite and finite power-law distributed networks. Here, discrepancies between analytical approaches and numerical results regarding the finite-size scaling of the epidemic threshold are a crucial open issue in the literature. We find that the scaling exponent assumes a nontrivial value during a long preasymptotic regime. We calculate this value, finding good agreement with numerical evidence. We also show that the crossover to the true asymptotic regime occurs for sizes much beyond currently feasible simulations. Our findings allow us to rationalize and reconcile all previously published results (both analytical and numerical), thus ending a long-standing debate.

[1]  Sergey N. Dorogovtsev,et al.  Localization and Spreading of Diseases in Complex Networks , 2012, Physical review letters.

[2]  Sergey N. Dorogovtsev,et al.  Critical phenomena in complex networks , 2007, ArXiv.

[3]  Romualdo Pastor-Satorras,et al.  Competing activation mechanisms in epidemics on networks , 2011, Scientific Reports.

[4]  P. Grassberger.  On the critical behavior of the general epidemic process and dynamical percolation , 1983 .

[5]  M. Newman.  Spread of epidemic disease on networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Alessandro Vespignani,et al.  Epidemic dynamics and endemic states in complex networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  R. Pastor-Satorras,et al.  Langevin approach for the dynamics of the contact process on annealed scale-free networks. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Romualdo Pastor-Satorras,et al.  Nature of the epidemic threshold for the susceptible-infected-susceptible dynamics in networks. , 2013, Physical review letters.

[9]  Mason A. Porter,et al.  Message-Passing Methods for Complex Contagions , 2017, ArXiv.

[10]  Alessandro Vespignani,et al.  Cut-offs and finite size effects in scale-free networks , 2003, cond-mat/0311650.

[11]  Mason A. Porter,et al.  Dynamical Systems on Networks: A Tutorial , 2014, ArXiv.

[12]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[13]  Thorsten Gerber,et al.  Handbook Of Mathematical Functions , 2016 .

[14]  P. Van Mieghem,et al.  Susceptible-infected-susceptible epidemics on the complete graph and the star graph: exact analysis. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Alessandro Vespignani,et al.  Epidemic spreading in scale-free networks. , 2000, Physical review letters.

[16]  Jean-Gabriel Young,et al.  Phase transition of the susceptible-infected-susceptible dynamics on time-varying configuration model networks. , 2017, Physical review. E.

[17]  Shankar Bhamidi,et al.  Survival and extinction of epidemics on random graphs with general degree , 2019, The Annals of Probability.

[18]  A. Roddam.  Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation O Diekmann and JAP Heesterbeek, 2000, Chichester: John Wiley pp. 303, £39.95. ISBN 0-471-49241-8 , 2001 .

[19]  Claudio Castellano,et al.  Thresholds for epidemic spreading in networks , 2010, Physical review letters.

[20]  Brian Karrer,et al.  Message passing approach for general epidemic models. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  J. Robins,et al.  Second look at the spread of epidemics on networks. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Qiang Yao,et al.  Metastable densities for the contact process on power law random graphs , 2013 .

[23]  R. Durrett,et al.  The Contact Process on Random Graphs and Galton Watson Trees , 2018, Latin American Journal of Probability and Mathematical Statistics.

[24]  O. Diekmann,et al.  Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation , 2000 .

[25]  Pyoung-Seop Shim,et al.  Epidemic threshold of the susceptible-infected-susceptible model on complex networks. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Yamir Moreno,et al.  Fundamentals of spreading processes in single and multilayer complex networks , 2018, Physics Reports.

[27]  Neil Genzlinger.  A. and Q , 2006 .

[28]  M. Serrano,et al.  Percolation and epidemic thresholds in clustered networks. , 2006, Physical review letters.

[29]  Angélica S. Mata,et al.  Pair quenched mean-field theory for the susceptible-infected-susceptible model on complex networks , 2013, 1305.5153.

[30]  P. Van Mieghem,et al.  Virus Spread in Networks , 2009, IEEE/ACM Transactions on Networking.

[31]  Christos Faloutsos,et al.  Epidemic spreading in real networks: an eigenvalue viewpoint , 2003, 22nd International Symposium on Reliable Distributed Systems, 2003. Proceedings..

[32]  Laurent M'enard,et al.  Percolation by cumulative merging and phase transition for the contact process on random graphs , 2015, 1502.06982.

[33]  Joel C. Miller,et al.  Mathematics of Epidemics on Networks: From Exact to Approximate Models , 2017 .

[34]  H. Poincaré,et al.  Percolation ? , 1982 .

[35]  Qiang Yao,et al.  Exponential extinction time of the contact process on finite graphs , 2012, 1203.2972.

[36]  Mark Newman,et al.  Dynamical systems on networks , 2018, Oxford Scholarship Online.

[37]  R. Pastor-Satorras,et al.  Generation of uncorrelated random scale-free networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  Piet Van Mieghem,et al.  Epidemic processes in complex networks , 2014, ArXiv.

[39]  R. Durrett,et al.  Contact processes on random graphs with power law degree distributions have critical value 0 , 2009, 0912.1699.

[40]  Romualdo Pastor-Satorras,et al.  Lifespan method as a tool to study criticality in absorbing-state phase transitions. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[41]  Reuven Cohen,et al.  Stability and topology of scale-free networks under attack and defense strategies. , 2005, Physical review letters.

[42]  R Pastor-Satorras,et al.  Dynamical and correlation properties of the internet. , 2001, Physical review letters.

[43]  J. Borge-Holthoefer,et al.  Discrete-time Markov chain approach to contact-based disease spreading in complex networks , 2009, 0907.1313.

[44]  Yong-Yeol Ahn.  Complex Spreading Phenomena in Social Systems , 2018 .

[45]  J. Gleeson.  High-accuracy approximation of binary-state dynamics on networks. , 2011, Physical review letters.