Abstract
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.
Highlights
Percolation and epidemic spreading are among the most interesting processes unfolding on complex network substrates, and their investigation has attracted a huge interest in the past 20 years [1,2,3,4,5]
For a specific class of cumulative merging percolation (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 theoretical picture presented in the previous sections can be applied to the CMP process associated to SIS dynamics, which is an instance of this class with ka 1⁄4 a=λ2 lnð1=λÞ, initial mass equal to the degree, and rðmÞ 1⁄4 m=ka; see Appendix A
Summary
Percolation and epidemic spreading are among the most interesting processes unfolding on complex network substrates, and their investigation has attracted a huge interest in the past 20 years [1,2,3,4,5]. It was later pointed out that, at the QMF level, the localization of activity around these hubs implies the existence, for small values of λ, of long-lived, but not stationary, states [22,23] Important progress in this debate is provided in Ref. For network sizes that can be currently simulated, a preasymptotic regime holds, whose nontrivial properties are determined, providing a prediction for the finite-size scaling of the SIS epidemic threshold in agreement with (previously unexplained) numerical results. Our work reconciles in a comprehensive way the different theories proposed to interpret the behavior of the SIS model, placing them in the proper context regarding the network size considered, and ends a long debate between the physics and mathematics communities. Several Appendixes provide some detailed analytical calculations and additional information
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