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

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. Summary Complex networks represent the interaction pattern for many real-world phenomena such as epidemic spreading. The simplest and most fundamental model for the diffusion of infectious diseases without acquired immunity predicts a vanishing epidemic threshold in the limit of large systems. In other words, no matter how small the infectiousness of the disease, there is always a finite fraction of the overall population which is infected for long times. So far, the detailed mechanism underlying this phenomenology has remained unclear. Here, we provide a complete and quantitative understanding of the model’s behavior. While the role of hubs (individuals with high connectivity) was already believed to be important, we fully clarify its nature, pointing out the highly nontrivial interplay among hubs. Each hub acts as an infection hotbed and the global epidemic emerges from hubs reinfecting each other. Surprisingly, this mechanism is at work even when the hubs are not in direct mutual contact but transmit the infection through chains of low-connectivity individuals. The survival of the epidemic is then a manifestation of a novel, long-range, percolation process among distant hubs, whose properties explain the behavior of the epidemic model. These findings rationalize and reconcile previously published results and point out long-range indirect interactions as potentially crucial ingredients for other collective phenomena in networked systems.

SUBMITTER: Castellano C 

PROVIDER: S-EPMC7226907 | biostudies-literature | 2020 Mar

REPOSITORIES: biostudies-literature