# Cumulative Merging Percolation and the epidemic transition of the Susceptible-Infected-Susceptible model in networks

## TL;DR

This paper introduces a generalized cumulative merging percolation process to analyze epidemic transitions in complex networks, providing a unified framework that reconciles previous conflicting results on epidemic thresholds.

## Contribution

It develops a scaling solution for CMP on uncorrelated networks and applies it to understand SIS epidemic thresholds, resolving discrepancies between analytical and numerical findings.

## Key findings

- Identifies diverse mechanisms for percolation transition.
- Derives a nontrivial scaling exponent during preasymptotic regime.
- Reconciles previous analytical and numerical results on epidemic thresholds.

## 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 (SIS) 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.