Viral processes by random walks on random regular graphs

TL;DR

This paper analyzes the spread of an epidemic modeled by random walks on random regular graphs, identifying thresholds for infection spread and providing infection timing insights based on edge weights.

Contribution

It introduces an edge-weighted graph reduction method to analyze epidemic thresholds and infection times in a random walk-based SIR model on regular graphs.

Findings

In the subcritical regime, about O(log k) particles are infected.

In the supercritical regime, a fraction β of particles are infected with probability β.

All particles can become infected in a certain regime.

Abstract

We study the SIR epidemic model with infections carried by $k$ particles making independent random walks on a random regular graph. Here we assume $k\leq n^{\epsilon}$, where $n$ is the number of vertices in the random graph, and $\epsilon$ is some sufficiently small constant. We give an edge-weighted graph reduction of the dynamics of the process that allows us to apply standard results of Erd\H{o}s-R\'{e}nyi random graphs on the particle set. In particular, we show how the parameters of the model give two thresholds: In the subcritical regime, $O(\ln k)$ particles are infected. In the supercritical regime, for a constant $\beta\in(0,1)$ determined by the parameters of the model, $\beta k$ get infected with probability $\beta$, and $O(\ln k)$ get infected with probability $(1-\beta)$. Finally, there is a regime in which all $k$ particles are infected. Furthermore, the edge weights give information about when a particle becomes infected. We exploit this to give a completion time of the process for the SI case.