Mixing time for an epidemic model on graphs with external sources of infection

TL;DR

This paper analyzes the mixing time of a noisy SIS epidemic model on various graphs, showing it is typically of order n log n, with detailed results on random graph families.

Contribution

It establishes the order of mixing time for the noisy SIS model on different graph classes, including Erdős–Rényi and random regular graphs, under typical structural conditions.

Findings

Mixing time is Θ(n log n) for the noisy SIS model on graphs.

High-probability structural properties ensure mixing time remains Θ(n log n).

Results apply to Erdős–Rényi, random regular, and Galton–Watson trees.

Abstract

We study the mixing time of a Susceptible--Infected--Susceptible (SIS) model on graphs with external sources of infection, which we refer to as the noisy SIS model. Under suitable assumptions on the parameters of the dynamics, we show that the mixing time is of the order $\Theta(n\log n)$ with respect to the number of vertices $n$. We further investigate the model on random graph families, including Erd{\"o}s--R{\'e}nyi graphs, random regular multigraphs, and Galton--Watson trees. By identifying high-probability structural properties of these graphs and conditioning on typical realizations, we prove that the mixing time remains of order $\Theta(n\log n)$ with high probability.