The contact process on the complete graph with random vertex-dependent infection rates

TL;DR

This paper analyzes the contact process on a complete graph with vertex-dependent infection rates, establishing a phase transition at a critical infection rate and providing formulas for critical values and infection probabilities.

Contribution

It introduces a model with random vertex weights affecting infection rates and derives the critical infection rate and infection probabilities, extending previous mean-field analyses.

Findings

Existence of a phase transition at a positive critical infection rate

Explicit formula for the critical infection rate $\,\lambda_c$

Precise approximation of infection probability in the quasi-stationary distribution

Abstract

We study the contact process on the complete graph on $n$ vertices where the rate at which the infection travels along the edge connecting vertices $i$ and $j$ is equal to $ \lambda w_i w_j / n$ for some $\lambda >0$, where $w_i$ are i.i.d. vertex weights. We show that when $E[w_1^2] < \infty$ there is a phase transition at $\lambda_c > 0$ so that for $\lambda<\lambda_c$ the contact process dies out in logarithmic time, and for $\lambda>\lambda_c$ the contact process lives for an exponential amount of time. Moreover, we give a formula for $\lambda_c$ and when $\lambda>\lambda_c$ we are able to give precise approximations for the probability a given vertex is infected in the quasi-stationary distribution. Our results are consistent with a non-rigorous mean-field analysis of the model. This is in contrast to some recent results for the contact process on power law random graphs where the mean-field calculations suggested that $\lambda_c>0$ when in fact $\lambda_c = 0$.