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

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 $ λw_i w_j / n$ for some $λ>0$, where $w_i$ are i.i.d. vertex weights. We show that when $E[w_1^2] < \infty$ there is a phase transition at $λ_c > 0$ so that for $λ<λ_c$ the contact process dies out in logarithmic time, and for $λ>λ_c$ the contact process lives for an exponential amount of time. Moreover, we give a formula for $λ_c$ and when $λ>λ_c$ 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 $λ_c>0$ when in fact $λ_c = 0$.