Subcritical epidemics on random graphs

Abstract

We study the contact process on random graphs with low infection rate λ. For random d-regular graphs, it is known that the survival time is $O(\log n)$ below the critical ${\lambda }_c$. By contrast, on the Erdős-Rényi random graphs $G(n,d/n)$, rare high-degree vertices result in much longer survival times. We show that the survival time is governed by high-density local configurations. In particular, we show that there is a long string of high-degree vertices on which the infection lasts for time $n^{{\lambda }^{2+o\left(1\right)}}$. To establish a matching upper bound, we introduce a modified version of the contact process which ignores infections that do not lead to further infections and allows for a sharper recursive analysis on branching process trees, the local-weak limit of the graph. Our methods, moreover, generalize to random graphs with given degree distributions that have exponential moments.