Critical value asymptotics for the contact process on random graphs

Abstract

Recent progress in the study of the contact process [2] has verified that the extinction-survival threshold ${\lambda }_1$ on a Galton-Watson tree is strictly positive if and only if the offspring distribution $\xi$ has an exponential tail. In this paper, we derive the first-order asymptotics of ${\lambda }_1$ for the contact process on Galton-Watson trees and its corresponding analog for random graphs. In particular, if $\xi$ is appropriately concentrated around its mean, we demonstrate that ${\lambda }_1(\xi )\sim 1/\mathbb{E}\xi$ as $\mathbb{E}\xi \to \infty$, which matches with the known asymptotics on the $d$-regular trees. The same result for the short-long survival threshold on the Erdős-Rényi and other random graphs are shown as well.

57 pages, 5 figures