Critical value asymptotics for the contact process on random graphs

TL;DR

This paper derives the first-order asymptotics of the critical infection rate for the contact process on Galton-Watson trees and random graphs, showing it behaves like the inverse of the mean offspring as the mean grows large.

Contribution

It provides the first asymptotic characterization of the contact process threshold on random graphs, extending known results from regular trees.

Findings

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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 $\mathbb{E}\xi\rightarrow \infty$, which matches with the known asymptotics on the $d$-regular trees. The same result for the short-long survival threshold on the Erd\H{o}s-R\'enyi and other random graphs are shown as well.