Phase transitions for contact processes on sparse random graphs via metastability and local limits

TL;DR

This paper studies phase transitions in contact processes on sparse random graphs, using local limits and metastability to characterize thresholds for extinction and survival, especially on scale-free networks.

Contribution

It introduces a new local-convergence based approach to analyze phase transitions and thresholds in contact processes on sparse graphs, linking finite graph behavior to local limits.

Findings

Metastable density characterizes phase boundary.

Thresholds for extinction and survival coincide under certain conditions.

Fast extinction can occur on stretched exponential time scales.

Abstract

We propose a new perspective on the asymptotic regimes of fast and slow extinction in the contact process on locally converging sequences of sparse finite graphs. We characterise the phase boundary by the existence of a metastable density, which makes the study of the phase transition particularly amenable to local-convergence techniques. We use this approach to derive general conditions for the coincidence of the critical threshold with the survival/extinction threshold in the local limit. We further argue that the correct time scale to separate fast extinction from slow extinction in sparse graphs is, in general, the exponential scale, by showing that fast extinction may occur on stretched exponential time scales in sparse scale-free spatial networks. Together with recent results by Nam, Nguyen and Sly (Trans. Am. Math. Soc. 375, 2022), our methods can be applied to deduce that the fast/slow threshold in sparse configuration models coincides with the survival/extinction threshold on the limiting Galton-Watson tree.