Metastability for the contact process on the preferential attachment graph

TL;DR

This paper analyzes the metastability of the contact process on preferential attachment graphs, providing sharp bounds on infection density near exponential times, confirming predictions about endemic behavior at small infection rates.

Contribution

It offers precise bounds on infection density over time, advancing understanding of contact process dynamics on complex networks.

Findings

Infection density remains high near exponential times.

Survival probability is polynomial in infection rate.

Results confirm metastability and endemic behavior predictions.

Abstract

We consider the contact process on the preferential attachment graph. The work of Berger, Borgs, Chayes and Saberi [BBCS1] confirmed physicists predictions that the contact process starting from a typical vertex becomes endemic for an arbitrarily small infection rate $\lambda$ with positive probability. More precisely, they showed that with probability $\lambda^{\Theta (1)}$, it survives for a time exponential in the largest degree. Here we obtain sharp bounds for the density of infected sites at a time close to exponential in the number of vertices (up to some logarithmic factor).