Exponential extinction time of the contact process on finite graphs

Abstract

We study the extinction time $\uptau$ of the contact process on finite trees of bounded degree. We show that, if the infection rate is larger than the critical rate for the contact process on $\mathbb{Z}$, then, uniformly over all trees of degree bounded by a given number, the expectation of $\uptau$ grows exponentially with the number of vertices. Additionally, for any sequence of growing trees of bounded degree, $\uptau$ divided by its expectation converges in distribution to the unitary exponential distribution. These also hold if one considers a sequence of graphs having spanning trees with uniformly bounded degree. Using these results, we consider the contact process on a random graph with vertex degrees following a power law. Improving a result of Chatterjee and Durrett \cite{CD}, we show that, for any infection rate, the extinction time for the contact process on this graph grows exponentially with the number of vertices.

References in corpus (2)

• Random graphs with arbitrary degree distributions and their applications
• Metastable Densities for Contact Processes on Power Law Random Graphs

Cited by in corpus (5)

• Analysis and Control of Epidemics: A survey of spreading processes on complex networks
• Phase Transitions in the Quadratic Contact Process on Complex Networks
• The contact process on finite homogeneous trees revisited
• Phase transition of the contact process on random regular graphs
• A Metastability Result for the Contact Process on a Random Regular Graph