Contact processes on random graphs with power law degree distributions have critical value 0

TL;DR

This paper proves that the critical infection rate for the contact process on power law random graphs is zero for all degree exponents greater than 3, contradicting previous mean field predictions and revealing complex infection dynamics.

Contribution

It establishes that the critical infection rate is zero for all >3, and characterizes the infection density near zero, challenging prior mean field results.

Findings

Critical value =0 for >3.

Infected density () scales as ^{eta} with eta between  and 2.

Infection persists for exponentially long times in large graphs.

Abstract

If we consider the contact process with infection rate $\lambda$ on a random graph on $n$ vertices with power law degree distributions, mean field calculations suggest that the critical value $\lambda_c$ of the infection rate is positive if the power $\alpha>3$. Physicists seem to regard this as an established fact, since the result has recently been generalized to bipartite graphs by G\'{o}mez-Garde\~{n}es et al. [Proc. Natl. Acad. Sci. USA 105 (2008) 1399--1404]. Here, we show that the critical value $\lambda_c$ is zero for any value of $\alpha>3$, and the contact process starting from all vertices infected, with a probability tending to 1 as $n\to\infty$, maintains a positive density of infected sites for time at least $\exp(n^{1-\delta})$ for any $\delta>0$. Using the last result, together with the contact process duality, we can establish the existence of a quasi-stationary distribution in which a randomly chosen vertex is occupied with probability $\rho(\lambda)$. It is expected that $\rho(\lambda)\sim C\lambda^{\beta}$ as $\lambda \to0$. Here we show that $\alpha-1\le\beta\le2\alpha-3$, and so $\beta>2$ for $\alpha>3$. Thus even though the graph is locally tree-like, $\beta$ does not take the mean field critical value $\beta=1$.