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

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 Gómez-Gardeñes 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 $\lambda \to 0$. Here we show that $\alpha -1\le \beta \le 2\alpha -3$, 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 $\beta =1$.

Published in at http://dx.doi.org/10.1214/09-AOP471 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Spreading of sexually transmitted diseases in heterosexual populations
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