If we consider the contact process with infection rate λ on a random graph on n vertices with power law degree distributions, mean field calculations suggest that the critical value λc of the infection rate is positive if the power α>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 λc is zero for any value of α>3, and the contact process starting from all vertices infected, with a probability tending to 1 as n→∞, maintains a positive density of infected sites for time at least exp(n1−δ) for any δ>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 ρ(λ). It is expected that ρ(λ)∼Cλβ as λ→0. Here we show that α−1≤β≤2α−3, and so β>2 for α>3. Thus even though the graph is locally tree-like, β does not take the mean field critical value β=1.
"Contact processes on random graphs with power law degree distributions have critical value 0." Ann. Probab. 37 (6) 2332 - 2356, November 2009. https://doi.org/10.1214/09-AOP471