The authors consider the contact process with infection rate \(\lambda\) on a random graph with \(n\) vertices and power law degree distributions. Based on mean field calculations, physicists seem to regard as an established fact that the critical value \(\lambda _c\) of the infection rate is positive if the power \(\alpha\) is larger than three; indeed, this result has recently been generalized to bipartite graphs by J. Gómez-Gardeñes et al. [Proc. Natl. Acad. Sci. USA 105, 1399–1404 (2008)]. However, it is shown in the present paper that the critical value \(\lambda _c\) is zero for any value of \(\alpha>3\). In addition, the contact process starting from all vertices infected, with a probability tending to one as \(n\) goes to infinity, maintains a positive density of infected sites for time at least exp\((n^{1-\delta})\) for any \(\delta>0\).
Thanks to the last result and the contact process duality, the authors also establish the existence of a quasi-stationary distribution in which a randomly chosen vertex is occupied with probability \(\rho (\lambda)\). One expects that \(\rho(\lambda)\sim C\lambda^\beta \) as \(\lambda\) tends to zero. The authors show that \(\alpha-1\leq\beta\leq 2\alpha-3\), so that in particular \(\beta\) is larger than two for \(\alpha\) lager than three. As a result, even if the graph is locally tree-like, \(\beta\) does not take the mean field critical value \(\beta=1\).