CONTACT PROCESSES ON RANDOM GRAPHS WITH POWER LAW DEGREE DISTRIBUTIONS HAVE CRITICAL VALUE 0

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 Gomez-Gardefies et al. [Proc. Nad. 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 -> infinity, 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 p(lambda) similar to C lambda(beta) as lambda -> 0. Here we show that alpha - 1 <= beta <= 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.