We investigate the contact process on random graphs generated from the configuration model for scale-free complex networks with the power law exponent β E （2, 3]. Using the neighborhood expansion method, we show that, with positive probability, any disease with an infection rate λ 〉 0 can survive for exponential time in the number of vertices of the graph. This strongly supports the view that stochastic scale-free networks are remarkably different from traditional regular graphs, such as, Z^d and classical Erdos-Renyi random graphs.
We prove that two independent continuous-time simple random walks on the infinite open cluster of a Bernoulli bond percolation in the lattice Z2 meet each other infinitely many times.An application to the voter model is also discussed.
