Exact solution of site and bond percolation on small-world networks

We study percolation on small-world networks, which has been proposed as a simple model of the propagation of disease. The occupation probabilities of sites and bonds correspond to the susceptibility of individuals to the dis ease, and the transmissibility of the disease respectively. We give an exac t solution of the model for both site and bond percolation, including the p osition of the percolation transition at which epidemic behavior sets in, t he values of the critical exponents governing this transition, the mean and variance of the distribution of cluster sizes (disease outbreaks) below th e transition, and the size of the giant component (epidemic) above the tran sition.