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

An exact solution of the model for both site and bond percolation is given, including the position of thePercolation transition at which epidemic behavior sets in, the values of the critical exponents governing this transition, the mean and variance of the distribution of cluster sizes below the transition, and the size of the giant component (epidemic) above the transition.

Abstract

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 disease, and the transmissibility of the disease respectively. We give an exact solution of the model for both site and bond percolation, including the position of the percolation transition at which epidemic behavior sets in, the values of the critical exponents governing this transition, the mean and variance of the distribution of cluster sizes (disease outbreaks) below the transition, and the size of the giant component (epidemic) above the transition.