Continuous time limits of the utterance selection model

The study of social networks, and in particular the spread of disease on networks, has attracted considerable recent attention in the physics community. In this paper, we show that a large class of standard epidemiological models, the so-called susceptible/infective/removed (SIR) models can be solved exactly on a wide variety of networks. In addition to the standard but unrealistic case of fixed infectiveness time and fixed and uncorrelated probability of transmission between all pairs of individuals, we solve cases in which times and probabilities are non-uniform and correlated. We also consider one simple case of an epidemic in a structured population, that of a sexually transmitted disease in a population divided into men and women. We confirm the correctness of our exact solutions with numerical simulations of SIR epidemics on networks.

We study the voter dynamics model on heterogeneous graphs. We exploit the non-conservation of the magnetization to characterize how consensus is reached on networks with different connectivity patterns. For a network of N sites with an arbitrary degree distribution, we show that the mean time to reach consensus T_N scales as N mu_1^2/mu_2, where mu_k is the kth moment of the degree distribution. For a power-law degree distribution n_k k^{-nu}, we thus find that T_N scales as N for nu>3, as N/ln N for nu=3, as N^{(2nu-4)/(nu-1)} for 2