Models of information spread in structured populations

Abstract

This paper presents a birth-death model of diffusion processes on graphs, making use of the full population state space consisting of 2N binary valued vectors together with a Markov process on this space with transition matrix defined by the edge weight matrix of the given population graph. A set of master equations is derived that allows computation of fixation probabilities for any given initial distribution of new information. The transition matrix of the Markov process gives information about most likely initial states, and preferred starting states. A simple example illustrates the apparently paradoxical fact that some population structures allow enhancement of fixation probabilities relative to random drift only for limited values of fitness (or, e.g., rumor believability). In addition, an exact solution is given for complete bipartite graphs. Results obtained are compared to results obtained from a probabilistic voter model update scheme. In addition, the edge-weight matrix of the population graph defines a graph Laplacian that provides information as to increasing or decreasing polarization in a population and this is illustrated with simple examples.