First-Passage Properties of Evolutionary Games in Structured Populations

Abstract

The statistics of the first time at which a fluctuating quantity reaches a given threshold is of great utility across a wide spectrum of science and engineering. We studied the fixation and extinction of cooperative alleles as a first-passage process in the context of social dilemma games in structured populations. In a variety of population graphs with controlled large-scale topological characteristics, we conducted social dilemma games to find that the heterogeneity in degree distribution is negatively correlated with the fixation probability whereas it accelerates the fixation once the process is conditioned on fixation. Furthermore, the degree-degree correlation also has negative impacts on fixation of cooperators. Despite the crucial role played by the seed cooperator for the eventual fixation of cooperators, the fixation time barely depends on the local characteristics, e.g. degree, of the seed. From the intrinsic symmetry between a seed cooperator in a sea of defectors and vice versa, we discuss the fixation or loss of cooperators in connection with the first-passage properties of epidemic spreading and consensus formation.