Noise-Induced Rare Events in Population Dynamics: The Effect of Spatial Heterogeneity

We study the influence of the network topology on the statistics of interest including rare events of interacting particle systems on complex graphs. As an example, we investigate the metastability and fixation properties of a set of evolutionary processes. In the framework of evolutionary game theory, where the fitness and selection are frequency dependent and vary with the population composition, we analyze the dynamics of snowdrift games (characterized by a metastable coexistence state) on scale-free networks. Using an effective diffusion theory in the weak selection limit, we demonstrate how the scale-free structure affects the system’s metastable state and leads to anomalous fixation compared to the non-spatial well-mixed case. In particular, we analytically and numerically show that the probability and mean time to fixation are characterized by stretched-exponential behaviors with exponents depending on the network’s degree distribution.