Replicator dynamics on heterogeneous networks

Networked evolutionary game theory is a well-established framework for modeling the evolution of social behavior in structured populations. Most of the existing studies in this field have focused on 2-strategy games on heterogeneous networks or n-strategy games on regular networks. In this paper, we consider n-strategy games on arbitrary networks under the pairwise comparison updating rule. We show that under the limit of weak selection, the short-run behavior of the stochastic evolutionary process can be approximated by replicator equations with a transformed payoff matrix that involves both the average value and the variance of the degree distribution. In particular, strongly heterogeneous networks can facilitate the evolution of the payoff-dominant strategy. We then apply our results to analyze the evolutionarily stable strategies in an n-strategy minimum-effort game and two variants of the prisoner's dilemma game. We show that the cooperative equilibrium becomes evolutionarily stable when the average degree of the network is low and the variance of the degree distribution is high. Agent-based simulations on quasi-regular, exponential, and scale-free networks confirm that the dynamic behaviors of the stochastic evolutionary process can be well approximated by the trajectories of the replicator equations.