Evolutionary Stability and the Evolution of Cooperation on Heterogeneous Graphs

Recent studies have revealed that graph heterogeneity can considerably affect evolutionary processes and that it promotes the emergence and maintenance of cooperation in social dilemmas. In this paper, we analytically derive the evolutionary dynamics and the evolutionarily stable strategy (ESS) condition for $2\times 2$ games on heterogeneous graphs based on ‘pairwise comparison’ updating. Using pair approximation, we introduce a new state variable to measure the evolutionary process. In the limit of weak selection, we show that the evolutionary dynamics can be approximated as a replicator equation with a transformed payoff matrix, and the ESS condition depends on both the mean value and the variance of the degree distribution. These results are subsequently applied to the Prisoner’s Dilemma game and the Stag Hunt game. In both games, we find that the variance plays a determinant role in the evolution of cooperation: Cooperative strategy cannot evolve in regular graphs, but it is favored by natural selection in strongly heterogeneous graphs.