Extended Pair Approximation of Evolutionary Game on Complex Networks

Evolutionary games on networks have recently attracted attention in evolutionary biology, behavioral science and statistical physics. In particular, an intensively studied question is how network structure influences the evolution of cooperative behavior in a dilemma situation (for regular lattices1)–6) and complex networks7)–14)). Nowak and May observed that cooperation behavior can be enhanced in the prisoner’s dilemma game on a lattice network.2) Contrastingly, Hauert and Doebeli found that network structure often inhibits cooperation behavior in the snowdrift game.6) Because most of these papers are based on numerical simulations, it is not clear how the network architecture affects the evolution in general cases. The purpose of this study is to establish a theoretical formula describing the network effect in evolutionary game theory. In this study, we focus on the influence of the average degree, degree fluctuations and clustering structure on the asymptotic result. To analyze these effects, we apply the pair approximation technique. The pair approximation is a useful tool for analysis of model ecosystems, because it can predict the population dynamics more accurately than the mean-field approximation.15)–18) The procedure for the pair approximation is often so complicated that it is carried out by numerical calculations. Here, to obtain an analytical solution, we propose novel procedures for the pair approximation.