Necessity of retaining spatial correlations in studying cooperative behavior in networked populations

The evolution of cooperation is studied within the context of an evolutionary snowdrift game on three special networks chosen to have the same uniform degree but with different extents of spatial correlations. The cooperative behaviors on the networks differ in where the phase transitions take place and the frequencies of cooperation in the mixed phase. With only the spatial correlations being different, the study allows us to gauge the accuracy of different theoretical approaches and shed light on the choice of theoretical approaches in handling spatial correlations. It is found that analyzing the last surviving patterns often provides an understanding in the transitions in the cooperative behavior. We also construct a theoretical framework to describe the dynamical process. For different approaches of incorporating the agents’ spatial correlations, the local configuration approximation captures all the features observed in numerical simulations, while the commonly used pair approximation is too crude for quantitative purposes in studying cooperation, especially on networks with more complicated spatial correlations.