A network criterion for the success of cooperation in an evolutionary prisoner's dilemma, and a variation on Hamilton's rule

We show that the success of cooperation in an evolutionary prisoner's dilemma on a complex network can be predicted by a simple, quantitative network analysis using mean field parameters. The criterion is shown to be accurate on a wide variety of networks with degree distributions ranging from regular to Poisson to scale-free. The network analysis uses a parameter that allows for comparisons of networks with both similar, and distinct, topologies. Furthermore, we establish the criterion here as a natural network analogue of Hamilton's classical rule for kin selection, despite arising in an entirely different context. The result is a network-based evolutionary rule for altruism that parallels Hamilton's genetic rule.