  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.
