We use the standardized variance $\left({\nu }_{\mathit{st}}\right)$ of the degree distribution of a random network as an analytic measure of its heterogeneity. We show that ${\nu }_{\mathit{st}}$ accurately predicts, quantitatively, the success of cooperators in an evolutionary prisoner’s dilemma. Moreover, we show how the generating functional expression for ${\nu }_{\mathit{st}}$ suggests an intrinsic interpretation for the heterogeneity of the network that helps explain local mechanisms through which cooperators thrive in heterogeneous populations. Finally, we give a simple relationship between ${\nu }_{\mathit{st}}$, the cooperation level, and the epidemic threshold of a random network that reveals an appealing connection between epidemic disease models and the evolutionary prisoner’s dilemma.

• Received 5 August 2008

DOI:https://doi.org/10.1103/PhysRevE.79.016107