Structure-coefficient theorem for evolutionary games on dynamic networks.

Evolution unfolds in populations of reproducing individuals, with the dynamics shaped by numerous factors, including population structure, update rules, mutation rates, traits ('strategies') and the reproductive success derived from traits ('payoffs'). Developing a unifying principle to predict which traits will evolve and when, remains a major challenge. While rigorous mathematical results exist for static population structures, the general case of dynamic structures, where connections between individuals change over time, is more difficult and less understood. Here we present a structure-coefficient theorem for evolutionary games on dynamic networks with exogenous stochastic transitions. The framework enables arbitrary Markovian network dynamics independent of strategic interactions, together with arbitrary update mechanisms, mutation rates, strategy spaces and payoff matrices. We demonstrate that the combined effects of structure, update rules and mutation on evolutionary outcomes can be fully captured by three universal structure coefficients. Further analysis reveals that, although most sampled dynamic networks do not promote prosocial behavior, a class of dynamic core-periphery networks consistently promotes multiple forms of prosocial behavior, including cooperation, altruism, fairness and trust.