Breaking the Symmetry between Interaction and Replacement in Evolutionary Dynamics on Graphs

We study the evolution of cooperation modeled as symmetric $2\times 2$ games in a population whose structure is split into an interaction graph defining who plays with whom and a replacement graph specifying evolutionary competition. We find it is always harder for cooperators to evolve whenever the two graphs do not coincide. In the thermodynamic limit, the dynamics on both graphs is given by a replicator equation with a rescaled payoff matrix in a rescaled time. Analytical results are obtained in the pair approximation and for weak selection. Their validity is confirmed by computer simulations.