We introduce a dynamic model for evolutionary games played on a network where strategy changes are correlated according to degree of influence between players. Unlike the notion of stochastic stability, which assumes mutations are stochastically independent and identically distributed, our framework allows for the possibility that agents correlate their strategies with the strategies of those they trust, or those who have influence over them. We show that the dynamical properties of evolutionary games, where such influence neighborhoods appear, differ dramatically from (...) those where all mutations are stochastically independent, and establish some elementary convergence results relevant for the evolution of social institutions. (shrink)