Evolutionary prisoner's dilemma in random graphs

We study an evolutionary version of the spatial prisoner's dilemma game (SPD), where the agents are placed in a random graph. For graphs with fixed connectivity, α, we show that for low values of α the final density of cooperating agents, ρc depends on the initial conditions. However, if the graphs have large connectivities ρc is independent of the initial conditions. We characterize the phase diagram of the system, using both, extensive numerical simulations and analytical computations. It is shown that two well defined behaviors are present: a Nash equilibrium, where the final density of cooperating agents ρc is constant, and a non-stationary region, where ρc fluctuates in time. Moreover, we study the SPD in Poisson random graphs and find that the phase diagram previously developed looses its meaning. In fact, only one regime may be defined. This regime is characterized by a non stationary final state where the density of cooperating agents varies in time.