  We study an evolutionary version of the spatial prisoner's dilemma game,
where the agents are placed in a random graph. For lattices with fixed
connectivity, $\alpha$, we show that for low values of $\alpha$ the final
density of cooperating agents depends on the initial conditions, while it does
not depend for high connectivity lattices. We fully characterized the phase
diagram of the system, using both, extensive numerical simulations and
analytical computations. It is shown that two different behaviors are well
defined: a Nash equilibrium one, where the density of cooperating agents
$\rho_c$ is fixed, and a non-stationary one, where $\rho_c$ fluctuates in time.
Moreover we study lattices with fluctuating connectivities and find that the
phase diagram previously developed looses its meaning. In fact, multiple
transitions appear and only one regime may be defined. This regime is
completely characterized by a non stationary state where the density of
cooperating agents varies in time.
