  Recent studies on the evolutionary dynamics of the Prisoner's Dilemma game in
scale-free networks have demonstrated that the heterogeneity of the network
interconnections enhances the evolutionary success of cooperation. In this
paper we address the issue of how the characterization of the asymptotic states
of the evolutionary dynamics depends on the initial concentration of
cooperators. We find that the measure and the connectedness properties of the
set of nodes where cooperation reaches fixation is largely independent of
initial conditions, in contrast with the behavior of both the set of nodes
where defection is fixed, and the fluctuating nodes. We also check for the
robustness of these results when varying the degree heterogeneity along a
one-parametric family of networks interpolating between the class of
Erdos-Renyi graphs and the Barabasi-Albert networks.
