Cooperation in the Prisoner's Dilemma game in Random Scale-Free Graphs

In this paper we study the cooperative behavior of agents playing the Prisoner's Dilemma game in random scale-free networks. We show that the survival of cooperation is enhanced with respect to random homogeneous graphs but, on the other hand, decreases when compared to that found in Barab\'asi-Albert scale-free networks. We show that the latter decrease is related with the structure of cooperation. Additionally, we present a mean field approximation for studying evolutionary dynamics in networks with no degree-degree correlations and with arbitrary degree distribution. The mean field approach is similar to the one used for describing the disease spreading in complex networks, making a further compartmentalization of the strategists partition into degree-classes. We show that this kind of approximation is suitable to describe the behavior of the system for a particular set of initial conditions, such as the placement of cooperators in the higher-degree classes, while it fails to reproduce the level of cooperation observed in the numerical simulations for arbitrary initial configurations.