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

TL;DR

This study examines how cooperation evolves among agents playing the Prisoner's Dilemma on different types of scale-free networks, revealing the influence of network structure and initial conditions on cooperative behavior.

Contribution

It introduces a mean field approximation for evolutionary dynamics in uncorrelated scale-free networks, linking network topology to cooperation levels.

Findings

Cooperation is more sustained in scale-free networks than in homogeneous graphs.

The mean field model accurately predicts cooperation for specific initial conditions.

Network structure significantly impacts the survival of cooperative behavior.

Abstract

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.