Cooperation in an evolutionary prisoner’s dilemma on networks with degree-degree correlations

Stephen Devlin,Thomas Treloar

doi:10.1103/physreve.80.026105

Abstract

We study the effects of degree-degree correlations on the success of cooperation in an evolutionary prisoner's dilemma played on a random network. When degree-degree correlations are not present, the standardized variance of the network's degree distribution has been shown to be an accurate analytical measure of network heterogeneity that can be used to predict the success of cooperation. In this paper, we use a local-mechanism interpretation of standardized variance to give a generalization to graphs with degree-degree correlations. Two distinct mechanisms are shown to influence cooperation levels on these types of networks. The first is an intrinsic measurement of base-line heterogeneity coming from the network's degree distribution. The second is the increase in heterogeneity coming from the degree-degree correlations present in the network. A strong linear relationship is found between these two parameters and the average cooperation level in an evolutionary prisoner's dilemma on a network.

Highlights

Agent-based game theoretical methods have become widely used tools in biology, social sciences, physics, and mathematics1͔
When the two plots are combined in Fig. 3͑c, cooperation levels seen on correlated networks are significantly higher than those expected on uncorrelated networks with the same heterogeneity
We have generalized methods used in21͔ to give analytical measures of heterogeneity for networks with degree-degree correlations such as Barabási-Albert scale-free networks generated via growth and preferential attachment

Summary

INTRODUCTION

Agent-based game theoretical methods have become widely used tools in biology, social sciences, physics, and mathematics1͔. Strategy updating depends on the success of an individual relative only to the success of that individual’s neighborhood of contacts as specified by the network In this framework, the evolutionary dynamics are strikingly different. Heterogeneity is widely understood to mean that the network contains considerable diversity in the numbers of agents’ contacts, resulting in a degree distribution that is significantly “spread out” and includes large vertices or “hubs.” Recent work has shown that mitigating the role played by these large vertices, through either payoff normalization16͔ or participation costs15͔, can dramatically reduce the success of cooperators, solidifying the notion of heterogeneity as a necessary ingredient in network cooperation. ␯c allows one to quantify the additional heterogeneity present in a correlated network coming from degree-degree correlations That such correlations can have considerable impact on cooperation has been seen in9͔ and is further evident in what follows. We isolate the contributions of both ␯st and ␯c to the success of cooperation on a network, helping to quantify and to clarify the relationship between these network parameters, and in particular, helping to isolate the specific role played by degreedegree correlations in the evolutionary dynamics on the network

GENERATING FUNCTIONS AND NETWORK HETEROGENEITY

NETWORKS AND EVOLUTIONARY GAMES

RESULTS AND DISCUSSION

CONCLUSIONS