Predicting transitions in cooperation levels from network connectivity

A Zhuk,I Sendiña-Nadal,S Boccaletti,M Perc,A M Raigorodskii,D Musatov,I Leyva

doi:10.1088/1367-2630/ac264d

Abstract

Networks determine our social circles and the way we cooperate with others. We know that topological features like hubs and degree assortativity affect cooperation, and we know that cooperation is favored if the benefit of the altruistic act divided by the cost exceeds the average number of neighbors. However, a simple rule that would predict cooperation transitions on an arbitrary network has not yet been presented. Here we show that the unique sequence of degrees in a network can be used to predict at which game parameters major shifts in the level of cooperation can be expected, including phase transitions from absorbing to mixed strategy phases. We use the evolutionary prisoner’s dilemma game on random and scale-free networks to demonstrate the prediction, as well as its limitations and possible pitfalls. We observe good agreements between the predictions and the results obtained with concurrent and Monte Carlo methods for the update of the strategies, thus providing a simple and fast way to estimate the outcome of evolutionary social dilemmas on arbitrary networks without the need of actually playing the game.

Highlights

In 1992 Nowak and May observed that cooperators form compact clusters and can thereby withstand invading defectors in an iterated prisoner’s dilemma game on a square lattice [1]
We show that the unique sequence of degrees in a network can be used to predict at which game parameters major shifts in the level of cooperation can be expected, including phase transitions from absorbing to mixed strategy phases
With the advent of network science at the turn of the 21st century [9,10,11], the field of evolutionary games on networks rapidly gained on popularity, and various complex networks have been studied for their impact on the evolution of cooperation, including scale-free [12,13,14,15,16,17,18,19,20,21,22,23], small-world [24,25,26,27,28,29,30], hierarchical [31, 32], coevolving [33,34,35,36,37,38], and empirical social networks [39,40,41]

Summary

Introduction

In 1992 Nowak and May observed that cooperators form compact clusters and can thereby withstand invading defectors in an iterated prisoner’s dilemma game on a square lattice [1]. This was a fascinating discovery because cooperators should have died out in agreement with the Nash equilibrium of the game [2, 3]. We first present the derivation of the conjecture and proceed with showing the results obtained for the evolutionary prisoner’s dilemma game on random and scale-free networks. We conclude by discussing the implications of our research to optimise large-scale simulations of evolutionary processes on complex networks and the possibilities for the generalisation of the conjecture to related subjects

Game and conjecture formulation

Results

Discussion