Evolution of cooperation in arbitrary complex networks

Abstract

This paper proposes a new model, based on the theory of nonlinear dynamical systems, to study the evolution of cooperation in arbitrary complex networks. We consider a large population of agents placed on some arbitrary network, interacting with their neighbors while trying to optimize their fitness over time. Each agent's strategy is continuous in nature, ranging from purely cooperative to purely defective behavior, where cooperation is costly but leads to shared benefits among the agent's neighbors. This induces a dilemma between social welfare and individual rationality. We show in simulation that our model clarifies why cooperation prevails in various regular and scale-free networks. Moreover we observe a relation between the network size and connectivity on the one hand, and the resulting level of cooperation in equilibrium on the other hand. These empirical findings are accompanied by an analytical study of stability of arbitrary networks. Furthermore, in the special case of regular networks we prove convergence to a specific equilibrium where all agents adopt the same strategy. Studying under which scenarios cooperation can prevail in structured societies of self-interested individuals has been a topic of interest in the past two decades. However, related work has been mainly restricted to either analytically studying a specific network structure, or empirically comparing different network structures. To the best of our knowledge we are the first to propose a dynamical model that can be used to analytically study arbitrary complex networks.