Evolution of Cooperation on Stochastic Block Models

TL;DR

This paper analyzes how cooperation evolves in networks modeled by Stochastic Block Models, providing analytical conditions and critical thresholds for cooperation to be favored, with implications for understanding social structures.

Contribution

It introduces analytical conditions for cooperation in SBM networks using a duality with coalescing random walks, and identifies critical inter-community link probabilities.

Findings

Existence of a critical inter-community link creation probability for cooperation

Mean-field solutions accurately predict critical benefit-to-cost ratios

Results generalize to arbitrary two-player games

Abstract

Cooperation is a major factor in the evolution of human societies. The structure of human social networks, which affects the dynamics of cooperation and other interpersonal phenomena, have common structural signatures. One of these signatures is the tendency to organize as groups. Among the generative models that network theorists use to emulate this feature is the Stochastic Block Model (SBM). In this paper, we study evolutionary game dynamics on SBM networks. Using a recently-discovered duality between evolutionary games and coalescing random walks, we obtain analytical conditions such that natural selection favors cooperation over defection. We calculate the transition point for each community to favor cooperation. We find that a critical inter-community link creation probability exists for given group density, such that the overall network supports cooperation even if individual communities inhibit it. As a byproduct, we present mean-field solutions for the critical benefit-to-cost ratio which perform with remarkable accuracy for diverse generative network models, including those with community structure and heavy-tailed degree distributions. We also demonstrate the generalizability of the results to arbitrary two-player games.