Perturbation theory for evolution of cooperation on networks

TL;DR

This paper develops a perturbation theory to predict how small changes in network structure, like adding or removing edges, affect the promotion of cooperation in social dilemma games, reducing computational complexity.

Contribution

It introduces a perturbation approach for a known threshold benefit-to-cost ratio, enabling efficient prediction of cooperation changes due to network modifications.

Findings

Removing edges generally lowers the threshold ratio, promoting cooperation.

The perturbation theory accurately predicts the impact of edge removal on cooperation.

Adding edges can increase the threshold ratio, but the theory is less accurate for large changes.

Abstract

Network structure is a mechanism for promoting cooperation in social dilemma games. In the present study, we explore graph surgery, i.e., to slightly perturb the given network, towards a network that better fosters cooperation. To this end, we develop a perturbation theory to assess the change in the propensity of cooperation when we add or remove a single edge to/from the given network. Our perturbation theory is for a previously proposed random-walk-based theory that provides the threshold benefit-to-cost ratio, $(b/c)^*$, which is the value of the benefit-to-cost ratio in the donation game above which the cooperator is more likely to fixate than in a control case, for any finite networks. We find that $(b/c)^*$ decreases when we remove a single edge in a majority of cases and that our perturbation theory captures at a reasonable accuracy which edge removal makes $(b/c)^*$ small to facilitate cooperation. In contrast, $(b/c)^*$ tends to increase when we add an edge, and the perturbation theory is not good at predicting the edge addition that changes $(b/c)^*$ by a large amount. Our perturbation theory significantly reduces the computational complexity for calculating the outcome of graph surgery.