A general network complexity reduction method for cooperative evolution in structured populations

Understanding the influence of the underlying population structure on evolutionary dynamics is a fundamental problem in many fields of both natural and social sciences. Despite significant progress, the complex spatial structure of networked populations with a large number of interacting components makes any direct analysis aimed at obtaining its evolutionary dynamics still very difficult. Here, we propose a complexity reduction method to transform the spatial structure of the original population into a simplified version and use it to capture the evolutionary dynamics of the original population. Using both computer-generated and empirical networks, we verify that population evolution follows a simple rule: cooperation is favored over defection by natural selection when the benefit-to-cost ratio (b/c) of cooperative behavior exceeds the effective degree (keff) of the simplified network. We then evaluate the impact of population structure on complexity reduction methods and demonstrate that our formalism accurately capture the evolutionary dynamics across populations without highly unbalanced degree distribution. Additionally, we consider four complexity reduction processes and find that the weighted process outperforms the unweighted process in accuracy, while the unweighted process exhibits broader applicability across population structures.