A unified framework for imitation dynamics on higher-order networks

Abstract

Cooperation is central to human societies and often unfolds within groups. Higher-order networks, such as hypergraphs, naturally represent these groups as hyperedges. Network structures and update rules, by which individuals revise their strategies, are the two fundamental components that shape the evolution of cooperation in structured populations. Yet while the effects of network structure have been studied extensively, update rules have been examined mostly through isolated models, leaving their relationships and the origins of their differing evolutionary outcomes poorly understood. Here we develop a unified framework for imitation dynamics on higher-order networks, parameterizing imitation-based update rules by the number of groups an individual samples and the number of peers consulted within each group. Under weak selection, we derive a closed-form condition for the success of cooperation in any multiplayer social dilemma on homogeneous hypergraphs, encompassing games with both linear and nonlinear payoff structures. The framework places previously disconnected update rules within a single family and reduces their effects on cooperation to one interpretable quantity, which we term information diversity. Update rules inducing higher information diversity promote cooperation more effectively, and we prove that this ordering holds strictly across the entire space of multiplayer social dilemmas. Simulations extend this principle to heterogeneous hypergraphs constructed both synthetically and from empirical data. Our framework provides a systematic way to represent, analyze, and compare update rules on higher-order networks, turning a fragmented collection of microscopic updating mechanisms into a tractable and interpretable theory.