Strategy evolution on temporal hypergraphs.

Individuals interact and cooperate in structured systems. Many studies represent this structure using static networks, where each link represents a permanent connection between two nodes. However, real interactions are generally not time-invariant and are often not pairwise. Recently, progress has been made in modeling higher-order interactions using hypergraphs, where a link may connect more than two individuals. Here, we study cooperation on temporal hypergraphs, capturing the time-varying, higher-order interactions seen in empirical systems. We find that temporal hypergraphs can promote cooperation compared with static networks, and the latter may even underestimate the cooperation-boosting effects of constrained, local interactions. We further show that cooperation can be facilitated by temporal hypergraphs with sparse components and higher-order interactions. Importantly, when the size of group interactions (hyperedges) is comparable to the population size, relatively small hyperedge sizes best facilitate cooperation. Synthetic and empirical hypergraphs alike affirm our findings, illuminating how temporal, higher-order interactions profoundly shape the evolution of cooperation.