Evolution of cooperation on hypergraphs with heterogeneous update dynamics

Evolutionary models on higher-order networks typically assume uniform update rates, neglecting the temporal heterogeneity inherent in collective decision-making. In this work, we depart from this assumption by investigating a public goods game on power random hypergraphs where the update rate of each node is coupled to its hyperdegree. Our results show that the interplay between hyperdegree heterogeneity and update dynamics yields outcomes that distinctly differ from established findings in pairwise networks. We confirm that the inverse update rule, assigning rates inversely proportional to hyperdegree, generally lowers the critical threshold for cooperation, but we observe a counter-intuitive reversal in high-synergy regimes where the proportional update rule, typically detrimental in pairwise settings, surpasses the inverse rule. This anomaly arises because different update rules distinctly modulate the evolutionary roles of hubs and peripheral nodes. Furthermore, we identify a trade-off between success and speed: while inverse updating promotes cooperation, it significantly prolongs the time to convergence. These findings demonstrate that the relative efficacy of update strategies is not invariant; rather, it emerges from the specific coupling between temporal dynamics and structural heterogeneity. Our study offers new insights into designing mechanisms for fostering pro-social behaviors, emphasizing the necessity of explicitly considering this coupling effect.