Indirect Reciprocity Enhances Collective Cooperation on Weighted Networks

Direct, indirect, and network reciprocities are established mechanisms that sustain cooperation in natural and artificial systems. Yet which mechanism most effectively promotes cooperation on a given network remains unclear. Here, we develop a game-theoretic model to explore the evolution of direct and indirect reciprocity on weighted networks. Unlike classical donor-recipient frameworks, we study symmetric repeated interactions on undirected weighted networks with bilateral reputation updates, capturing heterogeneous tie strengths and accelerating reputation spread. We derive a general condition for reciprocal cooperation that unifies unweighted and weighted cases. Across large ensembles of random and empirical networks, indirect reciprocity consistently enhances cooperation, whereas stronger interactions sharply lower the benefit-to-cost threshold under direct reciprocity. To test the robustness of these insights, we examine competition among six reciprocity strategies and find that indirect reciprocity dominates. Our findings demonstrate that choosing the right reciprocity can promote global cooperation on social networks.