Cooperation and the incentive power distribution on multi-layer hierarchical networks

Understanding how cooperation emerges and persists in populations remains a fundamental challenge in various academic disciplines. Spatial structures have great impacts on the evolution of cooperation, while existing studies have mainly focused on two-dimensional networks. To further investigate the evolution of cooperation on complex networks, this study takes a different approach by introducing hierarchical structures into multi-layer networks. Specifically, each layer in these networks represents a different hierarchical structure, wherein individuals possess different levels of punitive authority depending on their hierarchical positions. Moreover, we propose a top-down punishment mechanism within the framework. Individuals at higher layers are endowed with punitive rights over those below, whereas lower-layer individuals are subject to sanctions but can not retaliate. We implement the agent-based Monte Carlo simulations on both a multi-layer square lattice network and a multi-layer pyramid network. We find that this asymmetric punishment mechanism can significantly promote cooperative behaviors and exhibits robustness across different networks. The results reveal several striking phenomena: the level of cooperation is generally higher in pyramid networks than in square networks; an alternating-layer pattern emerges in multi-layer networks in some scenarios; and this alternating pattern becomes more pronounced when inter-layer interactions are more frequent. These findings suggest that hierarchical organization and top-down punishment play a crucial role in sustaining cooperation, and offer new insights into the design of mechanisms for fostering pro-social behaviors in complex systems.