The nested structures of higher-order interactions promote the cooperation in complex social networks

As a powerful tool for exploring non-pairwise interactions, higher-order networks deepen our understanding of how cooperation emerges in complex systems. However, nearly all researches concerning the cooperative evolution on higher-order networks have neglected the prevalence of nested structures in real-world interactions - for instance, students are members of classes, which are components of grades, and these grades, in turn, are part of schools and so on. In this paper, we propose a general and scalable method to investigate the influence of nested structures in higher-order interactions on group cooperation. Specifically, by adjusting the proportion of lower-order nested relations (e.g., pairwise links) within higher-order structures, we find that higher-order interactions are more favorable to the cooperation if they comprise a rich nested dyadic structure. In addition, for the first time, we introduce higher-order network motifs to examine the contributions of various nested structures to cooperation. It is interesting to observe that the more uniform the nested structure, the more it promotes the cooperation. Our study offers a novel perspective to examine the evolution of cooperative behaviors in more complex realistic systems.