Evolutionary game analysis of individual behaviors with coexisting dual relationship types

In complex social systems, individuals are often embedded in multiple types of relationships, including stable, long-term ties and dynamic, temporary connections. To investigate how these coexisting relationships shape behavioral evolution, we propose a two-layer evolutionary game model that integrates a weighted fixed relationship network with an activity-driven time-varying network. Through extensive agent-based simulations, we systematically analyze the effects of network topology, payoff parameters, and individual heterogeneity on the emergence and sustainability of cooperation. Our results reveal that heterogeneous networks significantly promote and stabilize cooperation, even under high temptation or frequent temporary interactions, while homogeneous networks are prone to the breakdown of cooperation. The adaptive adjustment of relationship weights further enhances the persistence of cooperative behavior, particularly when cooperation and defection are strongly reinforced or penalized. Additionally, we find that cooperation levels are highly sensitive to the integration of different relationship types and to individual activity rates, with highly active individuals less likely to maintain strong cooperative ties. This study highlights the critical role of multi-relational network structure and behavioral diversity in the evolution of cooperation, providing new insights for designing resilient and cooperative social systems.