Evolutionary game on mutually influenceing double-layer network

In recent years, coupled double-layer networks have played an increasingly critical role in evolutionary game theory. Research indicates that these networks more accurately reflect real-world relationships between individuals. However, current studies mainly focus on unidirectional influence within double-layer networks. Based on this, we propose a strongly coupled double-layer network cooperation evolution model. Strength individuals are located in the upper network layer, influencing the strategy choices of ordinary individuals in the lower layer, and vice versa. Monte Carlo simulations show that strength individuals can effectively enhance overall group cooperation. Under low temptation to defect, the group maintains a high cooperation rate; under high temptation, the presence of strength individuals prevents the group from falling into total defection, helping ordinary individuals escape the defection dilemma and improve cooperation levels.