This study investigates the coevolution of relationships and interactions in cooperative dynamical multiplex networks. The research explores how the strength of relationships between agents influences their strategic interactions and how these interactions, in turn, affect the relationship strengths. The model is based on a multiplex network with two layers: a relationship layer and an interaction layer. In the relationship layer, edge weights represent the strength of relationships between agents, while in the interaction layer, agents play the weak prisoner's dilemma (WPD) game. The weight of edges in the relationship layer is updated based on the strategies chosen by the agents in the interaction layer. Mutual cooperation increases edge weights, while mutual defection decreases them. The fitness of an agent, which determines the likelihood of strategy imitation, depends on both the payoff from interactions and the relationship index, which is derived from the edge weights in the relationship layer. The study finds that the average degree of the relationship graph is more beneficial for maintaining cooperation in regular graphs, but the randomness of links can be decisive in harsh situations. A stronger coupling between the relationship index and fitness discourages the evolution of cooperation by weakening the direct consequence of a strategy change. The results show that the distribution of the relationship index varies across different network topologies, and there is a strong correlation between the polarization of the relationship index and the general cooperation level. The research highlights the importance of considering both the structural relationships and the strategic interactions in understanding the evolution of cooperation in social dilemmas. The findings suggest that the coevolution of relationships and interactions can significantly influence the emergence and persistence of cooperative behavior in multiplex networks. The study also demonstrates that the fitness of agents is not only determined by their payoffs but also by their relationship index, which is influenced by the coevolutionary process. The results provide insights into how different network structures and parameter configurations affect the evolution of cooperative behaviors among agents. The study concludes that the coevolution of relationships and interactions in multiplex networks is a crucial factor in the emergence and maintenance of cooperation in social dilemmas.This study investigates the coevolution of relationships and interactions in cooperative dynamical multiplex networks. The research explores how the strength of relationships between agents influences their strategic interactions and how these interactions, in turn, affect the relationship strengths. The model is based on a multiplex network with two layers: a relationship layer and an interaction layer. In the relationship layer, edge weights represent the strength of relationships between agents, while in the interaction layer, agents play the weak prisoner's dilemma (WPD) game. The weight of edges in the relationship layer is updated based on the strategies chosen by the agents in the interaction layer. Mutual cooperation increases edge weights, while mutual defection decreases them. The fitness of an agent, which determines the likelihood of strategy imitation, depends on both the payoff from interactions and the relationship index, which is derived from the edge weights in the relationship layer. The study finds that the average degree of the relationship graph is more beneficial for maintaining cooperation in regular graphs, but the randomness of links can be decisive in harsh situations. A stronger coupling between the relationship index and fitness discourages the evolution of cooperation by weakening the direct consequence of a strategy change. The results show that the distribution of the relationship index varies across different network topologies, and there is a strong correlation between the polarization of the relationship index and the general cooperation level. The research highlights the importance of considering both the structural relationships and the strategic interactions in understanding the evolution of cooperation in social dilemmas. The findings suggest that the coevolution of relationships and interactions can significantly influence the emergence and persistence of cooperative behavior in multiplex networks. The study also demonstrates that the fitness of agents is not only determined by their payoffs but also by their relationship index, which is influenced by the coevolutionary process. The results provide insights into how different network structures and parameter configurations affect the evolution of cooperative behaviors among agents. The study concludes that the coevolution of relationships and interactions in multiplex networks is a crucial factor in the emergence and maintenance of cooperation in social dilemmas.