This paper explores the coevolution of relationships and interactions in cooperative dynamical multiplex networks, where actors can interact freely but social relations influence their preferences for certain individuals. The study uses a multiplex network model with two layers: a relationship layer and an interaction layer. In the relationship layer, edge weights change over time based on the outcomes of interactions in the interaction layer. Bilateral cooperation strengthens these weights, while mutual defection weakens them. Fitness, which determines strategy imitation, depends on both payoff from interactions and a relationship index calculated from edge weights. The research focuses on the weak prisoner's dilemma scenario and examines the impact of different relationship graph topologies on cooperation levels. Key findings include: 1. **Network Topology Impact**: Higher average degrees in regular graphs favor cooperation, but randomness in links is crucial in harsh conditions. 2. **Coupling Between Relationship Index and Fitness**: A stronger coupling between the relationship index and fitness discourages cooperation by reducing the direct effect of strategy changes. 3. **Polarization of Relationship Index**: The distribution of the relationship index is strongly related to the overall cooperation level, with cooperation leading to higher relationship index values and defection causing them to converge towards zero. The study concludes that the interplay between relationship and interaction dynamics significantly influences the evolution of cooperation in multiplex networks, highlighting the importance of both structural and behavioral factors in social dynamics.This paper explores the coevolution of relationships and interactions in cooperative dynamical multiplex networks, where actors can interact freely but social relations influence their preferences for certain individuals. The study uses a multiplex network model with two layers: a relationship layer and an interaction layer. In the relationship layer, edge weights change over time based on the outcomes of interactions in the interaction layer. Bilateral cooperation strengthens these weights, while mutual defection weakens them. Fitness, which determines strategy imitation, depends on both payoff from interactions and a relationship index calculated from edge weights. The research focuses on the weak prisoner's dilemma scenario and examines the impact of different relationship graph topologies on cooperation levels. Key findings include: 1. **Network Topology Impact**: Higher average degrees in regular graphs favor cooperation, but randomness in links is crucial in harsh conditions. 2. **Coupling Between Relationship Index and Fitness**: A stronger coupling between the relationship index and fitness discourages cooperation by reducing the direct effect of strategy changes. 3. **Polarization of Relationship Index**: The distribution of the relationship index is strongly related to the overall cooperation level, with cooperation leading to higher relationship index values and defection causing them to converge towards zero. The study concludes that the interplay between relationship and interaction dynamics significantly influences the evolution of cooperation in multiplex networks, highlighting the importance of both structural and behavioral factors in social dynamics.