Structural balance and evolution of cooperation in a population with hybrid interactions.

This study explores the evolution of cooperation in populations with mixed pairwise and three-body interactions, investigating the impact of higher-order interaction density ρ and individual interaction preference α. Our results reveal that sparse higher-order interactions markedly boost cooperation, exhibiting two critical phase transitions as ρ changes. These transitions underscore the delicate equilibrium needed for optimal cooperation, as excessive higher-order interactions can diminish returns. The preference parameter α significantly influences cooperation sustainability, with intermediate values maximizing cooperative outcomes, particularly when the temptation to defect r is not strong. Crucially, our findings demonstrate that hybrid social dilemmas structurally encode emergent cooperation pathways that are unattainable within homogeneous interaction frameworks, emphasizing the importance of modeling mixed interactions to capture real-world complexity. These insights offer valuable guidance for designing systems aimed at promoting cooperative behavior across social, ecological, and artificial domains.