Behavioral alignment in social networks.

The orderly behaviors observed in large-scale groups, such as fish schooling and the organized movement of crowds, are both ubiquitous and essential for the survival and stability of these systems. Understanding how such complex collective behaviors emerge from simple local interactions and behavioral adjustments is a significant scientific challenge. Historically, research has predominantly focused on imitation and social learning, where individuals adopt the strategies of more successful peers to refine their behavior. However, in recent years, an alternative learning approach based on self-exploration and introspective learning has garnered increasing attention. In this paradigm, individuals assess their own circumstances and select strategies that best align with their specific conditions. Two examples are coordination and anti-coordination, where individuals align with and diverge from the local majority, respectively. In this study, we analyze networked systems of coordinating and anti-coordinating individuals, exploring the combined effects of system dynamics, network structure and behavioral patterns. We address several practical questions, including the number of equilibria, their characteristics, the equilibrium time and the resilience of the system. We find that the number of equilibrium states can be extremely large, even increasing exponentially with minor alterations to the network structure. Moreover, the network structure has a significant impact on the average equilibrium time. Despite the complexity of these findings, we find that variations can be captured by a single, simple network characteristic (the average path length), which we illustrate in both synthetic and empirical networks.