Social balance in directed networks

Social networks inherently exhibit complex relationships that can be positive or negative, as well as directional. Understanding balance in these networks is crucial for unraveling social dynamics, yet current approaches struggle to incorporate directed interactions. Here, we present a comprehensive framework for understanding balance in signed directed networks, extending traditional balance theory to account for directed interactions, advancing our understanding of complex social systems and their dynamics. Balance is indicated by the enrichment of higher-order patterns like triads compared to an adequate null model, where the network is randomized with some key aspects being preserved. As a special case, we propose a “maximally” constrained signed directed null model based on the maximum-entropy principle that reveals consistent patterns of balance across large-scale social networks. We also consider directed generalizations of balance theory and find that the observed patterns are well aligned with two proposed directed notions of strong balance. Although social networks inherently exhibit directionality, current approaches struggle to generalize social balance theory to directed networks. This paper presents a directed null model for signed networks using the maximum entropy principle, revealing consistency with proposed notions of strong directed balance.