Interpersonal relationships are building blocks of human social groups and resulting social behavior. The dynamics of social groups can be critical in understanding the origins of polarization, cooperation, and conflict. Social balance is a classical notion that defines which configurations of pairwise relationships among three individuals are favored. Previous studies have found that the principle of social balance, when applied to all triads within a population, leads to two mutually antagonistic groups that are internally friendly or one giant friendly group. However, this body of theory assumes interpersonal relationships are binary—either friendly or unfriendly. Neutrality adds another layer of complexity. We develop and analyze two models of social balance with interpersonal relationships that are positive, negative, or neutral. We find that the population almost never reaches a fully polarized or paradise state: even though the overall frequencies of triad configurations reach an equilibrium, individuals continue to update their interpersonal relationships over time. We also identify the equilibrium triad frequencies, based on a mean-field analysis. Subsequently, we derive an effective mapping between these two alternative models, which clarifies a nontrivial connection between distinct social processes. We also find a phase transition when some parameters are small, which characterizes the emergence of a giant component connected by active (i.e., friendly or unfriendly) links; and we discuss how the critical regime is relevant in a growing society. Our analysis helps fill a theoretical gap in the literature, and reveals qualitatively new dynamics that arise from a theory of triadic social interactions.
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Dynamics of Social Balance with Ternary Interpersonal Relationships
Author Information
Received: 18 Oct 2025 | Revised: 10 Dec 2025 | Accepted: 12 Dec 2025 | Published: 18 Dec 2025
Abstract
Keywords
interpersonal relationships | balance theory | signed networks | complex networks | triadic interactions
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