Dynamic Social Balance and Convergent Appraisals via Homophily and Influence Mechanisms

Social balance theory describes allowable and forbidden configurations of the topologies of signed directed social appraisal networks. In this paper, we propose two discrete-time dynamical systems that explain how an appraisal network evolves towards social balance from an initially unbalanced configuration. These two models are based on two different socio-psychological mechanisms respectively: the homophily mechanism and the influence mechanism. Our main theoretical contribution is a comprehensive analysis for both models in three steps. First, we establish the well-posedness and bounded evolution of the interpersonal appraisals. Second, we characterize the set of equilibrium points as follows: for both models, each equilibrium network is composed by an arbitrary number of complete subgraphs satisfying structural balance. Third, under a technical condition, we establish convergence of the appraisal network to a final equilibrium network satisfying structural balance. In addition to our theoretical analysis, we provide numerical evidence that our technical condition for convergence holds for generic initial conditions in both models. Finally, adopting the homophily-based model, we present numerical results on the mediation and globalization of local conflicts, the competition for allies, and the asymptotic formation of a single versus two factions.