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 converges to 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 fully characterize the set of equilibrium points; for both models, each equilibrium network is composed of an arbitrary number of complete subgraphs satisfying structural balance. Third, we establish the equivalence among three distinct properties: non-vanishing appraisals, convergence to all-to-all appraisal networks, and finitetime achievement of social balance. In addition to theoretical analysis, Monte Carlo validations illustrate how the non-vanishing appraisal condition holds for generic initial conditions in both models. Moreover, a numerical comparison between the two models indicates that the homophily-based model might be a more universal explanation for the emergence of social balance. 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.

We propose a signed network formation game, in which pairs of individuals strategically change the signs of the edges in a complete network. These individuals are members of a social network who strategically reduce cognitive dissonances by changing their interpersonal appraisals. We characterize the best-response dynamics for this game and prove that its implementation can dynamically drive the network to a sociologically meaningful sign configuration called clustering balance. In this configuration, agents in the social network form one or more clusters that have positive relationships among their members but negative relationships among members of other clusters. In the past, various researchers in the fields of psycho-sociology, political science, and physics have looked at models that explain the generation of up to two clusters. Our work contributes to these fields by proposing a simple model that generates a broader class of signed networks.

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