Emergence of Balance from a model of Social Dynamics

Abstract: We propose a model for social dynamics on a network. In this model each actor holds a position on some issue, actors and their opinions being associated to vertices of the graph, and, additionally, the actors hold opinions of one another, with these opinions being associated to edges in the graph. These quantities are allowed to evolve according to the gradient flow of a natural free energy. We show that for a small spread in opinions the model converges to a consensus state, where all actors hold the same position. For a larger spread in opinion there is a phase transition marked by the birth of a second stable state: in addition to the consensus state there is a second polarized or partisan state. This state, when it exists, is conjectured to be global energy minimizer, with the consensus state being a local energy minimizer. We derive an energy inequality which supports this. Interestingly, all of the steady states we find, with the exception of the consensus state, are either balanced (in the sense of Heider) or are completely unbalanced states where all triangles are unbalanced. The latter solutions are, not surprisingly, always unstable.