Emergence of balance from a model of social dynamics

In this paper we propose and analyze a new model for social dynamics on a network. In the model, each actor holds a position on some issue with actors and their positions being asso- ciated to the vertices of a graph. Additionally, the actors have opinions of one another which are associated to edges in the graph. These quantities are assumed to be real valued and are allowed to evolve according to a constrained gradient ow of a natural free energy. We show that for a small spread in opinions, the model converges to a consensus state where all actors have the same position on the issue in question. 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 balanced state where the actors divide into two groups with opposing views. This state, when it exists, is a local energy minimizer and is conjectured to be the global energy minimizer|the consensus state being the other local energy minimizer. We derive an energy inequality which strongly supports but does not prove this. Interestingly, all of the steady states we Find, with the exception of the consensus state, either are balanced (in the sense of Heider) or are completely unbalanced states where all triangles are unbalanced. The latter solutions are, unsurprisingly, always unstable. In numerical experiments on Erd}osRenyi random graphs we Find that the steady states behave in a similar manner to those of the complete graph with the very interesting additional feature that the intensity of the opinions varies strongly with distance to the con ict|those actors who are in contact with actors having opposing views holding the most extreme opinions, while those further removed from the frontline holding more moderate opinions.