  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.
