Emergence of Balance from a model of Social Dynamics

TL;DR

This paper introduces a social dynamics model on networks showing how opinions evolve, leading to consensus or polarization depending on initial opinion spread, with stable states characterized by balanced or unbalanced relationships.

Contribution

The paper presents a novel gradient flow model for social opinion dynamics that predicts phase transitions between consensus and polarized states based on opinion spread.

Findings

Small opinion spread leads to consensus.

Large opinion spread causes a phase transition to polarization.

Stable states are either balanced or completely unbalanced, with unbalanced states being unstable.

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.