The effects of nonlinear interactions and network structure in small group opinion dynamics

We present a model of opinion dynamics in social networks in which an individualʹs opinion evolves under the action of (i) a linear force which tends to restore the opinion back towards the individualʹs natural bias that is his or her initial opinion and (ii) a nonlinear coupling with other individuals which acts to bring opinions closer together but wanes for high opinion discrepancies. Bifurcation analysis for the case of a two-person group shows that a critical value for the difference in natural biases exists which demarcates regimes of qualitatively different behavior. For low to moderate natural bias differences, the dynamics are qualitatively similar to linear theory. For high bias differences, the system takes on a binary nature and is marked by discontinuous transitions between deadlock and consensus as well as hysteresis as the coupling is varied. The coupling required to force consensus grows extremely rapidly with the natural bias difference indicating that trying to achieve group consensus solely via increasing the communications rate becomes fruitless as the biases become extremely divergent. We also show that, for high bias differences, a triad broker network topology can reduce group discord more effectively than a clique, contrary to linear theory.