Low-dimensional dynamics of the social compass model

The social compass model has been recently proposed as a model for depolarization in populations where individuals have multiple, possibly correlated, opinions. Previous work has focused on the steady state of this model but has not addressed the dynamics leading to depolarization. We show that the macroscopic dynamics of the social compass model can be described using the Ott-Antonsen Ansatz and that, for initially clustered opinions, the resulting equations reduce to a finite-dimensional system of ordinary differential equations. We study the linear stability of the polarized state and find a dispersion relation for the growth rate of perturbations from this state. We find that the critical coupling for depolarization depends only on the first inverse moment of the conviction distribution, whereas the rate of depolarization depends on higher moments. Consequently, conviction distributions with the same critical coupling can exhibit vastly different depolarization timescales. We also demonstrate how our analysis can be extended to study depolarization in the presence of community structure.