Continuum dynamics of the intention field under weakly cohesive social interactions

TL;DR

This paper analyzes the long-term behavior of an opinion formation model using a Fokker-Planck equation, comparing symmetric and non-symmetric interactions and their effects on consensus dynamics.

Contribution

It derives a macroscopic limit for opinion dynamics under weakly cohesive social interactions and compares different interaction rates' impacts on consensus formation.

Findings

Symmetric interactions lead to a conservative opinion density model.

Non-symmetric interactions produce a non-conservative equation.

Consensus speed varies significantly with interaction rate density dependence.

Abstract

We investigate the long-time dynamics of an opinion formation model inspired by a work by Borghesi, Bouchaud and Jensen. Firstly, we derive a Fokker-Planck type equation under the assumption that interactions between individuals produce little consensus of opinion (grazing collision approximation). Secondly, we study conditions under which the Fokker-Planck equation has non-trivial equilibria and derive the macroscopic limit (corresponding to the long-time dynamics and spatially localized interactions) for the evolution of the mean opinion. Finally, we compare two different types of interaction rates: the original one given in the work of Borghesi, Bouchaud and Jensen (symmetric binary interactions) and one inspired from works by Motsch and Tadmor (non-symmetric binary interactions). We show that the first case leads to a conservative model for the density of the mean opinion whereas the second case leads to a non-conservative equation. We also show that the speed at which consensus is reached asymptotically for these two rates has fairly different density dependence.