Abstract
We investigate the long-time dynamics of an opinion formation model inspired by a work by Borghesi, Bouchaud and Jensen. First, we derive a Fokker–Planck-type equation under the assumption that interactions between individuals produce little consensus of opinion (grazing collision approximation). Second, 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.
Highlights
The goal of this paper is the investigation of an opinion formation model inspired from the one presented in Ref. 10
We study the equilibria of these dynamics
Under the conditions of Proposition 3.2 we prove that the equilibrium for the corresponding mean-field equation is given by a Gaussian with fixed variance σ2 and undetermined mean φ
Summary
The goal of this paper is the investigation of an opinion formation model inspired from the one presented in Ref. 10. We obtain the mean-field equations for this model and approximate the dynamics under the assumption that interactions between individuals produce little convergence of opinions (weak consensus interaction). The final aim is to derive the equation for the evolution of the mean opinion φ in the spatially heterogeneous case when interactions become localized. During this analysis, we will consider two different cases corresponding to two different types of interaction rates: the original one given in Ref. 10 and one inspired from Refs. As far as we know, this is the first result that derives the macroscopic dynamics for these equations
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