Continuum dynamics of the intention field under weakly cohesive social interaction

• Published in: Mathematical models & methods in applied sciences Vol. 27; no. 1; pp. 159 - 182
• Main Authors: Degond, Pierre, Liu, Jian-Guo, Merino-Aceituno, Sara, Tardiveau, Thomas
• Format: Journal Article
• Language: English
• Published: Singapore World Scientific Publishing Company 01.01.2017; World Scientific Publishing Co. Pte., Ltd; World Scientific Publishing
• Subjects: Collision dynamics; Density; Dependence; Fokker-Planck equation; Mathematics; Social factors; non-symmetric rate; Opinion formation; continuum limit; Deffuant–Weisbuch model; grazing limit; Deffuant-Weisbuch model AMS Subject Classification: 82C21
• ISSN: 0218-2025; 1793-4060; 1793-6314; 1793-6314
• DOI: 10.1142/S021820251740005X

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.