Asymptotic and stability analysis of kinetic models for opinion formation on networks: an Allen-Cahn approach

TL;DR

This paper analyzes the stability of opinion formation models on social networks using kinetic equations and derives an Allen-Cahn type equation to study opinion dynamics and their stability.

Contribution

It introduces a novel approach linking kinetic Boltzmann models to Allen-Cahn equations for opinion dynamics on networks.

Findings

Derived an Allen-Cahn type equation from kinetic models.

Performed stability analysis using integral operators.

Applied entropy tools to binary interaction models.

Abstract

We present the analysis of the stationary equilibria and their stability in case of an opinion formation process in presence of binary opposite opinions evolving according to majority-like rules on social networks. The starting point is a kinetic Boltzmann-type model derived from microscopic interactions rules for the opinion exchange among individuals holding a certain degree of connectivity. The key idea is to derive from the kinetic model an Allen-Cahn type equation for the fraction of individuals holding one of the two opinions. The latter can be studied by means of a linear stability analysis and by exploiting integral operator analysis. While this is true for ternary interactions, for binary interactions the derived equation of interest is a linear scattering equation, that can be studied by means of General Relative Entropy tools and integral operators.