On the Steady State of Continuous Time Stochastic Opinion Dynamics with Power Law Confidence

TL;DR

This paper models continuous-time opinion dynamics with power-law dependent interaction rates, deriving explicit stationary solutions and analyzing their properties using Mellin transforms, advancing understanding of opinion formation with distance-dependent interactions.

Contribution

It introduces a novel non-linear opinion dynamics model with power-law interaction rates and provides explicit stationary solutions using Mellin transforms, a first in this context.

Findings

Explicit stationary solutions for two-agent case.

New qualitative insights into opinion dynamics with power-law interactions.

Applicability to mean-field and linear interaction models.

Abstract

This paper introduces a class of non-linear and continuous-time opinion dynamics model with additive noise and state dependent interaction rates between agents. The model features interaction rates which are proportional to a negative power of opinion distances. We establish a non-local partial differential equation for the distribution of opinion distances and use Mellin transforms to provide an explicit formula for the stationary solution of the latter, when it exists. Our approach leads to new qualitative and quantitative results on this type of dynamics. To the best of our knowledge these Mellin transform results are the first quantitative results on the equilibria of opinion dynamics with distance-dependent interaction rates. The closed form expressions for this class of dynamics are obtained for the two agent case. However the results can be used in mean-field models featuring several agents whose interaction rates depend on the empirical average of their opinions. The technique also applies to linear dynamics, namely with a constant interaction rate, on an interaction graph.