# Macroscopic Noisy Bounded Confidence Models with Distributed Radical Opinions

## TL;DR

This paper analyzes a nonlinear Fokker-Planck equation modeling opinion dynamics with environmental noise and radical opinions, establishing mathematical properties, stability conditions, and numerical methods to understand opinion clustering and transitions.

## Contribution

It provides the first rigorous analysis of the well-posedness, stability, and clustering behavior of macroscopic opinion models influenced by radicals and noise, with validated numerical schemes.

## Key findings

- Proved well-posedness and existence of stationary solutions.
- Derived explicit noise bounds for exponential convergence.
- Developed numerical schemes for clustering analysis.

## Abstract

In this article, we study the nonlinear Fokker-Planck (FP) equation that arises as a mean-field (macroscopic) approximation of bounded confidence opinion dynamics, where opinions are influenced by environmental noises and opinions of radicals (stubborn individuals). The distribution of radical opinions serves as an infinite-dimensional exogenous input to the FP equation, visibly influencing the steady opinion profile. We establish mathematical properties of the FP equation. In particular, we (i) show the well-posedness of the dynamic equation, (ii) provide existence result accompanied by a quantitative global estimate for the corresponding stationary solution, and (iii) establish an explicit lower bound on the noise level that guarantees exponential convergence of the dynamics to stationary state. Combining the results in (ii) and (iii) readily yields the input-output stability of the system for sufficiently large noises. Next, using Fourier analysis, the structure of opinion clusters under the uniform initial distribution is examined. Specifically, two numerical schemes for identification of order-disorder transition and characterization of initial clustering behavior are provided. The results of analysis are validated through several numerical simulations of the continuum-agent model (partial differential equation) and the corresponding discrete-agent model (interacting stochastic differential equations) for a particular distribution of radicals.