# Macroscopic Noisy Bounded Confidence Models with Distributed Radical Opinions

arXiv: https://arxiv.org/abs/1905.04057 · HTML: https://mlsanity.com/paper/1905.04057/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 May 2019.

## Built on by (most cited first, 1 of 1)

- [Optimal Transportation by Orthogonal Coupling Dynamics](https://mlsanity.com/paper/2410.08060/) (2024)

## Abstract (arXiv)

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.