Macroscopic Noisy Bounded Confidence Models With Distributed Radical Opinions

• 成果类型: Article
• 署名作者: Kolarijani, Mohamad Amin Sharifi; Proskurnikov, Anton, V; Esfahani, Peyman Mohajerin
• 署名单位: Delft University of Technology; Polytechnic University of Turin
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2020.2994284
• 发表日期: 2021
• 页码: 1174-1189
• 关键词: Mathematical model; Numerical models; Analytical models; Numerical stability; stability analysis; Boundary conditions; Noise level; Distributed parameter systems; Nonlinear systems; opinion dynamics; stability of nonlinear systems; Stochastic systems

摘要

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, first, show the well-posedness of the dynamic equation, second, provide existence result accompanied by a quantitative global estimate for the corresponding stationary solution, and, third, establish an explicit lower bound on the noise level that guarantees exponential convergence of the dynamics to stationary state. Combining the results in second and third 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. 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.