2019/05/10 by Mohamad Amin Sharifi Kolarijani, Kolarijani, M. A. S., A. V. Proskurnikov +3
Computer Science · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Electrical engineering #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Opinion Dynamics and Social Influence #Optimization and Control (math.OC) #Systems and Control (eess.SY) #Theoretical and Computational Physics #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.1905.04057
openalex publication_date 2019/05/10 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
In this article, we study the nonlinear Fokker-Planck (FP) equation that\narises as a mean-field (macroscopic) approximation of bounded confidence\nopinion dynamics, where opinions are influenced by environmental noises and\nopinions of radicals (stubborn individuals). The distribution of radical\nopinions serves as an infinite-dimensional exogenous input to the FP equation,\nvisibly influencing the steady opinion profile. We establish mathematical\nproperties of the FP equation. In particular, we (i) show the well-posedness of\nthe dynamic equation, (ii) provide existence result accompanied by a\nquantitative global estimate for the corresponding stationary solution, and\n(iii) establish an explicit lower bound on the noise level that guarantees\nexponential convergence of the dynamics to stationary state. Combining the\nresults in (ii) and (iii) readily yields the input-output stability of the\nsystem for sufficiently large noises. Next, using Fourier analysis, the\nstructure of opinion clusters under the uniform initial distribution is\nexamined. Specifically, two numerical schemes for identification of\norder-disorder transition and characterization of initial clustering behavior\nare provided. The results of analysis are validated through several numerical\nsimulations of the continuum-agent model (partial differential equation) and\nthe corresponding discrete-agent model (interacting stochastic differential\nequations) for a particular distribution of radicals.\n