Macroscopic Noisy Bounded Confidence Models With Distributed Radical Opinions

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. 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.Index Terms-Opinion dynamics, distributed parameter systems, stochastic systems, nonlinear systems, stability of NL systems. I. INTRODUCTIONR ECENT decades have witnessed enormous progress in study of complex systems and their system-theoretic properties [1], [2]. The main effort has been invested into the study of "self-organization" and "spontaneous order" phenomena [3] that have inspired the development of synchronization and consensus theory [4], [5]. Paradoxically, these regular behaviors arising from local interactions between subsystems (agents, nodes) of a complex system are studied much better than various "irregular" dynamic effects such as persistent disagreement and clustering, exhibited by many real-world systems. Although some culprits of this asynchrony and dissent (e.g. symmetries and other special structures in the coupling mechanisms, exogenous forces acting on some nodes, heterogeneous dynamics of nodes, etc.) have been revealed in the literature [6]-[10], only a few mathematical models have been proposed that are sufficiently "rich" to capture the

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