Macroscopic Noisy Bounded Confidence Models with Distributed Radical Opinions

Mohamad Amin Sharifi Kolarijani*, Anton V. Proskurnikov, Peyman Mohajerin Esfahani

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

4 Citations (Scopus)
12 Downloads (Pure)


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.

Original languageEnglish
Pages (from-to)1174-1189
JournalIEEE Transactions on Automatic Control
Issue number3
Publication statusPublished - 2021

Bibliographical note

Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.


  • Distributed parameter systems
  • nonlinear systems
  • opinion dynamics
  • stability of nonlinear systems
  • stochastic systems


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