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Well posedness and asymptotic consensus in the Hegselmann-Krause model with finite speed of information propagation

Author: Jan Haskovec
Journal: Proc. Amer. Math. Soc. 149 (2021), 3425-3437
MSC (2020): Primary 34K20, 34K60, 82C22, 92D50
Published electronically: May 12, 2021
MathSciNet review: 4273146
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Abstract: We consider a variant of the Hegselmann-Krause model of consensus formation where information between agents propagates with a finite speed $\mathfrak {c}>0$. This leads to a system of ordinary differential equations (ODE) with state-dependent delay. Observing that the classical well-posedness theory for ODE systems does not apply, we provide a proof of global existence and uniqueness of solutions of the model. We prove that asymptotic consensus is always reached in the spatially one-dimensional setting of the model, as long as agents travel slower than $\mathfrak {c}$. We also provide sufficient conditions for asymptotic consensus in the spatially multidimensional setting.

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Additional Information

Jan Haskovec
Affiliation: Computer, Electrical and Mathematical Sciences & Engineering, King Abdullah University of Science and Technology, 23955 Thuwal, Kingdom of Saudi Arabia
MR Author ID: 754324
ORCID: 0000-0003-3464-304X

Keywords: Hegselmann-Krause model, state-dependent delay, nite speed of information propagation, well-posedness, long-time behavior, asymptotic consensus.
Received by editor(s): May 11, 2020
Received by editor(s) in revised form: December 6, 2020
Published electronically: May 12, 2021
Additional Notes: The author was supported by the KAUST baseline funds
Communicated by: Wenxian Shen
Article copyright: © Copyright 2021 American Mathematical Society