Graph limit of the consensus model with self-delay

• Published in: Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2024-09, Vol.57 (34), p.345203
• Main Author: Haskovec, Jan
• Format: Article
• Language: English
• Subjects: asymptotic consensus; graph limit; graphon; long-time behavior; self-delay
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8121/ad6ab1

It is known that models of interacting agents with self-delay (reaction-type delay) do not admit, in general, the classical mean-field limit description in terms of a Fokker–Planck equation. In this paper we propose the graph limit of the nonlinear consensus model with self-delay as an alternative continuum description and study its mathematical properties. We establish the well-posedness of the resulting integro-differential equation in the Lebesgue L p space. We present a rigorous derivation of the graph limit from the discrete consensus system and derive a sufficient condition for reaching global asymptotic consensus. We also consider a linear variant of the model with a given interaction kernel, which can be interpreted as a dynamical system over a graphon. Here we derive an optimal (i.e. sufficient and necessary) condition for reaching global asymptotic consensus. Finally, we give a detailed explanation of how the presence of the self-delay term rules out a description of the mean-field limit in terms of a particle density governed by a Fokker–Planck-type equation. In particular, we show that the indistinguishability-of-particles property does not hold, which is one of the main ingredients for deriving the classical mean-field description.