Graph limit of the consensus model with self-delay

IF 2 3区 物理与天体物理 Q2 PHYSICS, MATHEMATICAL Journal of Physics A: Mathematical and Theoretical Pub Date : 2024-08-13 DOI:10.1088/1751-8121/ad6ab1
Jan Haskovec
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Abstract

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 Lp 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.
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具有自延迟的共识模型的图极限
众所周知,具有自延时(反应型延时)的相互作用者模型一般不采用福克-普朗克方程的经典均场极限描述。在本文中,我们提出了具有自延迟的非线性共识模型的图极限,作为另一种连续描述,并研究了它的数学特性。我们在 Lebesgue Lp 空间中建立了所得到的积分微分方程的好求解性。我们对离散共识系统的图极限进行了严格推导,并得出了达成全局渐近共识的充分条件。我们还考虑了具有给定交互核的线性模型变体,它可以被解释为图元上的动力系统。在这里,我们得出了达成全局渐近共识的最优(即充分和必要)条件。最后,我们详细解释了自延迟项的存在如何排除了用福克-普朗克方程式的粒子密度来描述均场极限。特别是,我们证明了粒子的不可分性特性不成立,而这正是推导经典均场描述的主要因素之一。
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来源期刊
CiteScore
4.10
自引率
14.30%
发文量
542
审稿时长
1.9 months
期刊介绍: Publishing 50 issues a year, Journal of Physics A: Mathematical and Theoretical is a major journal of theoretical physics reporting research on the mathematical structures that describe fundamental processes of the physical world and on the analytical, computational and numerical methods for exploring these structures.
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