A Simple Proof of Asymptotic Consensus in the Hegselmann--Krause and Cucker--Smale Models with Normalization and Delay

Abstract

We present a simple proof of asymptotic consensus in the discrete Hegselmann-Krause model and flocking in the discrete Cucker-Smale model with normalization and variable delay. It utilizes the convexity of the normalized communication weights and a Gronwall-Halanay-type inequality. The main advantage of our method, compared to previous approaches to the delay Hegselmann-Krause model, is that it does not require any restriction on the maximal time delay, or the initial data, or decay rate of the influence function. From this point of view the result is optimal. For the Cucker-Smale model it provides an analogous result in the regime of unconditonal flocking with sufficiently slowly decaying communication rate, but still without any restriction on the length of the maximal time delay. Moreover, we demonstrate that the method can be easily extended to the mean-field limits of both the Hegselmann-Krause and Cucker-Smale systems, using appropriate stability results on the measure-valued solutions.

Direct proof of unconditional asymptotic consensus in the Hegselmann–Krause model with transmission‐type delay

“…In contrast to this work, our paper offers a global consensus result under significantly weaker assumptions -namely, for any time delay length, without restrictions on the decay of the influence function (only assuming global positivity) and without smallness of the fluctuation of the initial datum. In [6] a simple proof of global consensus was given for the system (3), however, exclusively with the normalized communication weights (7). This is because the method of proof is based on a geometric argument, exploiting the convexity property of the weights (7), namely, that j =i ψ ij = 1 for all i = 1, .…”

Body-attitude coordination in arbitrary dimension

“…All these models have stimulated an intense literature which is impossible to cite exhaustively (see e.g. [15] for the three-zone model, [19,21,27,28,38,41,43,46,52,63,66] for the Vicsek model, and [1,4,5,16,48,49,50] for the Cucker-Smale model). Variants or combinations of these different models can be found in [8,9,10,54].…”

“…Traditionally, there are three levels of modelling for systems of interacting agents. The finer level of description is the "particle" level, by which each agent is followed in the course of time by means of ordinary differential equations or stochastic processes [3,15,19,21,23,50,53,54,64]. This is an appealing approach as the behavioral rules can be directed encoded in the equations.…”

Flocking in the Cucker-Smale model with self-delay and nonsymmetric interaction weights

“…Asymptotic convergence to global consensus in the system (1) is defined as lim t→∞ d x (t) = 0, with d x given by (4). It has been studied, for diverse variants of the Hegselmann-Krause system with delay, in, e.g., [2,5,7,6,11,14,13,9]. However, to our best knowledge, only the case without self-delay was considered in the literature so far, i.e., with x i in the right-hand side of (1) being evaluated at the present time t.…”