Asymptotic consensus with transmission and reaction delay: an overview

TL;DR

This paper reviews asymptotic consensus results for Hegselmann-Krause models with transmission and reaction delays, highlighting conditions under which consensus is achieved and discussing various analytical methods used.

Contribution

It provides a systematic overview of consensus results with delays, including new theorems and analysis for different delay types and communication weights.

Findings

Consensus can be achieved with any transmission delay length.

Reaction delay requires sufficiently small delays for consensus.

Different analytical tools are used depending on the delay scenario.

Abstract

The aim of this paper is to provide a systematic overview of results on asymptotic consensus for the Hegselmann-Krause-type model with delay and discuss the corresponding analytical tools. We explain that two types (sources) of delay - transmission and reaction - are justifiable from the modeling point of view. We consider both classical and normalized communication weights. Studying a toy model with two agents only, we develop an intuitive insight into what type of dynamics we can expect from the systems. In particular, we stress that with transmission-type delay, asymptotic consensus can be reached with any length of the delay (i.e., without smallness assumptions). In contrast, the systems with reaction-type delay can only reach asymptotic consensus if the delay is sufficiently small. We formulate four theorems that establish asymptotic consensus in the following scenarios: (1) transmission-type delay with classical communication weights, (2) transmission-type delay with normalized communication weights, (3) reaction-type delay with symmetric communication weights, (4) reaction-type delay with non-symmetric communication weights. We explain how the methods of proof depend on the particular scenario: direct estimates for (1), convexity arguments for (2), Lyapunov functional for (3) and generalized Gronwall-Halanay inequality for (4).