Asymptotic consensus with transmission and reaction delay: an overview

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).