Continuous-Time Consensus under Non-Instantaneous Reciprocity

TL;DR

This paper proves that continuous-time consensus systems with non-instantaneous reciprocity always converge, extending known results from instantaneous reciprocity to more general interaction scenarios, including random and endogenous systems.

Contribution

It introduces convergence conditions for systems with time-extended reciprocity, broadening the understanding of consensus dynamics beyond instantaneous interactions.

Findings

Systems always converge regardless of interaction specifics

Simple conditions ensure agents reach consensus

Applicable to endogenous and random interaction systems

Abstract

We consider continuous-time consensus systems whose interactions satisfy a form or reciprocity that is not instantaneous, but happens over time. We show that these systems have certain desirable properties: They always converge independently of the specific interactions taking place and there exist simple conditions on the interactions for two agents to converge to the same value. This was until now only known for systems with instantaneous reciprocity. These result are of particular relevance when analyzing systems where interactions are a priori unknown, being for example endogenously determined or random. We apply our results to an instance of such systems.