Consensus Formation in First-Order Graphon Models with Time-Varying Topologies

TL;DR

This paper studies how consensus forms in large multi-agent systems modeled by time-varying graphon dynamics, providing conditions for exponential convergence and exploring the relation between different norms.

Contribution

It introduces a framework for analyzing consensus in infinite-dimensional graphon models, extending classical notions like scrambling coefficient and algebraic connectivity.

Findings

Exponential consensus conditions in $L^{inity}$-norm for time-dependent graphon dynamics.

Three consensus results for symmetric, balanced, and strongly connected topologies.

Numerical simulations illustrating the theoretical results.

Abstract

In this article, we investigate the asymptotic formation of consensus for several classes of time-dependent cooperative graphon dynamics. After motivating the use of this type of macroscopic models to describe multi-agent systems, we adapt the classical notion of scrambling coefficient to this setting, leverage it to establish sufficient conditions ensuring the exponential convergence to consensus with respect to the $L^{\infty}$-norm topology. We then shift our attention to consensus formation expressed in terms of the $L^2$-norm, and prove three different consensus result for symmetric, balanced and strongly connected topologies, which involve a suitable generalisation of the notion of algebraic connectivity to this infinite-dimensional framework. We then show that, just as in the finite-dimensional setting, the notion of algebraic connectivity that we propose encodes information about the connectivity properties of the underlying interaction topology. We finally use the corresponding results to shed some light on the relation between $L^2$- and $L^{\infty}$-consensus formation, and illustrate our contributions by a series of numerical simulations.