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

Benoît Bonnet (1, 2, 3) , Nastassia Pouradier Duteil (3, 2) , Mario Sigalotti (3, 1)

1 CaGE - Control And GEometry
2 MAMBA - Modelling and Analysis for Medical and Biological Applications
3 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions

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 model to describe multi-agent systems, we adapt the classical notion of scrambling coefficient to this setting, and leverage it to establish sufficient conditions ensuring the exponential convergence to consensus with respect to the L∞-norm topology. We then shift our attention to consensus formation expressed in terms of the L2-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 L2-and L∞-consensus formation, and illustrate our contributions by a series of numerical simulations.

Mots clés: Multi-Agent Systems; Graphon Dynamics; Consensus Formation; Algebraic Connectivity; Scrambling Coefficient; Persistence Conditions

Domaines: Optimisation et contrôle [math.OC]; Mathématiques [math]

Benoît Bonnet, Nastassia Pouradier Duteil, Mario Sigalotti. Consensus Formation in First-Order Graphon Models with Time-Varying Topologies. 2021. ⟨hal-03418399v1⟩