Consensus formation in first-order graphon models with time-varying topologies

Source abstract

In this paper, 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, and leverage it to establish sufficient conditions ensuring the exponential convergence to consensus with respect to the [Formula: see text]-norm topology. We then shift our attention to consensus formation expressed in terms of the [Formula: see text]-norm, and prove three different results 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 [Formula: see text]- and [Formula: see text]-consensus formation, and illustrate our contributions by a series of numerical simulations.