Vlasov equations on digraph measures

Abstract

Many science phenomena are described as interacting particle systems (IPS). The mean field limit (MFL) of large all-to-all coupled deterministic IPS is given by the solution of a PDE, the Vlasov Equation (VE). Yet, many applications demand IPS coupled on networks/graphs. In this paper, we are interested in IPS on directed graphs, or digraphs for short. It is interesting to know, how the limit of a sequence of digraphs associated with the IPS influences the macroscopic MFL of the IPS. This paper studies VEs on a generalized digraph, regarded as limit of a sequence of digraphs, which we refer to as a digraph measure (DGM) to emphasize that we work with its limit via measures. We provide (i) unique existence of solutions of the VE on continuous DGMs, and (ii) discretization of the solution of the VE by empirical distributions supported on solutions of an IPS via ODEs coupled on a sequence of digraphs converging to the given DGM. The result substantially extends results on one-dimensional Kuramoto-type models and we allow the underlying digraphs to be not necessarily dense. The technical contribution of this paper is a generalization of Neunzert's in-cell-particle approach from a measure-theoretic viewpoint, which is different from the known techniques in L p -functions using graphons and their generalization via harmonic analysis of locally compact Abelian groups. Finally, we apply our results to various models in higher-dimensional Euclidean spaces in epidemiology, ecology, and social sciences.

Mean field limits of co-evolutionary heterogeneous networks

“…The techniques have led to a quite complete derivation of the MFL for the Kuramoto model for all-to-all coupling by Lancellotti [29]. Later, MFL of Kuramoto oscillators on a sequence of dense heterogeneous (deterministic or random) graphs with and without Lipschitz continuity were studied, e.g., in [23,13,14,25,17,26]. Recently, results were extended to sparse graphs using different approaches [35,28,25,17,26,21].…”

“…Later, MFL of Kuramoto oscillators on a sequence of dense heterogeneous (deterministic or random) graphs with and without Lipschitz continuity were studied, e.g., in [23,13,14,25,17,26]. Recently, results were extended to sparse graphs using different approaches [35,28,25,17,26,21]. So far, all the above network models are given on a static network.…”

“…It is worth mentioning that the approaches in [35,28,21] in dealing with sparsity are more graph-theoretic, the ones in [23] is based on analysis of L p -functions while restricted to graphon type graph limits, the one in [17] is more operator-theoretic combined with harmonic analysis techniques, and [26] more measuretheoretic. It is noteworthy that results for graph limits of a sequence of intermediate/low density are also covered by the approaches in [17,26]. The fast development of mean field theory of IPS coupled on heterogeneous networks owes much to the development of graph limits [6,30,27,3].…”

“…The density of the MFL is captured by the solution of a transport type PDE, the so-called Vlasov equation, cf. [33,23,25,17,26].…”

“…Therefore, we equivalently represent the Kuramoto model as an integral equation coupled on a static initial graph. Then one may be able to utilize the techniques in establishing the MFL of an IPS on a graph limit (e.g., [23,17,26]). Nevertheless, the added effects coming from the co-evolutionary terms bring new difficulties in establishing continuous dependence of the solutions on the initial conditions, since the standard technique by Gronwall inequalities fails.…”