Vlasov equations on digraph measures

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 Neunzerts 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.