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 a PDE, Vlasov Equation (VE). Yet, many applications demand on networks/graphs. In this paper, we interested in sequence directed graphs, or digraphs for short. It interesting to know, how associated with influences macroscopic MFL. This paper studies VEs generalized digraph, regarded digraphs, which refer digraph measure (DGM) emphasize that work its via measures. We provide (i) unique existence solutions VE continuous DGMs, and (ii) discretization empirical distributions supported an ODEs converging DGM. Our result extends existing results one-dimensional Kuramoto-type networks dense graphs. Here allow underlying be not necessarily include graphical structures such stars, trees rings, have been frequently used sparse network models finance, telecommunications, physics, genetics, neuroscience, social sciences. A key contribution nontrivial generalization Neunzert's in-cell-particle approach indistinguishable global Lipschitz continuity Euclidean spaces distinguishable heterogeneous local continuity, measure-theoretic viewpoint. together metrics different from known techniques Lp-functions using graphons their means harmonic analysis locally compact Abelian groups. Finally, demonstrate wide applicability, apply our various higher-dimensional epidemiology, ecology,