On a Class of Nonlocal Continuity Equations on Graphs

TL;DR

This paper investigates nonlocal continuity equations on graphs motivated by data science, establishing existence and uniqueness of solutions and highlighting structural differences from Euclidean spaces based on the interpolation functions used.

Contribution

It introduces a fixed-point approach to prove existence and uniqueness for a broad class of nonlocal PDEs on graphs, considering various interpolation functions.

Findings

Existence and uniqueness of solutions established

Structural differences from Euclidean spaces identified

Analysis depends on the choice of interpolation functions

Abstract

Motivated by applications in data science, we study partial differential equations on graphs. By a classical fixed-point argument, we show existence and uniqueness of solutions to a class of nonlocal continuity equations on graphs. We consider general interpolation functions, which give rise to a variety of different dynamics, e.g., the nonlocal interaction dynamics coming from a solution-dependent velocity field. Our analysis reveals structural differences with the more standard Euclidean space, as some analogous properties rely on the interpolation chosen.