On a Class of Nonlocal Continuity Equations on Graphs

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.

References in corpus (5)

• Convergence of a Fully Discrete and Energy-Dissipating Finite-Volume Scheme for Aggregation-Diffusion Equations
• Continuum Limit of Lipschitz Learning on Graphs
• The Scharfetter--Gummel scheme for aggregation-diffusion equations
• Cosh gradient systems and tilting
• Homogenisation of dynamical optimal transport on periodic graphs

Cited by in corpus (2)

• Variational convergence of the Scharfetter-Gummel scheme to the aggregation-diffusion equation and vanishing diffusion limit
• Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs