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