  Fokker-Planck equations represent a suitable description of the finite-time
behavior for a large class of particle systems as the size of the population
tends to infinity. Recently, the theory of graph limits has been introduced in
the classical mean-field framework to account for heterogeneous interactions
among particles. In many instances, such network heterogeneity is preserved in
the limit which turns from being a single Fokker-Planck equation (also known as
McKean-Vlasov) to an infinite system of non-linear partial differential
equations (PDE) coupled by means of a graphon. While appealing from an applied
viewpoint, few rigorous results exist on the graphon particle system. This note
addresses such limit system focusing on the relation between interaction
network and initial conditions: if the system initial datum and the graphon
degrees satisfy a suitable condition, a significantly simpler representation of
the solution is available. Interesting conclusions can be drawn from this
result: substantially different graphons can give rise to exactly the same
particle behavior, e.g., if the initial condition is independent of the
graphon, the particle behavior is indistinguishable from the well-known
mean-field limit as long as the graphon has constant degree; step-kernels, a
building block of graph limit theory, can be used to approximate the graphon
particle system with a finite system of Fokker-Planck equations and an explicit
rate of convergence.
