  The Kuramoto model (KM) of coupled phase oscillators on graphs provides the
most influential framework for studying collective dynamics and
synchronization. It exhibits a rich repertoire of dynamical regimes. Since the
work of Strogatz and Mirollo, the mean field equation derived in the limit as
the number of oscillators in the KM goes to infinity, has been the key to
understanding a number of interesting effects, including the onset of
synchronization and chimera states. In this work, we study the mathematical
basis of the mean field equation as an approximation of the discrete KM.
Specifically, we extend the Neunzert's method of rigorous justification of the
mean field equation to cover interacting dynamical systems on graphs. We then
apply it to the KM on convergent graph sequences with non-Lipschitz limit. This
family of graphs includes many graphs that are of interest in applications,
e.g., nearest-neighbor and small-world graphs.
