  One of a classical synchronization model is the Kuramoto model. We propose
both first and second order Kuramoto dynamical models on graphs using discrete
optimal transport dynamics. We analyze the synchronization behaviors for some
examples of Kuramoto models on graphs. We also provide a generalized Hopf-Cole
transformation for discrete optimal transport systems. Focus on the two points
graph, we derive analytical formulas of the Kuramoto dynamics with various
potential induced from entropy functionals. Several numerical examples for the
Kuramoto model on general graphs are presented.
