  We study the Wasserstein Hamiltonian flow with a common noise on the density
manifold of a finite graph. Under the framework of stochastic variational
principle, we first develop the formulation of stochastic Wasserstein
Hamiltonian flow and show the local existence of a unique solution. We also
establish a sufficient condition for the global existence of the solution.
Consequently, we obtain the global well-posedness for the nonlinear
Schr\"odinger equations with common noise on graph. In addition, using
Wong-Zakai approximation of common noise, we prove the existence of the
minimizer for an optimal control problem with common noise. We show that its
minimizer satisfies the stochastic Wasserstein Hamiltonian flow on graph as
well.
