  We study the nonlinear Fokker-Planck equation on graphs, which is the
gradient flow in the space of probability measures supported on the nodes with
respect to the discrete Wasserstein metric. The energy functional driving the
gradient flow consists of a Boltzmann entropy, a linear potential and a
quadratic interaction energy. We show that the solution converges to the Gibbs
measures exponentially fast with a rate that can be given analytically. The
continuous analog of this asymptotic rate is related to the Yano's formula.
