## Wasserstein Eigendistances

Wasserstein eigendistances are intrinsic distances of a Markov chain. Here I considered simple random walks on various finite graphs. The distance between two vertices is a measure for how easy it is for the random walk to distinguish the two vertices.

## Simulations

The simulations show convergence to Wasserstein eigendistances starting from the trivial distance, where any two distinct nodes have distance 1. That is, we consider the map $$W(\rho)(x,y) = \inf_{\pi_{xy}}\int \rho \;d\pi,$$ where \(\pi_{xy}\) is a coupling of \(P_x\) and \(P_y\), the transition kernel of the random walk when starting in \(x\) or \(y\). Then we let \(\rho_0(x,y)=\delta_x(y)\) and \(\rho_{n+1}=\frac{W(\rho_n)}{\sup_{x,y}W(\rho_n)(x,y)}\). The numerical solutions are then embedded in the plane so that distances across an edge \(xy\) are close to \(\rho_n(x,y)\). The number of iterations increases in each time step to better illustrate the convergence.