The mean first passage time as a natural diffusion distance

Given an irreducible finite Markov chain, we propose the mean first passage time (MFPT) as a diffusion distance. We motivate this definition by considering a compact Riemannian manifold, and the submanifold resulting from removing the closure of a small ball. The steady-state solution to an associated inhomogeneous heat flow problem on the submanifold is non-negative and can be viewed as having large values at locations which are “far” away from the removed ball. The same function is also shown, at any point, to “count”, via probability, the paths starting at that point which miss the ball. As a third viewpoint, the same function gives the expected value of the first hitting time of the removed ball. The latter interpretation leads to our proposing the MFPT as a diffusion distance for a given finite set of states (samples) and an associated transition matrix. Even if the transition matrix does not arise from heat flow, and may in fact be non-symmetric and non-bistochastic, we note that the MFPT satisfies the triangle inequality. Moreover, various efficient ways to compute the MFPT, and approximations to the MFPT, have been proposed in the literature.Additionally, we establish a novel connection between certain mean first passage times and the sum of squares of Coifman-Lafon diffusion distances across all scales.