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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 shown to give the expected value of the first hitting time of the removed ball from any location in the submanifold.  

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 have been proposed in the literature.

Zoom link: https://yale.zoom.us/j/99114648888