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Journal of Applied Mathematics
Volume 2010 (2010), Article ID 464815, 17 pages
Research Article

Some Remarks on Diffusion Distances

1Theoretical and Applied Science, Ramapo College of NJ, 505 Ramapo Valley Road, Mahwah, NJ 07430, USA
2Mathematics Department, SUNY Rockland Community College, 145 College Road, Suffern, NY 10901, USA

Received 14 June 2010; Revised 31 July 2010; Accepted 3 September 2010

Academic Editor: Andrew Pickering

Copyright © 2010 Maxim J. Goldberg and Seonja Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


As a diffusion distance, we propose to use a metric (closely related to cosine similarity) which is defined as the 𝐿 2 distance between two 𝐿 2 -normalized vectors. We provide a mathematical explanation as to why the normalization makes diffusion distances more meaningful. Our proposal is in contrast to that made some years ago by R. Coifman which finds the 𝐿 2 distance between certain 𝐿 1 unit vectors. In the second part of the paper, we give two proofs that an extension of mean first passage time to mean first passage cost satisfies the triangle inequality; we do not assume that the underlying Markov matrix is diagonalizable. We conclude by exhibiting an interesting connection between the (normalized) mean first passage time and the discretized solution of a certain Dirichlet-Poisson problem and verify our result numerically for the simple case of the unit circle.