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The diffusion state distance (DSD) was introduced byCao-Zhang-Park-Daniels-Crovella-Cowen-Hescott [{\em PLoS ONE, 2013}] to capturefunctional similarity in protein-protein interaction networks. They proved theconvergence of DSD for non-bipartite graphs. In this paper, we extend the DSDto bipartite graphs using lazy-random walks and consider the general$L_q$-version of DSD. We discovered the connection between the DSD$L_q$-distance and Green's function, which was studied by Chung and Yau [{\emJ. Combinatorial Theory (A), 2000}]. Based on that, we computed the DSD$L_q$-distance for Paths, Cycles, Hypercubes, as well as random graphs $G(n,p)$and $G(w_1,..., w_n)$. We also examined the DSD distances of two biologicalnetworks.

Computing Diffusion State Distance using Green's Function and Heat Kernel on Graphs