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

TL;DR

This paper extends the diffusion state distance (DSD) to bipartite graphs, links it to Green's function, and computes DSD distances for various graph types and biological networks, enhancing understanding of functional similarity in networks.

Contribution

It introduces a new extension of DSD to bipartite graphs using lazy-random walks and connects DSD to Green's function, enabling explicit calculations for several graph classes.

Findings

Extended DSD to bipartite graphs using lazy-random walks

Connected DSD $L_q$-distance to Green's function

Computed DSD distances for paths, cycles, hypercubes, and biological networks

Abstract

The diffusion state distance (DSD) was introduced by Cao-Zhang-Park-Daniels-Crovella-Cowen-Hescott [{\em PLoS ONE, 2013}] to capture functional similarity in protein-protein interaction networks. They proved the convergence of DSD for non-bipartite graphs. In this paper, we extend the DSD to 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 [{\em J. 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 biological networks.