The resistance distance is a diffusion distance on a graph

The resistance distance is a squared Euclidean metric on the vertices of a graph derived
from the consideration of a graph as an electrical circuit. Its connection with the commute
time of a random walker on the graph has made it particularly appealing for the analysis
of networks. Here, we prove that the resistance distance is given by a difference of “mass
concentrations” obtained at the vertices of a graph by a diffusive process. The nature of
this diffusive process is characterized here by means of an operator corresponding to the
matrix logarithm of a Perron-like matrix based on the pseudoinverse of the graph Laplacian.
We prove also that this operator is indeed the Laplacian matrix of a signed version of the
original graph, in which nonnearest neighbors’ “interactions” are also considered. In this
way, the resistance distance is part of a family of squared Euclidean distances emerging
from diffusive dynamics on graphs.