The resistance distance is a diffusion distance on a graph
Estrada, Ernesto
Mathematics 13, 2380 (2025)
Mathematics 13, 2380 (2025)
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.
Related research projects
SNANDOG
Spatiotemporal Nonlocal and Non-conservative Diffusion on Graphs
P.I.: Ernesto Estrada
Complex systems permeate biological life, society, and modern infrastructures. Their functioning depends primarily on the dynamic processes that take place between interacting entities that form network structures. Such structures can be appropriately …
Related News
Ernesto Estrada’s research recognized with Editor’s Choice and Feature Paper in Mathematics
March 6, 2026
The study, published in Mathematics, introduces a new theoretical perspective linking resistance distance in networks with diffusion processes on graphs.The research is the result of work carried out by Ernesto Estrada at the Institute for Cross-Disciplinary Physics and Complex Systems …
Una recerca d'Ernesto Estrada és reconeguda com a Editor’s Choice i Feature Paper en Mathematics
March 6, 2026
L'estudi, publicat en Mathematics, introdueix una nova perspectiva teòrica que connecta la distància de resistència en xarxes amb processos de difusió en grafs.La recerca és el resultat del treball realitzat per Ernesto Estrada en l'Institut de Física Interdisciplinària i Sistemes …
Una investigación de Ernesto Estrada es reconocida como Editor’s Choice y Feature Paper en Mathematics
March 6, 2026
El estudio, publicado en Mathematics, introduce una nueva perspectiva teórica que conecta la distancia de resistencia en redes con procesos de difusión en grafos.La investigación es el resultado del trabajo realizado por Ernesto Estrada en el Instituto de Física Interdisciplinar …