Resistance Distances in Directed Graphs: Definitions, Properties, and Applications

TL;DR

This paper extends the concept of resistance distance to strongly connected directed graphs, establishing its properties, introducing a Laplacian matrix for directed graphs, and proposing an approximation algorithm for related optimization problems.

Contribution

It provides a physically interpretable, metric-based resistance distance for directed graphs and introduces a Laplacian matrix framework, along with an approximation algorithm for node group selection.

Findings

Resistance distance in directed graphs is a metric.

The Laplacian matrix for directed graphs is introduced and utilized.

A greedy approximation algorithm for node group selection is validated.

Abstract

Resistance distance has been studied extensively in the past years, with the majority of previous studies devoted to undirected networks, in spite of the fact that various realistic networks are directed. Although several generalizations of resistance distance on directed graphs have been proposed, they either have no physical interpretation or are not a metric. In this paper, we first extend the definition of resistance distance to strongly connected directed graphs based on random walks and show that the two-node resistance distance on directed graphs is a metric. Then, we introduce the Laplacian matrix for directed graphs that subsumes the Laplacian matrix of undirected graphs as a particular case and use its pseudoinverse to express the two-node resistance distance, and many other relevant quantities derived from resistance distances. Moreover, we define the resistance distance between a vertex and a vertex group on directed graphs and further define a problem of optimally selecting a group of fixed number of nodes, such that their resistance distance is minimized. Since this combinatorial optimization problem is NP-hard, we present a greedy algorithm with a proved approximation ratio, and conduct experiments on model and realistic networks to validate the performance of this approximation algorithm.