Kron Reduction and Effective Resistance of Directed Graphs

TL;DR

This paper extends Kron reduction and effective resistance concepts to directed graphs, preserving key structural properties and introducing new metrics related to commute and covering times, with applications in stochastic network analysis.

Contribution

It proposes a generalized Kron reduction for directed graphs and a novel effective resistance measure that captures important network dynamics.

Findings

The reduction preserves strong connectivity and weight balance.

Effective resistance induces new graph metrics for directed graphs.

In stochastic cases, resistance relates to hitting probabilities and ergodic Markov chains.

Abstract

In network theory, the concept of effective resistance is a distance measure on a graph that relates the global network properties to individual connections between nodes. In addition, the Kron reduction method is a standard tool for reducing or eliminating the desired nodes, which preserves the interconnection structure and the effective resistance of the original graph. Although these two graph-theoretic concepts stem from the electric network on an undirected graph, they also have a number of applications throughout a wide variety of other fields. In this study, we propose a generalization of a Kron reduction for directed graphs. Furthermore, we prove that this reduction method preserves the structure of the original graphs, such as the strong connectivity or weight balance. In addition, we generalize the effective resistance to a directed graph using Markov chain theory, which is invariant under a Kron reduction. Although the effective resistance of our proposal is asymmetric, we prove that it induces two novel graph metrics in general strongly connected directed graphs. In particular, the effective resistance captures the commute and covering times for strongly connected weight balanced directed graphs. Finally, we compare our method with existing approaches and relate the hitting probability metrics and effective resistance in a stochastic case. In addition, we show that the effective resistance in a doubly stochastic case is the same as the resistance distance in an ergodic Markov chain.