Effective Resistance Preserving Directed Graph Symmetrization

TL;DR

This paper introduces a novel symmetrization method for directed graphs that preserves effective resistances, enabling spectral analysis and applications like graph partitioning and Kron reduction.

Contribution

A new symmetrization technique that maintains effective resistances and spectral properties of directed graphs, facilitating analysis and reduction methods.

Findings

Effective resistance is preserved in the symmetrization process.

Spectral properties of the directed graph are interpretable through the symmetrized version.

The method enables applications in spectral partitioning and Kron reduction.

Abstract

This work presents a new method for symmetrization of directed graphs that constructs an undirected graph with equivalent pairwise effective resistances as a given directed graph. Consequently a graph metric, square root of effective resistance, is preserved between the directed graph and its symmetrized version. It is shown that the preservation of this metric allows for interpretation of algebraic and spectral properties of the symmetrized graph in the context of the directed graph, due to the relationship between effective resistance and the Laplacian spectrum. Additionally, Lyapunov theory is used to demonstrate that the Laplacian matrix of a directed graph can be decomposed into the product of a projection matrix, a skew symmetric matrix, and the Laplacian matrix of the symmetrized graph. The application of effective resistance preserving graph symmetrization is discussed in the context of spectral graph partitioning and Kron reduction of directed graphs.