Contribution of directedness in graph spectra

TL;DR

This paper investigates how ignoring edge directions in graphs affects their spectral properties, revealing that under certain conditions, the spectrum remains largely conserved despite simplification.

Contribution

It introduces the concept of random directization and analytically demonstrates its effect on the spectral structure of directed graphs.

Findings

Random directization often preserves spectral structure

Spectrum conservation occurs in the perturbative regime

Ignoring directions may not significantly alter spectral properties

Abstract

In graph analyses, directed edges are often approximated to undirected ones so that the adjacency matrices may be symmetric. However, such simplification has not been thoroughly verified. In this study, we investigate how directedness affects the graph spectra by introducing random directization, which is an opposite operation of neglecting edge directions. We analytically reveal that uniformly random directization typically conserves the relative spectral structure of the adjacency matrix in the perturbative regime. The result of random directization implies that the spectrum of the adjacency matrix can be conserved after the directedness is ignored.