Localized eigenvectors of the non-backtracking matrix

TL;DR

This paper investigates the localization of eigenvectors in the non-backtracking matrix used for graph partitioning, revealing that localized eigenvectors can exist outside the spectral band, affecting the method's effectiveness.

Contribution

It demonstrates the existence of localized eigenvectors outside the spectral band of the non-backtracking matrix, challenging previous assumptions about its spectral properties.

Findings

Non-backtracking matrix eigenvectors are generally not localized within the spectral band.

Localized eigenvectors can exist outside the spectral band, impacting partitioning performance.

The spectral method's failure can be linked to these localized eigenvectors outside the spectral band.

Abstract

In the case of graph partitioning, the emergence of localized eigenvectors can cause the standard spectral method to fail. To overcome this problem, the spectral method using a non-backtracking matrix was proposed. Based on numerical experiments on several examples of real networks, it is clear that the non-backtracking matrix does not exhibit localization of eigenvectors. However, we show that localized eigenvectors of the non-backtracking matrix can exist outside the spectral band, which may lead to deterioration in the performance of graph partitioning.