Limiting empirical spectral distribution for the non-backtracking matrix of an Erd\H{o}s-R\'enyi random graph

TL;DR

This paper characterizes the limiting spectral distribution of non-backtracking matrices for Erdős-Rényi graphs when the average degree grows faster than log n, using derandomization and spectral comparison techniques.

Contribution

It provides a precise description of the spectral distribution for non-backtracking matrices in Erdős-Rényi graphs under specific growth conditions, employing derandomization and spectral analysis methods.

Findings

The limiting empirical spectral distribution is explicitly described.

Derandomization simplifies the spectrum significantly.

The spectrum of the derandomized matrix closely approximates the original spectrum.

Abstract

In this note, we give a precise description of the limiting empirical spectral distribution (ESD) for the non-backtracking matrices for an Erd\H{o}s-R\'{e}nyi graph assuming $np/\log n$ tends to infinity. We show that derandomizing part of the non-backtracking random matrix simplifies the spectrum considerably, and then we use Tao and Vu's replacement principle and the Bauer-Fike theorem to show that the partly derandomized spectrum is, in fact, very close to the original spectrum.