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Limiting empirical spectral distribution for the non-backtracking matrix of an Erdős-Rényi random graph

Ke Wang, Philip Matchett Wood

Abstract

In this note, we give a precise description of the limiting empirical spectral distribution (ESD) for the non-backtracking matrices for an Erdős-Rényi 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.

Limiting empirical spectral distribution for the non-backtracking matrix of an Erdős-Rényi random graph

Abstract

In this note, we give a precise description of the limiting empirical spectral distribution (ESD) for the non-backtracking matrices for an Erdős-Rényi graph assuming 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.

Paper Structure

This paper contains 12 sections, 15 theorems, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Results
  3. Outline
  4. Background definitions
  5. The spectrum of $H_0$
  6. Spectrum of $\widetilde{H}_0$: Proof of Proposition \ref{['t:H0spectrum']}
  7. Real eigenvalues of $\widetilde{H}_0$ when $p\le n^{-1/2}$
  8. $\widetilde{H}_0$ is diagonalizable
  9. The bulk distribution: proving Theorem \ref{['t:bulkESD-intro']}
  10. Proof of Theorem \ref{['t:bulkESD']}
  11. Perturbation theory: proving Theorem \ref{['thm:main-dense']}
  12. Proof of Lemma \ref{['lem:sing_min']}

Key Result

Proposition 1.2

Let $H_0$ be defined as in e:def_H_0, and let $0< p\le p_0<1$ for a constant $p_0$. If $p\ge C/\sqrt{n}$ for some large constant $C>0$, then, with probability $1-o(1)$, $\frac{1}{\sqrt{\alpha}} H_0$ has two real eigenvalues $\mu_1$ and $\mu_2$ satisfying $\mu_1=\sqrt{\alpha} (1+o(1))$ and $\mu_2 = 1

Figures (1)

  • Figure 1: The eigenvalues of $H/\sqrt{\alpha}$ defined in \ref{['e:def_H']} and $H_0/\sqrt{\alpha}$ defined in \ref{['e:def_H_0']} for a sample of $G(n,p)$ with $n=500$ and different values of $p$. The blue circles are the eigenvalues of $H/\sqrt{\alpha}$ and the red x's are for $H_0/\sqrt{\alpha}$. For comparison, the black dashed line is the unit circle. For the figures from top to bottom and from left to right, the values of $p$ are taken to be $p=0.5, p=0.1, p=0.08$ and $p=0.05$ respectively.

Theorems & Definitions (23)

  • Proposition 1.2: Spectrum of the partly averaged matrix
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 2.1: KS03LS18
  • proof
  • proof : Proof of Proposition \ref{['t:H0spectrum']}
  • Theorem 3.1
  • Theorem 3.2: Replacement principle TaoVu2010
  • ...and 13 more