The completely delocalized region of the Erd\H{o}s-R\'enyi graph

TL;DR

This paper characterizes the full range of parameters for the Erdős-Rényi graph where eigenvectors of the adjacency matrix are completely delocalized, identifying critical thresholds for delocalization depending on the average degree.

Contribution

It precisely determines the critical values of the average degree ratio rac{d}{ ext{log} N} for complete eigenvector delocalization in Erdős-Rényi graphs.

Findings

Eigenvectors are fully delocalized when d/log N > 1/(log 4 - 1)

Eigenvectors away from spectral edges are delocalized for d/log N > 1

Localized eigenvectors exist below the critical thresholds

Abstract

We analyse the eigenvectors of the adjacency matrix of the Erd\H{o}s-R\'enyi graph on $N$ vertices with edge probability $\frac{d}{N}$. We determine the full region of delocalization by determining the critical values of $\frac{d}{\log N}$ down to which delocalization persists: for $\frac{d}{\log N} > \frac{1}{\log 4 - 1}$ all eigenvectors are completely delocalized, and for $\frac{d}{\log N} > 1$ all eigenvectors with eigenvalues away from the spectral edges are completely delocalized. Below these critical values, it is known [arXiv:2005.14180, arXiv:2106.12519] that localized eigenvectors exist in the corresponding spectral regions.