Noise sensitivity for the top eigenvector of a sparse random matrix

TL;DR

This paper studies how the top eigenvector of a sparse random symmetric matrix reacts to small changes, showing it remains stable under few modifications but becomes orthogonal with many, depending on the degree and size of perturbation.

Contribution

It extends noise sensitivity analysis to sparse matrices, establishing thresholds for eigenvector stability and orthogonality based on matrix sparsity and perturbation size.

Findings

Eigenvector stability when perturbations are small (k bc N^{5/3})

Eigenvector orthogonality when perturbations are large (k b7 N^{5/3})

Results apply to adjacency matrices of Erd
51s-Re9nyi graphs with degree b7 N^{2/9}

Abstract

We investigate the noise sensitivity of the top eigenvector of a sparse random symmetric matrix. Let $v$ be the top eigenvector of an $N\times N$ sparse random symmetric matrix with an average of $d$ non-zero centered entries per row. We resample $k$ randomly chosen entries of the matrix and obtain another realization of the random matrix with top eigenvector $v^{[k]}$. Building on recent results on sparse random matrices and a noise sensitivity analysis previously developed for Wigner matrices, we prove that, if $d\geq N^{2/9}$, with high probability, when $k \ll N^{5/3}$, the vectors $v$ and $v^{[k]}$ are almost collinear and, on the contrary, when $k\gg N^{5/3}$, the vectors $v$ and $v^{[k]}$ are almost orthogonal. A similar result holds for the eigenvector associated to the second largest eigenvalue of the adjacency matrix of an Erd\H{o}s-R\'enyi random graph with average degree $d \geq N^{2/9}$.