# Noise sensitivity for the top eigenvector of a sparse random matrix

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 $d \geq N^{2/9}$.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-03447814
Contributor : Charles Bordenave Connect in order to contact the contributor
Submitted on : Wednesday, November 24, 2021 - 8:26:03 PM
Last modification on : Thursday, November 25, 2021 - 3:45:36 AM

### Identifiers

• HAL Id : hal-03447814, version 1
• ARXIV : 2106.09570

### Citation

Charles Bordenave, Jaehun Lee. Noise sensitivity for the top eigenvector of a sparse random matrix. 2021. ⟨hal-03447814⟩

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