Noise sensitivity for the top eigenvector of a sparse random matrix

We investigate the noise sensitivity of the top eigenvector of a sparse random symmetric matrix. Let v be the top eigenvector of an N x 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 >= N-2/9, with high probability, when k << N-5/3, the vectors v and v([k]) are almost collinear and, on the contrary, when k >> 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 Erdos-Renyi random graph with average degree d >= N-2/9.

Automated summary

This study examines the noise sensitivity of the top eigenvector of a sparse random symmetric matrix. By analyzing the noise sensitivity of sparse random matrices and Wigner matrices, the study proves that when the average number of non-zero centered entries in the matrix is large, the change in the top eigenvector leads to the vectors v and v([k]) being nearly collinear or nearly orthogonal.