We investigate the noise sensitivity of the top eigenvector of a sparse random symmetric matrix. Let v be the top eigenvector of an 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 . Building on recent results on sparse random matrices and a noise sensitivity analysis previously developed for Wigner matrices, we prove that, if , with high probability, when , the vectors v and are almost collinear and, on the contrary, when , the vectors v and are almost orthogonal. A similar result holds for the eigenvector associated to the second largest eigenvalue of the adjacency matrix of an Erdős-Rényi random graph with average degree .
CB was supported by the research grant ANR-16-CE40-0024-01. JL was supported by the National Research Foundation of Korea (NRF-2017R1A2B2001952; NRF-2019R1A5A1028324).
We thank the referees for their careful reading of the manuscript and many helpful suggestions.
Charles Bordenave. Jaehun Lee. "Noise sensitivity for the top eigenvector of a sparse random matrix." Electron. J. Probab. 27 1 - 50, 2022. https://doi.org/10.1214/22-EJP770