Noise sensitivity for the top eigenvector of a sparse random matrix

We investigate the noise sensitivity of top eigenvector a sparse random symmetric matrix. Let v be an N×N matrix with average d non-zero centered entries per row. resample k randomly chosen and obtain another realization v[k]. Building on recent results matrices analysis previously developed for Wigner matrices, we prove that, if d?N2?9, high probability, when k?N5?3, vectors v[k] are almost collinear and, contrary, k?N5?3, orthogonal. A similar result holds associated to second largest eigenvalue adjacency Erd?s-Rényi graph degree d?N2?9.