  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 $d \geq N^{2/9}$.
