Noise sensitivity of second-top eigenvectors of Erd\H{o}s-R\'{e}nyi graphs and sparse matrices

We consider eigenvectors of adjacency matrices of Erd\H{o}s-R\'{e}nyi graphs and study the variation of their directions by resampling the entries randomly. Let $\mathbf{v}$ be the eigenvector associated with the second-largest eigenvalue of the Erd\H{o}s-R\'{e}nyi graphs. After choosing $k$ entries of the given matrix randomly and resampling them, we obtain another eigenvector $\mathbf{w}$ corresponding to the second-largest eigenvalue of the matrix obtained from the resampling procedure. We prove that, in a certain sparsity regime, $\mathbf{w}$ is "almost" orthogonal to $\mathbf{v}$ with high probability if $k\gg N^{5/3}$. On the other hand, if $k\ll q^2 N^{2/3}$, where $q$ is the sparsity parameter, we observe that $\mathbf{v}$ and $\mathbf{w}$ are "almost" collinear. This extends the recent work of Bordenave, Lugosi and Zhivotovskiy to the Erd\H{o}s-R\'{e}nyi model.