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Noise sensitivity of second-top eigenvectors of Erdős-Rén...
Jaehun Lee · 2020-01-10 · via math.PR updates on arXiv.org

We consider eigenvectors of adjacency matrices of Erdős-Rényi 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 Erdős-Rényi 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 Erdős-Rényi model.