Non-backtracking eigenvalues and eigenvectors of random regular graphs and hypergraphs

Abstract: The non-backtracking operator of a graph is a powerful tool in spectral graph theory and random matrix theory. Most existing results for the non-backtracking operator of a random graph concern only eigenvalues or top eigenvectors. In this paper, we take the first step in analyzing its bulk eigenvector behaviors. We demonstrate that for the non-backtracking operator $B$ of a random $d$-regular graph, its eigenvectors corresponding to nontrivial eigenvalues are completely delocalized with high probability. Additionally, we show complete delocalization for a reduced $2n \times 2n$ non-backtracking matrix $\tilde{B}$. By projecting all eigenvalues of $\tilde{B}$ onto the real line, we obtain an empirical measure that converges weakly in probability to the Kesten-McKay law for fixed $d\geq 3$ and to a semicircle law as $d \to\infty$ with $n \to\infty$. We extend our analysis to random regular hypergraphs, including the limiting measure of the real part of the spectrum for $\tilde{B}$, $\ell_{\infty}$-norm bounds for the eigenvectors of $\tilde{B}$ and $B$, and a deterministic relation between eigenvectors of $B$ and the eigenvectors of the adjacency matrix. As an application, we analyze the non-backtracking spectrum of the regular stochastic block model (RSBM) and provide a spectral method based on eigenvectors of $\tilde{B}$ to recover the community structure exactly. We also show that there exists an isolated real eigenvalue with an informative eigenvector inside the circle of radius $\sqrt{d_1+d_2-1}$ in the spectrum of $B$, analogous to the "eigenvalue insider" phenomenon for the Erd\H{o}s-R\'{e}nyi stochastic block model conjectured in Dall'Amico et al. (2019).