Finding Nearly-Periodic Components in Digraphs and Markov Chains from the Spectrum of Rotated Laplacian Matrices

Inspired by recent advances in notions of spectral approximation of digraphs [Ahm+20], we study spectral algorithms for finding periodic structures in digraphs via the spectrum of a class of rotated Laplacian matrices. This class of Laplacian matrices was previously studied by Lange, Liu, Peyerimhoff, and Post [Lan+15]. We consider a notion of periodicity ratio that generalizes the bipartiteness ratio of Trevisan [Tre09], and show that it is closely related to the spectrum of rotated Laplacian matrices. In particular, if the digraph is strongly connected and represents a Markov chain, this periodicity ratio for a given $p \in \mathbb{N}$ is a quantitative measure of how close this Markov chain is to having periodicity $p$. We propose and analyze a periodicity-ratio variant of the spectral algorithm by Louis, Raghavendra, Tetali and Vempala [Lou+12]. We show that the algorithm runs in randomized polynomial time and can find many nearly periodic components (i.e, components with small periodicity ratio). This also implies a new higher-order Cheeger-type inequality for periodicity in the spirit of that in [Lou+12; LOT14]. As part of our analysis, we prove a new theorem that upper bounds the probability that the largest magnitudes of two sequences of coordinate-wise correlated complex Gaussian random variables occur at different indices, which may be of independent interest. Previously, an analogous result was known only for real Gaussian random variables.