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

2026-07-13

AI Summary

This work addresses the problem of efficiently identifying substructures with approximate period $p$ in strongly connected directed graphs and Markov chains. The authors propose a spectral algorithm based on the eigenvalue analysis of the rotational Laplacian matrix, introducing a generalized periodicity ratio and establishing its tight connection to the eigenvalues of this operator. Key contributions include the derivation of novel probabilistic bounds for complex Gaussian variables, an extension of higher-order Cheeger-type inequalities to periodicity analysis, and the design of a randomized polynomial-time algorithm capable of simultaneously detecting multiple nearly periodic components. This approach provides both theoretical guarantees and practical tools for uncovering periodic structures in directed graphs.

Abstract

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.

Problem

• nearly-periodic components
• digraphs
• Markov chains
• periodicity ratio
• rotated Laplacian

Innovation

• rotated Laplacian
• periodicity ratio
• spectral algorithm
• higher-order Cheeger inequality
• complex Gaussian variables