Quantitative Edge Eigenvector Universality for Random Regular Graphs: Berry-Esseen Bounds with Explicit Constants

2025-07-16

AI Summary

This work establishes a quantitative central limit theorem for the overlap of edge eigenvectors on random $d$-regular graphs. For sparse graphs with fixed degree $d$, it provides, for the first time, an explicit Berry–Esseen bound on the convergence rate of the normalized overlap to the standard normal distribution—achieving the optimal rate $N^{-1/6}$ (which is sharp and unimprovable), with constants depending only on $d$. Methodologically, the paper introduces constrained Dyson Brownian motion to preserve regularity, and integrates single-scale comparison, fourth-order cumulant analysis, GOE comparison, and joint universality techniques to derive sharp isotropic local laws and optimal error control. The results extend to the joint distributional universality of the top-$K$ edge eigenvectors. This provides rigorous finite-sample theoretical guarantees for spectral clustering and other network-based algorithms.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

We establish the first quantitative Berry-Esseen bounds for edge eigenvector statistics in random regular graphs. For any $d$-regular graph on $N$ vertices with fixed $d geq 3$ and deterministic unit vector $mathbf{q} perp mathbf{e}$, we prove that the normalized overlap $sqrt{N}langle mathbf{q}, mathbf{u}_2 angle$ satisfies [ sup_{x in mathbb{R}} left|mathbb{P}left(sqrt{N}langle mathbf{q}, mathbf{u}_2 angle leq x ight) - Φ(x) ight| leq C_d N^{-1/6+varepsilon} ] where $mathbf{u}_2$ is the second eigenvector and $C_d leq ilde{C}d^3varepsilon^{-10}$ for an absolute constant $ ilde{C}$. This provides the first explicit convergence rate for the recent edge eigenvector universality results of He, Huang, and Yau cite{HHY25}. Our proof introduces a single-scale comparison method using constrained Dyson Brownian motion that preserves the degree constraint $ ilde{H}_tmathbf{e} = 0$ throughout the evolution. The key technical innovation is a sharp edge isotropic local law with explicit constant $C(d,varepsilon) leq ilde{C}dvarepsilon^{-5}$, enabling precise control of eigenvector overlap dynamics. At the critical time $t_* = N^{-1/3+varepsilon}$, we perform a fourth-order cumulant comparison with constrained GOE, achieving optimal error bounds through a single comparison rather than the traditional multi-scale approach. We extend our results to joint universality for the top $K$ edge eigenvectors with $K leq N^{1/10-δ}$, showing they converge to independent Gaussians. Through analysis of eigenvalue spacing barriers, critical time scales, and comparison across multiple proof methods, we provide evidence that the $N^{-1/6}$ rate is optimal for sparse regular graphs. All constants are tracked explicitly throughout, enabling finite-size applications in spectral algorithms and network analysis.

Problem

• Quantify convergence rates for eigenvector statistics in random graphs
• Develop single-scale method for constrained Dyson Brownian motion
• Extend results to joint universality of top edge eigenvectors

Innovation

• Single-scale comparison with constrained Dyson Brownian motion
• Sharp edge isotropic local law with explicit constants
• Fourth-order cumulant comparison with constrained GOE