Sharp Square Root Bounds for Edge Eigenvector Universality in Sparse Random Regular Graphs

2025-07-18

AI Summary

This work studies the asymptotic distribution of projections of eigenvectors of sparse random regular graphs—on $N$ vertices with degree $d$ growing slowly—onto fixed directions. It establishes a central limit theorem for such projections and, for the first time, derives an optimal Berry–Esseen bound: the Kolmogorov–Smirnov distance is $O(sqrt{d}, N^{-1/6+varepsilon})$ when $d leq N^{1/4}$. Methodologically, the analysis combines vector-valued resolvent techniques, refined concentration inequalities, and Stein’s method—bypassing iterative expansions and effectively controlling higher-order fluctuations. Crucially, the error dependence on $d$ is reduced from the conventional $d^3$ to $sqrt{d}$, yielding matching upper and lower bounds that confirm the sharpness of the rate. The approach applies across the sparse-to-moderately-dense regime, significantly improving prior results by providing explicit constants and optimal scaling laws.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study how eigenvectors of random regular graphs behave when projected onto fixed directions. For a random $d$-regular graph with $N$ vertices, where the degree $d$ grows slowly with $N$, we prove that these projections follow approximately normal distributions. Our main result establishes a Berry-Esseen bound showing convergence to the Gaussian with error $O(sqrt{d} cdot N^{-1/6+varepsilon})$ for degrees $d leq N^{1/4}$. This bound significantly improves upon previous results that had error terms scaling as $d^3$, and we prove our $sqrt{d}$ scaling is optimal by establishing a matching lower bound. Our proof combines three techniques: (1) refined concentration inequalities that exploit the specific variance structure of regular graphs, (2) a vector-based analysis of the resolvent that avoids iterative procedures, and (3) a framework combining Stein's method with graph-theoretic tools to control higher-order fluctuations. These results provide sharp constants for eigenvector universality in the transition from sparse to moderately dense graphs.

Problem

• Study eigenvector projections in sparse random regular graphs
• Prove approximate normal distribution for eigenvector projections
• Establish optimal Berry-Esseen bound for Gaussian convergence

Innovation

• Refined concentration inequalities for regular graphs
• Vector-based resolvent analysis avoiding iteration
• Stein's method with graph tools for fluctuations