Eigenvector fluctuations and limit results for random graphs with infinite rank kernels

TL;DR

This paper analyzes the behavior of leading eigenvectors in complex random graphs with infinite rank kernels, providing new bounds and tests for graph inference, advancing understanding of spectral methods in high-dimensional graph models.

Contribution

It introduces uniform error bounds and a rank-adaptive testing procedure for eigenvectors in general latent position models with infinite rank kernels, extending beyond low-rank or positive semidefinite assumptions.

Findings

Derived uniform error bounds in two-to-infinity norm

Developed a rank-adaptive test for latent position equality

Quantified spectral embedding properties in high-dimensional graphs

Abstract

This paper systematically studies the behavior of the leading eigenvectors for independent edge undirected random graphs generated from a general latent position model whose link function is possibly infinite rank and also possibly indefinite. We first derive uniform error bounds in the two-to-infinity norm as well as row-wise normal approximations for the leading sample eigenvectors. We then build on these results to tackle two graph inference problems, namely (i) entrywise bounds for graphon estimation and (ii) testing for the equality of latent positions, the latter of which is achieved by proposing a rank-adaptive test statistic that converges in distribution to a weighted sum of independent chi-square random variables under the null hypothesis. Our fine-grained theoretical guarantees and applications differ from the existing literature which primarily considers first order upper bounds and more restrictive low rank or positive semidefinite model assumptions. Further, our results collectively quantify the statistical properties of eigenvector-based spectral embeddings with growing dimensionality for large graphs.