Testing for high-dimensional geometry in random graphs

TL;DR

This paper investigates methods to detect high-dimensional geometric structures in random graphs, proposing efficient tests in dense regimes and exploring detection boundaries and dimension estimation in sparse regimes.

Contribution

It introduces a new signed triangles statistic for efficient detection and provides bounds on detection limits, advancing understanding of geometric structure detection in graphs.

Findings

Proposes a near-optimal testing procedure based on signed triangles.

Establishes a new bound on total variation distance between Wishart and GOE matrices.

Conjectures on the detection boundary in sparse regimes.

Abstract

We study the problem of detecting the presence of an underlying high-dimensional geometric structure in a random graph. Under the null hypothesis, the observed graph is a realization of an Erd\H{o}s-R\'enyi random graph $G(n,p)$. Under the alternative, the graph is generated from the $G(n,p,d)$ model, where each vertex corresponds to a latent independent random vector uniformly distributed on the sphere $\mathbb{S}^{d-1}$, and two vertices are connected if the corresponding latent vectors are close enough. In the dense regime (i.e., $p$ is a constant), we propose a near-optimal and computationally efficient testing procedure based on a new quantity which we call signed triangles. The proof of the detection lower bound is based on a new bound on the total variation distance between a Wishart matrix and an appropriately normalized GOE matrix. In the sparse regime, we make a conjecture for the optimal detection boundary. We conclude the paper with some preliminary steps on the problem of estimating the dimension in $G(n,p,d)$.