Random geometric graphs with smooth kernels: sharp detection threshold and a spectral conjecture

TL;DR

This paper investigates the detection threshold for random geometric graphs with smooth kernels, revealing a lower critical dimension for distinguishability than previously known and proposing a spectral conjecture relating the dimension to kernel properties.

Contribution

It provides the first sharp threshold analysis for smooth kernel RGGs, extends results to signal-dependent kernels, and introduces a new posterior analysis approach for latent point estimation.

Findings

Critical dimension for detection is $d = n^{3/4}$ for smooth kernels.

Established $d = \sqrt{n}$ as the threshold for latent vector estimation.

Proposed a spectral conjecture linking the critical dimension to kernel operator trace.

Abstract

A random geometric graph (RGG) with kernel $K$ is constructed by first sampling latent points $x_1,\ldots,x_n$ independently and uniformly from the $d$-dimensional unit sphere, then connecting each pair $(i,j)$ with probability $K(\langle x_i,x_j\rangle)$. We study the sharp detection threshold, namely the highest dimension at which an RGG can be distinguished from its Erd\H{o}s--R\'enyi counterpart with the same edge density. For dense graphs, we show that for smooth kernels the critical scaling is $d = n^{3/4}$, substantially lower than the threshold $d = n^3$ known for the hard RGG with step-function kernels \cite{bubeck2016testing}. We further extend our results to kernels whose signal-to-noise ratio scales with $n$, and formulate a unifying conjecture that the critical dimension is determined by $n^3 \mathop{\rm tr}^2(\kappa^3) = 1$, where $\kappa$ is the standardized kernel operator on the sphere. Departing from the prevailing approach of bounding the Kullback-Leibler divergence by successively exposing latent points, which breaks down in the sublinear regime of $d=o(n)$, our key technical contribution is a careful analysis of the posterior distribution of the latent points given the observed graph, in particular, the overlap between two independent posterior samples. As a by-product, we establish that $d=\sqrt{n}$ is the critical dimension for non-trivial estimation of the latent vectors up to a global rotation.