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

TL;DR

This work resolves the sharp detection threshold for soft random geometric graphs with smooth kernels, showing that for fixed kernels the detection threshold scales as $d_*=n^{3/4}$, markedly lower than the hard-RGG $n^3$ benchmark. By analyzing the full posterior distribution of latent points and the overlap between independent posterior samples, the authors derive tight detection bounds across low- and high-SNR regimes, extend results to polynomial and general kernels, and formulate a unifying spectral conjecture based on the standardized kernel operator $oldsymbol{ kappa}$ via ${ m tr}(oldsymbol{ kappa}^3)$. The paper also establishes a latent-point recovery threshold at $d= ilde{O}( oot 2 ) $ (up to constants) with a simple spectral method achieving consistency when $1\nle d def \sqrt{n}$ and non-trivial recovery otherwise. Together, these contributions connect kernel smoothness, spectral properties, and information-theoretic limits to provide a comprehensive picture of detection and estimation in soft RGGs, with a conjectured universal criterion $n^3 { m tr}^2(oldsymbol{ kappa}^3)=1$ governing the tenability of detection across kernel families.

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 Erdős--Rényi 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 $n^3 \mathop{\rm tr}^2(κ^3) = 1$, where $κ$ 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.

Theorems & Definitions (78)

• Theorem 1
• Remark 1
• Theorem 2
• Theorem 3: Linear kernel
• Theorem 4
• Theorem 5
• Theorem 6
• Conjecture 1
• Theorem 7
• proof