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

2026-02-16

AI Summary

This study addresses the statistical distinguishability between random geometric graphs with smooth kernels and Erdős–Rényi graphs with matching edge densities, aiming to characterize the detection threshold. By introducing a novel analytical framework based on posterior overlap, the work overcomes limitations of traditional Kullback–Leibler divergence methods in low dimensions. The approach integrates trace analysis of normalized kernel operators on the sphere, posterior overlap estimation, and high-dimensional probability techniques to establish a unified spectral-type conjecture. The main contributions include proving that, under smooth kernels, the critical dimension for reliable detection of graph structure is \(d = n^{3/4}\), and identifying \(d = \sqrt{n}\) as the phase transition threshold beyond which nontrivial estimation of latent position vectors becomes possible.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• random geometric graph
• detection threshold
• Erdős–Rényi graph
• sharp threshold
• kernel

Innovation

• random geometric graphs
• sharp detection threshold
• smooth kernels
• posterior overlap
• spectral conjecture