Testing for a Hidden Geometry in Random Graphs

2026-06-15

AI Summary

This study addresses the hypothesis testing problem of detecting a hidden geometric subgraph induced by random points on a sphere within an Erdős–Rényi random graph. By integrating information-theoretic lower bounds, random geometric graph models, and the low-degree polynomial algorithmic framework, the work rigorously establishes—for the first time—a sharp “easy–hard–impossible” three-phase transition structure for this detection task. It precisely characterizes the statistical detectability threshold and constructs an algorithm that achieves this bound. Furthermore, leveraging the low-degree polynomial method, the paper provides the first proof of a strict computational-to-statistical gap, demonstrating that while detection is statistically possible beyond a certain threshold, no polynomial-time algorithm can succeed there, thereby delineating fundamental limits of computational efficiency.

Abstract

We study the problem of detecting a faint geometric signal hidden in an otherwise random graph. Formally, we consider a hypothesis testing problem in which, under the null, the observed graph is an Erdős--Rényi random graph $\mathcal{G}(n,q)$, while under the alternative a random geometric graph $\mathcal{G}(k,q,d)$ is planted on $k\le n$ vertices. The planted subgraph is generated from independent random points on the unit sphere $\mathbb{S}^{d-1}$, with edges determined by latent geometric proximity and calibrated to have edge density $q$. Our goal is to characterize the statistical and computational limits of detecting this hidden geometry. We derive sharp information-theoretic lower bounds that identify regimes where detection is impossible and provide algorithms that achieve these limits whenever detection is feasible. We further investigate the computational complexity of the problem and determine when efficient polynomial-time tests exist. The model exhibits an \emph{easy--hard--impossible} phase transition: some regimes allow efficient detection, others permit detection only with computationally intractable procedures, and still others render detection impossible even with unlimited computational power. As evidence for the computational barrier, we prove that all low-degree polynomial algorithms fail throughout the conjecturally hard regime, demonstrating a sharp gap between statistical and computational feasibility.

Problem

• hidden geometry
• random graphs
• hypothesis testing
• statistical detection
• computational complexity

Innovation

• hidden geometry
• random geometric graph
• statistical-computational gap
• phase transition
• low-degree polynomial algorithm