Detection of local geometry in random graphs: information-theoretic and computational limits

2026-03-25

AI Summary

This work investigates the detection of local geometric structure induced by spherical random geometric graphs within a hidden community, under the constraint that the edge distribution is indistinguishable from that of an Erdős–Rényi random graph. Focusing on the mixed model 𝒢(n, p, d, k), the study establishes—for the first time—the information-theoretic and computational limits simultaneously, revealing a significant statistical-to-computational gap and extending the known optimal bounds for the full graph to the sparse community regime. By leveraging signed triangle counting, a truncated second-moment method based on Wishart–GOE comparison, tensorized KL divergence, and the low-degree polynomial framework, the authors prove that for fixed p, the detectability threshold is d = Θ̃(k² ∨ k⁶/n³). Moreover, they demonstrate that signed cycle counts of length at least four are computationally suboptimal.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Computational social science

Abstract

We study the problem of detecting local geometry in random graphs. We introduce a model $\mathcal{G}(n, p, d, k)$, where a hidden community of average size $k$ has edges drawn as a random geometric graph on $\mathbb{S}^{d-1}$, while all remaining edges follow the Erdős--Rényi model $\mathcal{G}(n, p)$. The random geometric graph is generated by thresholding inner products of latent vectors on $\mathbb{S}^{d-1}$, with each edge having marginal probability equal to $p$. This implies that $\mathcal{G}(n, p, d, k)$ and $\mathcal{G}(n, p)$ are indistinguishable at the level of the marginals, and the signal lies entirely in the edge dependencies induced by the local geometry. We investigate both the information-theoretic and computational limits of detection. On the information-theoretic side, our upper bounds follow from three tests based on signed triangle counts: a global test, a scan test, and a constrained scan test; our lower bounds follow from two complementary methods: truncated second moment via Wishart--GOE comparison, and tensorization of KL divergence. These results together settle the detection threshold at $d = \widetildeΘ(k^2 \vee k^6/n^3)$ for fixed $p$, and extend the state-of-the-art bounds from the full model (i.e., $k = n$) for vanishing $p$. On the computational side, we identify a computational--statistical gap and provide evidence via the low-degree polynomial framework, as well as the suboptimality of signed cycle counts of length $\ell \geq 4$.

Problem

• local geometry
• random graphs
• hidden community
• detection
• random geometric graph

Innovation

• random geometric graph
• information-theoretic limits
• computational-statistical gap
• signed triangle counts
• low-degree polynomial method