Open Access
2023 Phase transition in noisy high-dimensional random geometric graphs
Suqi Liu, Miklós Z. Rácz
Author Affiliations +
Electron. J. Statist. 17(2): 3512-3574 (2023). DOI: 10.1214/23-EJS2162


We study the problem of detecting latent geometric structure in random graphs. To this end, we consider the soft high-dimensional random geometric graph G(n,p,d,q), where each of the n vertices corresponds to an independent random point distributed uniformly on the sphere Sd1, and the probability that two vertices are connected by an edge is a decreasing function of the Euclidean distance between the points. The probability of connection is parametrized by q[0,1], with smaller q corresponding to weaker dependence on the geometry; this can also be interpreted as the level of noise in the geometric graph. In particular, the model smoothly interpolates between the hard spherical random geometric graph G(n,p,d) (corresponding to q=1) and the Erdős–Rényi model G(n,p) (corresponding to q=0). We focus on the dense regime (i.e., p is a constant).

We show that if nq0 or d»n3q2, then geometry is lost: G(n,p,d,q) is asymptotically indistinguishable from G(n,p). On the other hand, if d«n3q6, then the signed triangle statistic provides an asymptotically powerful test for detecting geometry. These results generalize those of Bubeck, Ding, Eldan, and Rácz (2016) for G(n,p,d), and give quantitative bounds on how the noise level affects the dimension threshold for losing geometry. We also prove analogous results under a related but different distributional assumption which corresponds to the random dot product graph.

Funding Statement

Research supported in part by NSF grant DMS-1811724.


S. L. thanks Cong Ma, Kaizheng Wang, and Jiacheng Zhang for numerous insightful discussions throughout the years. We thank the editors and anonymous referees for their valuable comments and help to improve the manuscript. The authors acknowledge the generous support from NSF grant DMS-1811724.


Download Citation

Suqi Liu. Miklós Z. Rácz. "Phase transition in noisy high-dimensional random geometric graphs." Electron. J. Statist. 17 (2) 3512 - 3574, 2023.


Received: 1 September 2022; Published: 2023
First available in Project Euclid: 29 November 2023

arXiv: 2103.15249
Digital Object Identifier: 10.1214/23-EJS2162

Primary: 60K35
Secondary: 05C80 , 60B20 , 60C05

Keywords: high-dimensional geometric structure , Hypothesis testing , Random geometric graph , random graph , signed triangle

Vol.17 • No. 2 • 2023
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