Sandwiching Random Geometric Graphs and Erdos-Renyi with Applications: Sharp Thresholds, Robust Testing, and Enumeration
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3717823.3718125.pdf
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Author(s)
Bangachev, Kiril
Bresler, Guy
Date Issued
June 15, 2025
Publisher
ACM|Proceedings of the 57th Annual ACM Symposium on Theory of Computing
Citation
Kiril Bangachev and Guy Bresler. 2025. Sandwiching Random Geometric Graphs and Erdos-Renyi with Applications: Sharp Thresholds, Robust Testing, and Enumeration. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing (STOC '25). Association for Computing Machinery, New York, NY, USA, 310–321.
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Final published version
Abstract
The distribution RGG(n,Sd−1,p) is formed by sampling independent vectors {Vi}i = 1n uniformly on Sd−1 and placing an edge between pairs of vertices i and j for which ⟨ Vi,Vj⟩ ≥ τdp, where τdp is such that the expected density is p. Our main result is a poly-time implementable coupling between Erdős-Rényi and RGG such that G(n,p(1 − O(√np/d)))⊆ RGG(n,Sd−1,p)⊆ G(n,p(1 + O(√np/d))) edgewise with high probability when d≫ np. We apply the result to: 1) Sharp Thresholds: We show that for any monotone property having a sharp threshold with respect to the Erdős-Rényi distribution and critical probability pnc, random geometric graphs also exhibit a sharp threshold when d≫ npnc, thus partially answering a question of Perkins. 2) Robust Testing: The coupling shows that testing between G(n,p) and RGG(n,Sd−1,p) with є n2p adversarially corrupted edges for any constant є>0 is information-theoretically impossible when d≫ np. We match this lower bound with an efficient (constant degree SoS) spectral refutation algorithm when d≪ np. 3) Enumeration: We show that the number of geometric graphs in dimension d is at least exp(dnlog−7n), recovering (up to the log factors) the sharp result of Sauermann.
Description
STOC ’25, Prague, Czechia
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
https://doi.org/10.1145/3717823.3718125
