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Sandwiching Random Geometric Graphs and Erdos-Renyi with Applications: Sharp Thresholds, Robust Testing, and Enumeration
Probability · Submitted 2 Aug 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The distribution $\mathsf{RGG}(n,\mathbb{S}^{d-1},p)$ is formed by sampling independent vectors $\{V_i\}_{i = 1}^n$ uniformly on $\mathbb{S}^{d-1}$ and placing an edge between pairs of vertices $i$ and $j$ for which $\langle V_i,V_j\rangle \ge \tau^p_d,$ where $\tau^p_d$ is such that the expected density is $p.$ Our main result is a poly-time implementable coupling between Erd\H{o}s-R\'enyi and $\mathsf{RGG}$ such that $\mathsf{G}(n,p(1 - \tilde{O}(\sqrt{np/d})))\subseteq \mathsf{RGG}(n,\mathbb{S}^{d-1},p)\subseteq \mathsf{G}(n,p(1 + \tilde{O}(\sqrt{np/d})))$ edgewise with high probability when $d\gg np.$ We apply the result to: 1) Sharp Thresholds: We show that for any monotone property having a sharp threshold with respect to the Erd\H{o}s-R\'enyi distribution and critical probability $p^c_n,$ random geometric graphs also exhibit a sharp threshold when $d\gg np^c_n,$ thus partially answering a question of Perkins. 2) Robust Testing: The coupling shows that testing between $\mathsf{G}(n,p)$ and $\mathsf{RGG}(n,\mathbb{S}^{d-1},p)$ with $\epsilon n^2p$ adversarially corrupted edges for any constant $\epsilon>0$ is information-theoretically impossible when $d\gg np.$ We match this lower bound with an efficient (constant degree SoS) spectral refutation algorithm when $d\ll np.$ 3) Enumeration: We show that the number of geometric graphs in dimension $d$ is at least $\exp(dn\log^{-7}n)$, recovering (up to the log factors) the sharp result of Sauermann.
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The distribution $\mathsf{RGG}(n,\mathbb{S}^{d-1},p)$ is formed by sampling independent vectors $\{V_i\}_{i = 1}^n$ uniformly on $\mathbb{S}^{d-1}$ and placing an edge between pairs of vertices $i$ and $j$ for which $\langle V_i,V_j\rangle \ge \tau^p_d,$ where $\tau^p_d$ is such that the expected density is $p.$ Our main result is a poly-time implementable coupling between Erd\H{o}s-R\'enyi and $\mathsf{RGG}$ such that $\mathsf{G}(n,p(1 - \tilde{O}(\sqrt{np/d})))\subseteq \mathsf{RGG}(n,\mathbb{S}^{d-1},p)\subseteq \mathsf{G}(n,p(1 + \tilde{O}(\sqrt{np/d})))$ edgewise with high probability when $d\gg np.$ We apply the result to: 1) Sharp Thresholds: We show that for any monotone property having a sharp threshold with respect to the Erd\H{o}s-R\'enyi distribution and critical probability $p^c_n,$ random geometric graphs also exhibit a sharp threshold when $d\gg np^c_n,$ thus partially answering a question of Perkins. 2) Robust Testing: The coupling shows that testing between $\mathsf{G}(n,p)$ and $\mathsf{RGG}(n,\mathbb{S}^{d-1},p)$ with $\epsilon n^2p$ adversarially corrupted edges for any constant $\epsilon>0$ is information-theoretically impossible when $d\gg np.$ We match this lower bound with an efficient (constant degree SoS) spectral refutation algorithm when $d\ll np.$ 3) Enumeration: We show that the number of geometric graphs in dimension $d$ is at least $\exp(dn\log^{-7}n)$, recovering (up to the log factors) the sharp result of Sauermann.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The distribution $\mathsf{RGG}(n,\mathbb{S}^{d-1},p)$ is formed by sampling independent vectors $\{V_i\}_{i = 1}^n$ uniformly on $\mathbb{S}^{d-1}$ and placing an edge between pairs of vertices $i$ and $j$ for which $\langle V_i,V_j\rangle \ge \tau^p_d,$ where $\tau^p_d$ is such that the expected density is $p.$ Our main result is a poly-time implementable coupling between Erd\H{o}s-R\'enyi and $\mathsf{RGG}$ such that $\mathsf{G}(n,p(1 - \tilde{O}(\sqrt{np/d})))\subseteq \mathsf{RGG}(n,\mathbb{S}^{d-1},p)\subseteq \mathsf{G}(n,p(1 + \tilde{O}(\sqrt{np/d})))$ edgewise with high probability when $d\gg np.$ We apply the result to: 1) Sharp Thresholds: We show that for any monotone property having a sharp threshold with respect to the Erd\H{o}s-R\'enyi distribution and critical probability $p^c_n,$ random geometric graphs also exhibit a sharp threshold when $d\gg np^c_n,$ thus partially answering a question of Perkins. 2) Robust Testing: The coupling shows that testing between $\mathsf{G}(n,p)$ and $\mathsf{RGG}(n,\mathbb{S}^{d-1},p)$ with $\epsilon n^2p$ adversarially corrupted edges for any constant $\epsilon>0$ is information-theoretically impossible when $d\gg np.$ We match this lower bound with an efficient (constant degree SoS) spectral refutation algorithm when $d\ll np.$ 3) Enumeration: We show that the number of geometric graphs in dimension $d$ is at least $\exp(dn\log^{-7}n)$, recovering (up to the log factors) the sharp result of Sauermann. | contribution 0.70 | 0.94 | 0.95 | asserted 1.00 |
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Original abstract
The distribution $\mathsf{RGG}(n,\mathbb{S}^{d-1},p)$ is formed by sampling independent vectors $\{V_i\}_{i = 1}^n$ uniformly on $\mathbb{S}^{d-1}$ and placing an edge between pairs of vertices $i$ and $j$ for which $\langle V_i,V_j\rangle \ge \tau^p_d,$ where $\tau^p_d$ is such that the expected density is $p.$ Our main result is a poly-time implementable coupling between Erd\H{o}s-R\'enyi and $\mathsf{RGG}$ such that $\mathsf{G}(n,p(1 - \tilde{O}(\sqrt{np/d})))\subseteq \mathsf{RGG}(n,\mathbb{S}^{d-1},p)\subseteq \mathsf{G}(n,p(1 + \tilde{O}(\sqrt{np/d})))$ edgewise with high probability when $d\gg np.$ We apply the result to: 1) Sharp Thresholds: We show that for any monotone property having a sharp threshold with respect to the Erd\H{o}s-R\'enyi distribution and critical probability $p^c_n,$ random geometric graphs also exhibit a sharp threshold when $d\gg np^c_n,$ thus partially answering a question of Perkins. 2) Robust Testing: The coupling shows that testing between $\mathsf{G}(n,p)$ and $\mathsf{RGG}(n,\mathbb{S}^{d-1},p)$ with $\epsilon n^2p$ adversarially corrupted edges for any constant $\epsilon>0$ is information-theoretically impossible when $d\gg np.$ We match this lower bound with an efficient (constant degree SoS) spectral refutation algorithm when $d\ll np.$ 3) Enumeration: We show that the number of geometric graphs in dimension $d$ is at least $\exp(dn\log^{-7}n)$, recovering (up to the log factors) the sharp result of Sauermann.
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Spectral properties of spherical RGGs have recently gained a lot of attention in a series of works [ref] , where they were further used algorithmically for latent vector recovery, community detection, and robust testing.
Testing Thresholds and Spectral Properties of High-Dimensional Random Toroidal Graphs via Edgeworth-Style Expansions · Our Results · 0.98
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