Sandwiching Random Geometric Graphs and Erdos-Renyi with Applications: Sharp Thresholds, Robust Testing, and Enumeration
Kiril Bangachev, Guy Bresler
TL;DR
The paper develops an algorithmic coupling between high-dimensional spherical random geometric graphs and Erdős-Rényi graphs, showing that in the regime ${d \gg np}$ one can tightly sandwich ${\mathsf{RGG}(n,\mathbb{S}^{d-1},p)}$ between ${\mathsf{G}}(n,p(1 - \widetilde{O}(\sqrt{np/d})))$ and ${\mathsf{G}}(n,p(1 + \widetilde{O}(\sqrt{np/d})))$. This coupling enables a suite of results: (i) transfer of sharp thresholds for monotone properties from ER to RGG with precise critical probabilities, (ii) information-theoretic impossibility of robust testing between ER and RGG under adversarial edge corruptions when ${d \gg np}$ and an efficient SOS-based refutation when ${d \ll np}$, and (iii) a lower bound on the number of geometric graphs in dimension ${d}$ via a recursive planted-ER representation that recovers Sauermann’s bounds up to log factors. The dimension ${d \asymp np}$ emerges as a natural entropic and spectral threshold, tying together the size of the support, eigenvalue structure, and computational tractability, and unifying geometric dependence with classical random-graph phenomena. The results provide both conceptual insight into the geometry of random graphs and practical algorithmic tools for robust testing, embedding, and enumeration in high dimensions.
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 $\langle V_i,V_j\rangle \ge τ^p_d,$ where $τ^p_d$ is such that the expected density is $p.$ Our main result is a poly-time implementable coupling between Erdős-Rényi 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 Erdős-Rényi 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 $εn^2p$ adversarially corrupted edges for any constant $ε>0$ 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.