Sandwiching Random Geometric Graphs and Erdos-Renyi with Applications: Sharp Thresholds, Robust Testing, and Enumeration
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917739602378752 |
|---|---|
| author | Bangachev, Kiril Bresler, Guy |
| author_facet | Bangachev, Kiril Bresler, Guy |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00995 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sandwiching Random Geometric Graphs and Erdos-Renyi with Applications: Sharp Thresholds, Robust Testing, and Enumeration Bangachev, Kiril Bresler, Guy Probability Discrete Mathematics Combinatorics Statistics Theory 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. |
| title | Sandwiching Random Geometric Graphs and Erdos-Renyi with Applications: Sharp Thresholds, Robust Testing, and Enumeration |
| topic | Probability Discrete Mathematics Combinatorics Statistics Theory |
| url | https://arxiv.org/abs/2408.00995 |