First order distinguishability of sparse random graphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911877102043136 |
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| author | Hershko, Tal Zhukovskii, Maksim |
| author_facet | Hershko, Tal Zhukovskii, Maksim |
| contents | We study the problem of distinguishing between two independent samples $\mathbf{G}_n^1,\mathbf{G}_n^2$ of a binomial random graph $G(n,p)$ by first order (FO) sentences. Shelah and Spencer proved that, for a constant $α\in(0,1)$, $G(n,n^{-α})$ obeys FO zero-one law if and only if $α$ is irrational. Therefore, for irrational $α\in(0,1)$, any fixed FO sentence does not distinguish between $\mathbf{G}_n^1,\mathbf{G}_n^2$ with asymptotical probability 1 (w.h.p.) as $n\to\infty$. We show that the minimum quantifier depth $\mathbf{k}_α$ of a FO sentence $φ=φ(\mathbf{G}_n^1,\mathbf{G}_n^2)$ distinguishing between $\mathbf{G}_n^1,\mathbf{G}_n^2$ depends on how closely $α$ can be approximated by rationals: (1) for all non-Liouville $α\in(0,1)$, $\mathbf{k}_α=Ω(\ln\ln\ln n)$ w.h.p.; (2) there are irrational $α\in(0,1)$ with $\mathbf{k}_α$ that grow arbitrarily slowly w.h.p.; (3) $\mathbf{k}_α=O_p(\frac{\ln n}{\ln\ln n})$ for all $α\in(0,1)$. The main ingredients in our proofs are a novel randomized algorithm that generates asymmetric strictly balanced graphs as well as a new method to study symmetry groups of randomly perturbed graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09146 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | First order distinguishability of sparse random graphs Hershko, Tal Zhukovskii, Maksim Combinatorics Discrete Mathematics Logic in Computer Science We study the problem of distinguishing between two independent samples $\mathbf{G}_n^1,\mathbf{G}_n^2$ of a binomial random graph $G(n,p)$ by first order (FO) sentences. Shelah and Spencer proved that, for a constant $α\in(0,1)$, $G(n,n^{-α})$ obeys FO zero-one law if and only if $α$ is irrational. Therefore, for irrational $α\in(0,1)$, any fixed FO sentence does not distinguish between $\mathbf{G}_n^1,\mathbf{G}_n^2$ with asymptotical probability 1 (w.h.p.) as $n\to\infty$. We show that the minimum quantifier depth $\mathbf{k}_α$ of a FO sentence $φ=φ(\mathbf{G}_n^1,\mathbf{G}_n^2)$ distinguishing between $\mathbf{G}_n^1,\mathbf{G}_n^2$ depends on how closely $α$ can be approximated by rationals: (1) for all non-Liouville $α\in(0,1)$, $\mathbf{k}_α=Ω(\ln\ln\ln n)$ w.h.p.; (2) there are irrational $α\in(0,1)$ with $\mathbf{k}_α$ that grow arbitrarily slowly w.h.p.; (3) $\mathbf{k}_α=O_p(\frac{\ln n}{\ln\ln n})$ for all $α\in(0,1)$. The main ingredients in our proofs are a novel randomized algorithm that generates asymmetric strictly balanced graphs as well as a new method to study symmetry groups of randomly perturbed graphs. |
| title | First order distinguishability of sparse random graphs |
| topic | Combinatorics Discrete Mathematics Logic in Computer Science |
| url | https://arxiv.org/abs/2405.09146 |