Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

First order distinguishability of sparse random graphs

Tal Hershko, Maksim Zhukovskii

TL;DR

We study distinguishing two independent sparse random graphs G(n, n^{-α}) by first-order sentences, showing the minimum FO depth k_α depends on α's rational approximability. The key tool is a random balanced graph construction that yields a large family of asymmetric strictly balanced subgraphs, enabling existential FO witnesses to separate the samples with depth k ∼ O_p( ln n / ln ln n ). We prove Ω(ln ln ln n) lower bounds for irrational α that are not Liouville-like, while showing for almost all α the depth is at least (1-o(1))/ln 2 · ln ln ln n, with refined results for well-approximable α. The analysis blends Ehrenfeucht-Fraïssé games, Poisson convergence of subgraph counts, and a randomized sampling of asymmetric graphs, revealing a deep link between Diophantine approximation and logical distinguishability in sparse random graphs and informing average-case complexity aspects of graph isomorphism testing.

Abstract

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 $\varphi=\varphi(\mathbf{G}_n^1,\mathbf{G}_n^2)$ 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.

First order distinguishability of sparse random graphs

TL;DR

We study distinguishing two independent sparse random graphs G(n, n^{-α}) by first-order sentences, showing the minimum FO depth k_α depends on α's rational approximability. The key tool is a random balanced graph construction that yields a large family of asymmetric strictly balanced subgraphs, enabling existential FO witnesses to separate the samples with depth k ∼ O_p( ln n / ln ln n ). We prove Ω(ln ln ln n) lower bounds for irrational α that are not Liouville-like, while showing for almost all α the depth is at least (1-o(1))/ln 2 · ln ln ln n, with refined results for well-approximable α. The analysis blends Ehrenfeucht-Fraïssé games, Poisson convergence of subgraph counts, and a randomized sampling of asymmetric graphs, revealing a deep link between Diophantine approximation and logical distinguishability in sparse random graphs and informing average-case complexity aspects of graph isomorphism testing.

Abstract

We study the problem of distinguishing between two independent samples of a binomial random graph by first order (FO) sentences. Shelah and Spencer proved that, for a constant , obeys FO zero-one law if and only if is irrational. Therefore, for irrational , any fixed FO sentence does not distinguish between with asymptotical probability 1 (w.h.p.) as . We show that the minimum quantifier depth of a FO sentence distinguishing between depends on how closely can be approximated by rationals: (1) for all non-Liouville , w.h.p.; (2) there are irrational with that grow arbitrarily slowly w.h.p.; (3) for all . 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.
Paper Structure (26 sections, 36 theorems, 160 equations, 3 figures)

This paper contains 26 sections, 36 theorems, 160 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The random balanced graph
  3. Definition
  4. Properties
  5. The family $\mathcal{H}(n,m)$
  6. The general upper bound
  7. A suitable family of subgraphs
  8. Proof of Proposition \ref{['prop:enhanced_balanceness']}
  9. Low upper bounds for well-approximable $\alpha$
  10. Diophantine approximations
  11. Asymptotic Poisson behavior
  12. A lower bound for almost every $\alpha$
  13. The Ehrenfeucht-Fraïssé game
  14. Extensions and the closure
  15. Look-ahead strategy
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $\alpha\in(0,1)$. Then $\mathbf{G}_n\sim G(n,n^{-\alpha})$ obeys a FO zero-one law if and only if $\alpha$ is irrational.

Figures (3)

  • Figure 1: A sample of ${\bf H}$ with $n=16$ and $r=5$. Vertices of $R$ are marked by big black circles.
  • Figure 2: Three types of reddish cycles, $\ell=9$. In types 2 and 3, an arrow points at the “ special edge” whose choice adds a factor of $\Theta(\ell)$ to the bound. The cycles are drawn such the red edges are fixed, thus emphasizing the process of choosing the blue edges.
  • Figure 3: All possible types of alternating paths connecting a given pair of vertices $x,y$. Vertices of $R$ are marked by big black circles (excluding the endpoints in the red cases, as described in the proof). An arrow marks the “ special edge” , which is again either a red edge with no $R$-endpoints or a blue edge with two $R$-endpoints.

Theorems & Definitions (106)

  • Theorem 1.1: S. Shelah, J. Spencer spencershelah
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • ...and 96 more