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A logical approach to concentration

Michael Benedikt, Maksim Zhukovskii

TL;DR

This work develops a unifying framework linking concentration phenomena in Erdős--Rényi random graphs with a real-valued logic that supports aggregate operators. It defines a rich term language Agg with Avg, LAvg, Max, and Sigma, and identifies classes of connective functions (relative Lipschitz and asymptotically polynomial) that preserve concentration, enabling a general meta-theorem: every closed term concentrates in dense $G(n,p)$ ($p=\text{const}$) and in sparse $G(n,n^{-\alpha})$ when $\alpha$ is irrational. The dense results yield concentration and explicit growth patterns for subgraph- and extension-count statistics, while the sparse irrational regime extends zero-one laws and establishes convergence of a broad class of terms, including averages and maxima. The framework subsumes and extends classical FO zero-one/convergence laws, provides new concentration bounds for a wide range of graph parameters, and offers tools for deriving further concentration results through a logical lens. The work also discusses limitations, asymptotic behavior, and potential extensions to broader relational settings and weighted models, opening avenues for both theory and applications in graph algorithms and probabilistic combinatorics.

Abstract

Concentration results say that a sequence of random variables becomes progressively concentrated around the mean. Such results are common in the study of functions of random graphs. We introduce a real-valued logic with various aggregate operators on graphs, including summation, and prove that every term in the language, seen as a random variable on random graphs within the classical Erdős-Rényi random graph model, is concentrated. We prove this for dense and sparse variants of Erdős-Rényi graphs. On the one hand, our results extend the line of work originating with Fagin and Glebskii et al. on zero-one laws for dense random graphs, as well as the zero-one law of Shelah and Spencer for sparse random graphs. On the other hand, they can be seen as a meta-theorem for inferring concentration results on random graphs, and we give examples of such applications.

A logical approach to concentration

TL;DR

This work develops a unifying framework linking concentration phenomena in Erdős--Rényi random graphs with a real-valued logic that supports aggregate operators. It defines a rich term language Agg with Avg, LAvg, Max, and Sigma, and identifies classes of connective functions (relative Lipschitz and asymptotically polynomial) that preserve concentration, enabling a general meta-theorem: every closed term concentrates in dense () and in sparse when is irrational. The dense results yield concentration and explicit growth patterns for subgraph- and extension-count statistics, while the sparse irrational regime extends zero-one laws and establishes convergence of a broad class of terms, including averages and maxima. The framework subsumes and extends classical FO zero-one/convergence laws, provides new concentration bounds for a wide range of graph parameters, and offers tools for deriving further concentration results through a logical lens. The work also discusses limitations, asymptotic behavior, and potential extensions to broader relational settings and weighted models, opening avenues for both theory and applications in graph algorithms and probabilistic combinatorics.

Abstract

Concentration results say that a sequence of random variables becomes progressively concentrated around the mean. Such results are common in the study of functions of random graphs. We introduce a real-valued logic with various aggregate operators on graphs, including summation, and prove that every term in the language, seen as a random variable on random graphs within the classical Erdős-Rényi random graph model, is concentrated. We prove this for dense and sparse variants of Erdős-Rényi graphs. On the one hand, our results extend the line of work originating with Fagin and Glebskii et al. on zero-one laws for dense random graphs, as well as the zero-one law of Shelah and Spencer for sparse random graphs. On the other hand, they can be seen as a meta-theorem for inferring concentration results on random graphs, and we give examples of such applications.
Paper Structure (43 sections, 14 theorems, 142 equations)

This paper contains 43 sections, 14 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Organisation.
  3. Notation and conventions.
  4. Preliminaries
  5. Graphs.
  6. Probability theory background.
  7. Erdős--Rényi random graphs.
  8. A real-valued logic with aggregate operators.
  9. Examples of the term language.
  10. Related work
  11. Connective functions
  12. Properties of relative Lipschitz functions.
  13. Limitations of relative Lipschitz.
  14. Asymptotically polynomial functions.
  15. Dense Erdős--Rényi graphs
  16. ...and 28 more sections

Key Result

Theorem 4.3

Let $f:\mathbb{R}^m_{\geq 0}\to\mathbb{R}_{\geq 0}\in{\mathcal{F}}_{\mathrm{rellip}}$. There exists constants $a_f,b_f>1$ such that the following holds. Let $\xi_1,\ldots,\xi_m\geq 0$ be random variables, $\varphi_1,\ldots,\varphi_m\geq 0$, $\delta,\delta'>0$, and $K>2$ be constants such that, for e Let $\xi=f(\xi_1,\ldots,\xi_m)$ and $\varphi=f(\varphi_1,\ldots,\varphi_m)$. Then Moreover $\varph

Theorems & Definitions (67)

  • Definition 2.1: Multi-rooted graphs $F{[\overline{x}]}$
  • Definition 2.2: Induced subgraphs
  • Definition 2.3: With high probability
  • Definition 2.4: Convergence
  • Definition 2.5: Concentration around the mean
  • Definition 2.6: Aggregate real-valued logic
  • Definition 2.7: Interpretation of terms
  • Definition 4.1: Relative Lipschitz functions
  • Remark 4.2
  • Theorem 4.3: Functions preserve concentration
  • ...and 57 more