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First Order Logic of Sparse Graphs with Given Degree Sequences

Alberto Larrauri, Guillem Perarnau

TL;DR

The proof of the existence of limit probabilities for first order properties in random graphs with a given degree sequence is amended; this result was already claimed by Lynch~[IEEE LICS 2003] but his proof contained some inaccuracies.

Abstract

We consider limit probabilities of first order properties in random graphs with a given degree sequence. Under mild conditions on the degree sequence, we show that the closure set of limit probabilities is a finite union of closed intervals. Moreover, we characterize the degree sequences for which this closure set is the interval $[0,1]$, a property that is intimately related with the probability that the random graph is acyclic. As a side result, we compile a full description of the cycle distribution of random graphs and study their fragment (disjoint union of unicyclic components) in the subcritical regime. Finally, we amend the proof of the existence of limit probabilities for first order properties in random graphs with a given degree sequence; this result was already claimed by Lynch~[IEEE LICS 2003] but his proof contained some inaccuracies.

First Order Logic of Sparse Graphs with Given Degree Sequences

TL;DR

The proof of the existence of limit probabilities for first order properties in random graphs with a given degree sequence is amended; this result was already claimed by Lynch~[IEEE LICS 2003] but his proof contained some inaccuracies.

Abstract

We consider limit probabilities of first order properties in random graphs with a given degree sequence. Under mild conditions on the degree sequence, we show that the closure set of limit probabilities is a finite union of closed intervals. Moreover, we characterize the degree sequences for which this closure set is the interval , a property that is intimately related with the probability that the random graph is acyclic. As a side result, we compile a full description of the cycle distribution of random graphs and study their fragment (disjoint union of unicyclic components) in the subcritical regime. Finally, we amend the proof of the existence of limit probabilities for first order properties in random graphs with a given degree sequence; this result was already claimed by Lynch~[IEEE LICS 2003] but his proof contained some inaccuracies.
Paper Structure (16 sections, 27 theorems, 105 equations, 1 figure)

This paper contains 16 sections, 27 theorems, 105 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Multigraphs
  5. Configuration Model
  6. Exchange of Limit and Sum Operators
  7. Probability preliminaries
  8. Sets of Partial Sums
  9. Outline of the proof of .
  10. Cycle distribution
  11. Previous results
  12. Fragment Distribution
  13. First part of Theorem 1.3: A finite union of intervals
  14. Second part of .
  15. Remarks About the Convergence Law
  16. ...and 1 more sections

Key Result

Theorem 1.2

Suppose that ${\bm{d}}$ satisfies assump:main. Then for any property $\varphi$ in the $\mathop{\mathrm{\textsc{FO}}}\nolimits$ language of graphs, the following limit exists

Figures (1)

  • Figure 1: Example of the map $H\mapsto F_H$.

Theorems & Definitions (54)

  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4: Discussion on \ref{['assump:main']}
  • Remark 1.5: The configuration model
  • Remark 1.6: Probability of being acyclic
  • Theorem 2.1: Bollobás bollobas1980probabilistic; Janson janson2009probability
  • Definition 2.2
  • Lemma 2.3
  • Theorem 2.4: Method of Moments for Poisson random variables
  • Lemma 2.5: Kakeya's Criterion
  • ...and 44 more