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Infinite random graphs

Ziemowit Kostana, Jarosław Swaczyna, Agnieszka Widz

TL;DR

The paper extends the theory of the Rado graph by introducing drawable graphs: countable graphs that arise almost surely from a random edge-drawing process with potentially biased, nonuniform edge probabilities. It develops a robust probabilistic framework using product measures on $2^{[\,\omega]^2}$, defines p‑random, weakly/strongly drawable notions, and analyzes how permutations of probabilities affect drawable outcomes. A key result is a two‑element basis: every drawable graph contains either $\mathcal{U}_{FIN}$ or its complement, establishing a strong structural dichotomy for weakly universal drawable graphs. The work also outlines a broader landscape, including conjectures (ISC) about when weakly drawable graphs are strongly drawable, and sketches a program for classifying drawable graphs via accumulation properties of probability sequences. Together with the appendix on Cantor-space product measures, the paper lays groundwork for a systematic study of universal homogeneous graphs generated by biased random processes and suggests rich avenues for future research.

Abstract

We study countable graphs that -- up to isomorphism and with probability one -- arise from a random process, in a similar fashion as the Rado graph. Unlike in the classical case, we do not require that probabilities assigned to pairs of points are all equal. We give examples of such generalized random graphs, and show that the class of graphs under consideration has a two-element basis.

Infinite random graphs

TL;DR

The paper extends the theory of the Rado graph by introducing drawable graphs: countable graphs that arise almost surely from a random edge-drawing process with potentially biased, nonuniform edge probabilities. It develops a robust probabilistic framework using product measures on , defines p‑random, weakly/strongly drawable notions, and analyzes how permutations of probabilities affect drawable outcomes. A key result is a two‑element basis: every drawable graph contains either or its complement, establishing a strong structural dichotomy for weakly universal drawable graphs. The work also outlines a broader landscape, including conjectures (ISC) about when weakly drawable graphs are strongly drawable, and sketches a program for classifying drawable graphs via accumulation properties of probability sequences. Together with the appendix on Cantor-space product measures, the paper lays groundwork for a systematic study of universal homogeneous graphs generated by biased random processes and suggests rich avenues for future research.

Abstract

We study countable graphs that -- up to isomorphism and with probability one -- arise from a random process, in a similar fashion as the Rado graph. Unlike in the classical case, we do not require that probabilities assigned to pairs of points are all equal. We give examples of such generalized random graphs, and show that the class of graphs under consideration has a two-element basis.
Paper Structure (13 sections, 30 theorems, 55 equations)

This paper contains 13 sections, 30 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs
  4. Isolated Unions
  5. Isomorphism classes
  6. Sequences of probabilities
  7. General facts about drawable graphs
  8. Properties of $\textbf{p}$-random graphs
  9. Permutations acting on a given sequence
  10. General procedure for generating $\textbf{p}$-random graphs.
  11. A basis theorem for weakly universal graphs
  12. Landscape on drawable graphs
  13. APPENDIX -- product measures on the Cantor space

Key Result

Theorem 2.3

Let $\mathbf{p}=(p_n)_{n\in \omega}$. The Rado graph is strongly $\{ \mathbf{p} \}$-drawable if and only if $\mathbf{p}$ is Borel--Cantelli.

Theorems & Definitions (75)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Theorem 6, coregliano2024random
  • Lemma 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • ...and 65 more