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Poisson-Dirichlet graphons and permutons

Benedikt Stufler

TL;DR

This work identifies and constructs universal limiting objects for large random graphs and permutations—Poisson--Dirichlet graphons and permutons—arising when complex structures are built by substituting head components with heavy-tailed mass partitions. It develops a unified framework using the two-parameter Poisson--Dirichlet process, Gibbs partitions, and graphon/permuton limits to prove invariance principles in a dilute regime, and to map out comprehensive phase diagrams (dense, condensation, mixture) describing when head or component structures dominate. The results include explicit constructions of PD-based limit objects, iterated constructions for cographs and separable permutations, and the Brownian graphon/permuton as notable special cases, thereby bridging graph limits and permutation limits under a common probabilistic mechanism. The findings illuminate universal limiting phenomena, offer precise asymptotics for component sizes, and reveal rich phase transitions with potential applications to random combinatorial models and data-structure analysis where heavy-tailed decompositions occur.

Abstract

We introduce classes of supergraphs and superpermutations with novel universal graphon and permuton limiting objects whose construction involves the two-parameter Poisson-Dirichlet process introduced by Pitman and Yor (1997). We demonstrate the universality of these limiting objects through general invariance principles in a heavy-tailed regime and establish a comprehensive phase diagram for the asymptotic shape of superstructures.

Poisson-Dirichlet graphons and permutons

TL;DR

This work identifies and constructs universal limiting objects for large random graphs and permutations—Poisson--Dirichlet graphons and permutons—arising when complex structures are built by substituting head components with heavy-tailed mass partitions. It develops a unified framework using the two-parameter Poisson--Dirichlet process, Gibbs partitions, and graphon/permuton limits to prove invariance principles in a dilute regime, and to map out comprehensive phase diagrams (dense, condensation, mixture) describing when head or component structures dominate. The results include explicit constructions of PD-based limit objects, iterated constructions for cographs and separable permutations, and the Brownian graphon/permuton as notable special cases, thereby bridging graph limits and permutation limits under a common probabilistic mechanism. The findings illuminate universal limiting phenomena, offer precise asymptotics for component sizes, and reveal rich phase transitions with potential applications to random combinatorial models and data-structure analysis where heavy-tailed decompositions occur.

Abstract

We introduce classes of supergraphs and superpermutations with novel universal graphon and permuton limiting objects whose construction involves the two-parameter Poisson-Dirichlet process introduced by Pitman and Yor (1997). We demonstrate the universality of these limiting objects through general invariance principles in a heavy-tailed regime and establish a comprehensive phase diagram for the asymptotic shape of superstructures.
Paper Structure (39 sections, 24 theorems, 188 equations)

This paper contains 39 sections, 24 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Two-parameter processes and partition structures
  3. The two-parameter Poisson-Dirichlet process
  4. The two-parameter Chinese restaurant process
  5. Pitman's coagulation-fragmentation duality
  6. Gibbs partitions in the dilute regime
  7. Gibbs partitions
  8. The dilute regime
  9. Background on graph limits
  10. Encoding graphs by homomorphism densities
  11. Encoding graphons by homomorphism densities
  12. Graph limits
  13. Convergence of random supergraphs
  14. Supergraphs
  15. Poisson-Dirichlet graphons in a dilute regime
  16. ...and 24 more sections

Key Result

Theorem 3.1

Suppose that conditions eq:crit, eq:condw and eq:condv are met. Define the following point process $\Upsilon_n = \sum_{\substack{1 \le i \le N_n \\ K_{i} > 0}} \delta_{K_{i} / n}$ on $]0,1]$, with $\delta$ referring to the Dirac measure. Then as $n \to \infty$.

Theorems & Definitions (45)

  • Theorem 3.1
  • Proposition 3.2: MR1440141
  • proof : Proof of Theorem \ref{['te:gibbsdilutegene']}
  • Proposition 4.1: MR2463439
  • Lemma 4.2: MR2463439
  • Definition 5.1: Poisson-Dirichlet graphon
  • Lemma 5.2
  • proof
  • Theorem 5.3
  • proof
  • ...and 35 more