Poisson-Dirichlet graphons and permutons

TL;DR

This paper introduces new universal graphon and permuton models based on the Poisson-Dirichlet process, demonstrating their universality and phase behavior in supergraph and superpermutation structures.

Contribution

It presents novel supergraph and superpermutation classes with universal limits constructed via the two-parameter Poisson-Dirichlet process, expanding understanding of asymptotic structures.

Findings

Universal limiting objects for supergraphs and superpermutations.

Invariance principles in a heavy-tailed regime.

A comprehensive phase diagram for superstructure asymptotics.

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.