On the equivalence of quasirandomness and exchangeable representations independent from lower-order variables

TL;DR

This paper explores when quasirandomness properties in exchangeable graph representations imply simpler, independent representations, introducing a new $ ext{ extasterisk}$-representation concept that extends the classical theon framework.

Contribution

It introduces $ ext{ extasterisk}$-representations with added random permutations and proves their equivalence to quasirandomness conditions, resolving a question by Coregliano and Razborov.

Findings

UCouple[$ ext{ extasterisk}$] implies $ ext{ extasterisk}$-$ ext{ extasterisk}$-independence

$ ext{ extasterisk}$-representations are necessary for certain quasirandomness properties

The results generalize the Aldous--Hoover theorem to more complex structures.

Abstract

It is often convenient to represent a process for randomly generating a graph as a graphon. (More precisely, these give \emph{vertex exchangeable} processes -- those processes in which each vertex is treated the same way.) Other structures can be treated by generalizations like hypergraphons, permutatons, and, for a very general class, theons. These representations are not unique: different representations can lead to the same probability distribution on graphs. This naturally leads to questions (going back at least to Hoover's proof of the Aldous--Hoover Theorem on the existence of such representations) that ask when quasirandomness properties on the distribution guarantee the existence of particularly simple representations. We extend the usual theon representation by adding an additional datum of a random permutation to each tuple, which we call a $\ast$-representation. We show that if a process satisfies the \emph{unique coupling} property UCouple[$\ell$], which says roughly that all $\ell$-tuples of vertices ``look the same'', then the process is $\ast$-$\ell$-independent: there is a $\ast$-representation that does not make use of any random information about $\ell$-tuples (including tuples of length $<\ell$). Simple examples show that the use of $\ast$-representations is necessary. This resolves a question of Coregliano and Razborov, since it easily follows that UCouple[l] implies Independence[\ell'] (the existence of an $\ell'$-independent ordinary representation) for $\ell'<\ell$.