Finitely forcible graphons and permutons

TL;DR

This paper explores conditions under which graphons and permutons are uniquely determined by finitely many substructure densities, revealing differences in their forcibility and establishing new classes of finitely forcible permutons.

Contribution

It demonstrates that some permutons are finitely forcible without their associated graphons being so, and characterizes finitely forcible permutons as finite combinations of monotone and quasirandom permutons.

Findings

Some permutons are finitely forcible but their associated graphons are not.

Permutons as finite combinations of monotone and quasirandom permutons are finitely forcible.

The permuton counterpart of Lovasz and Sos's result for graphons is established.

Abstract

We investigate when limits of graphs (graphons) and permutations (permutons) are uniquely determined by finitely many densities of their substructures, i.e., when they are finitely forcible. Every permuton can be associated with a graphon through the notion of permutation graphs. We find permutons that are finitely forcible but the associated graphons are not. We also show that all permutons that can be expressed as a finite combination of monotone permutons and quasirandom permutons are finitely forcible, which is the permuton counterpart of the result of Lovasz and Sos for graphons.