Probability graphons: the right convergence point of view

TL;DR

This paper extends the theory of probability graphons by establishing their convergence properties from the 'right convergence' perspective, linking it with existing notions like cut-metric and homomorphism densities.

Contribution

It generalizes overlay functionals and quotient sets to probability graphons and proves the equivalence of different convergence notions, unifying 'left' and 'right' convergence perspectives.

Findings

Proves the equivalence of 'left' and 'right' convergence for probability graphons.

Generalizes the concepts of overlay functionals and quotient sets.

Shows the convergence in terms of global quantities like cut-metric and homomorphism densities.

Abstract

We extend the theory of probability graphons, continuum representations of edge-decorated graphs arising in graph limits theory, to the 'right convergence' point of view. First of all, we generalise the notions of overlay functionals and quotient sets to the case of probability graphons. Furthermore, we characterise the convergence of probability graphons in terms of these global quantities. In particular, we show the equivalence of these two notions of convergence with the unlabelled cut-metric convergence (and thus also with the homomorphism densities convergence and the subgraph sampling convergence). In other words, we prove the equivalence of the 'left convergence' and the 'right convergence' views on probability graphons convergence, generalising the corresponding result for (real-valued) graphons (the classical continuum representation for simple graphs).