Probability graphons and P-variables: two equivalent viewpoints for dense weighted graph limits

TL;DR

This paper establishes an equivalence between probability graphons and P-variables as two perspectives for representing dense weighted graph limits, enabling transfer of results and generalizations to other structures.

Contribution

It introduces P-variables as an alternative to probability graphons for dense weighted graph limits and proves their equivalence, facilitating new insights and extensions.

Findings

Proves the equivalence between P-variables and probability graphons.

Defines a metric for P-variables and shows convergence equivalence.

Extends P-variables convergence to bipartite graphs and hypergraphs.

Abstract

We develop further the graph limit theory for dense weighted graph sequences. In particular, we consider probability graphons, which have recently appeared in graph limit theory as continuum representations of weighted graphs, and we introduce P-variables, which also appear in the context of the Aldous-Hoover theorem for exchangeable infinite random arrays, as an alternative continuum representation for weighted graphs. In particular, we explain how P-variables are related to probability graphons in a similar way in which random variables are related to probability measures. We define a metric for P-variables (inspired by action convergence in the graph limit theory of sparse graph sequences) and show that convergence of P-variables in this metric is equivalent to probability graphons convergence. We exploit this equivalence to translate several results from the theory of probability graphons to P-variables. In addition, we prove several properties of P-variables convergence, thus showing new properties also for probability graphons convergence and demonstrating the power of the connection between probability graphons and P-variables. Furthermore, we show how P-variables convergence can be easily modified and generalised to cover other combinatorial structures such as bipartite graphs and hypergraphs.