Multigraph limits and exchangeability

TL;DR

This paper extends the theory of dense graph limits to multigraphs, utilizing exchangeable arrays and Aldous' representation theorem to construct limit objects, generalizing the foundational work of Lovasz and Szegedy.

Contribution

It introduces a framework for multigraph limits based on exchangeability and Aldous' theorem, broadening the scope of graph limit theory.

Findings

Established a multigraph limit framework

Connected graph limits with exchangeable array representations

Provided a construction method for multigraph limit objects

Abstract

The theory of limits of dense graph sequences was initiated by Lovasz and Szegedy. We give a possible generalization of this theory to multigraphs. Our proofs are based on the correspondence between dense graph limits and countable, exchangeable arrays of random variables observed by Diaconis and Janson. The main ingredient in the construction of the limit object is Aldous' representation theorem for exchangeable arrays.