Categorical approach to graph limits

TL;DR

This paper introduces a categorical framework for graph limits using measures and morphisms, establishing a convergence notion and proving the compactness of the space of all such limits.

Contribution

It defines a new category of graph limits with measure-theoretic objects and morphisms, and proves the compactness of this space using category theory tools.

Findings

Defined a category of graph limits with measure-based objects and morphisms

Introduced a convergence notion inspired by s-convergence

Proved the compactness of the space of all graph limits

Abstract

We define and study a natural category of graph limits. The objects are pairs $(\pi,\mu)$, where $\pi$ (the distribution of vertices) is an abstract probability measure on some abstract measurable space $(X,\mathcal{A})$ and $\mu$ (the distribution of edges) is an abstract finite measure on the square $(X,\mathcal{A})^2$. Morphisms are random maps between the underlying measurable spaces which preserve the distribution of vertices as well as the distribution of edges. We also define a convergence notion (inspired by s-convergence) for sequences of graph limits. We apply tools from category theory to prove the compactness of the space of all graph limits.