Flows on measurable spaces

TL;DR

This paper extends core concepts of flow theory from finite directed graphs to the setting of measurable spaces, particularly Markov spaces, broadening the scope of graph limit theory.

Contribution

It demonstrates that flow theory can be generalized to measurable spaces, requiring only a standard Borel space with a measure on its square, thus expanding the applicability of graph flow concepts.

Findings

Flow theory extends to measurable spaces.

Markov space structure is not fully necessary.

Results generalize flow theory for directed graphs.

Abstract

The theory of graph limits is only understood to any nontrivial degree in the cases of dense graphs and of bounded degree graphs. There is, however, a lot of interest in the intermediate cases. It appears that the most important constituents of graph limits in the general case will be Markov spaces (Markov chains on measurable spaces with a stationary distribution). This motivates our goal to extend some important theorems from finite graphs to Markov spaces or, more generally, to measurable spaces. In this paper, we show that much of flow theory, one of the most important areas in graph theory, can be extended to measurable spaces. Surprisingly, even the Markov space structure is not fully needed to get these results: all we need a standard Borel space with a measure on its square. Our results may be considered as extensions of flow theory for directed graphs to the measurable case.