Graphons, cut norm and distance, couplings and rearrangements

TL;DR

This paper surveys fundamental results on the cut norm and cut metric for graphons, emphasizing the equivalence problem, and provides new proofs and technical complements, including results for special classes of graphons.

Contribution

It offers a comprehensive survey with new technical results and a novel proof of the uniqueness theorem for graphons on general probability spaces.

Findings

Provides a new proof of the uniqueness theorem for graphons.

Includes technical complements to existing results.

Extends results to {0,1}-valued and pure graphons.

Abstract

We give a survey of basic results on the cut norm and cut metric for graphons (and sometimes more general kernels), with emphasis on the equivalence problem. The main results are not new, but we add various technical complements, and a new proof of the uniqueness theorem by Borgs, Chayes and Lov\'asz. We allow graphons on general probability spaces whenever possible. We also give some new results for {0,1}-valued graphons and for pure graphons.