## Graphons, cut norm and distance, couplings and rearrangements.(English)Zbl 1268.05001

NYJM Monographs 4. Albany, NY: State University of New York, University at Albany. 76 p., electronic only, open access (2013).
Publisher’s description: 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 C. Borgs et al. [Geom. Funct. Anal. 19(2009), No. 6, 1597–1619 (2010; Zbl 1223.05193)]. We allow graphons on general probability spaces whenever possible. We also give some new results for $$\{0,1\}$$-valued graphons and for pure graphons.

### MSC:

 05-02 Research exposition (monographs, survey articles) pertaining to combinatorics 05C99 Graph theory 28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence

Zbl 1223.05193
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