Continuous graph

In graph theory and statistics, a graphon is a symmetric measurable function ${\displaystyle W:[0,1]^{2}\to [0,1]}$, that is important in the study of dense graphs. Graphons arise as the fundamental objects in two areas: as the defining objects of exchangeable random graph models and as a natural notion of limit for sequences of dense graphs. Graphons are tied to dense graphs by the following pair of observations: the random graph models defined by graphons give rise to dense graphs almost surely, and, by the regularity lemma, graphons capture the structure of arbitrary large dense graphs.

Graphons are sometimes referred to as “continuous graphs”, but this is bad practice because there are many distinct objects that this label might be applied to. In particular, there are generalizations of graphons to the sparse graph regime that could just as well be called “continuous graphs.”

A graphon is a symmetric measurable function ${\displaystyle W:[0,1]^{2}\to [0,1]}$. Usually a graphon is understood as defining an exchangeable random graph model according to the following scheme:

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