A graphon is a symmetric, bounded, measurable function

ie, it is a bounded kernel (weighting function).

We can think of a graphon as a graph with an uncountable number of nodes $u\in [0,1]$ and edges $(u,v)$ with weights $W(u,v)$. Alternatively, we can think of a graphon as a model that we can use to sample graphs.

A more general definition is

where $\mathrm{\Omega }$ is a sample space with some probability measure $f$. We can always map $\mathrm{\Omega }$ onto $[0,1]$ using a measure-preserving map (as long as the CDF of $f$ is strictly monotone), so we can use $[0,1{]}^2$ WLOG.

Show that if the CDF of $f$ is strictly monotone, we can map $\mathrm{\Omega }$ to the interval $[0,1]$ with a measure-preserving map.

The codomain of $W$ may be $(0,B]$ where $B\in \mathbb{R}<+\mathrm{\infty }$. However, we usually have $B=1$ since $W(u,v)$ often represents a probability.

^note-2

From left to right:

1. block model with same community sizes
2. different community sizes, and
3. $w(u,v)=\exp \left(\frac{(u-v{)}^2}{{\sigma }^2}\right)$
   ^example

Graphon as a limiting object for some sequence of graphs

Graphon as generating object for finite graphs