kernel cut metric

The cut metric for two graphons (kernels) is given by

Where $||\cdot |{|}_{◻}$ is the cut norm and $W^{\phi }(u,v)=W(\phi (u),\phi (v))$ and $\phi$ are measure-preserving bijections (on the unit interval)
^definition

We can think of the $\phi$ as "permutations"