 To formalize this, for any symmetric, measurable function formula_177, define
the cut norm of formula_178 to be the quantity

formula_179

taken over all measurable subsets formula_180 of the unit interval.This norm captures our earlier notion of distance, because for two graphs formula_59 and formula_127 with same vertex set formula_183 of size formula_184, the cut norm with the associated graphons

formula_185

lets us compute the maximum discrepancy of the edge densities between formula_59 and formula_127.Note that this definition can still be used even when the graphs being compared do not share a vertex set.