  The theory of graphons comes with the so-called cut norm and the derived cut
distance. The cut norm is finer than the weak* topology (when considering the
predual of $L^{1}$-functions). Dole\v{z}al and Hladk\'y [J. Combin. Theory Ser.
B 137 (2019), 232-263] showed, that given a sequence of graphons, a cut
distance accumulation graphon can be pinpointed in the set of weak*
accumulation points as a minimizer of the entropy. Motivated by this, we study
graphon parameters with the property that their minimizers or maximizers
identify cut distance accumulation points over the set of weak* accumulation
points. We call such parameters cut distance identifying.
  Of particular importance are cut distance identifying parameters coming from
homomorphism densities, $t(H,\cdot)$. This concept is closely related to the
emerging field of graph norms, and the notions of the step Sidorenko property
and the step forcing property introduced by Kr\'a\v{l}, Martins, Pach and
Wrochna [J. Combin. Theory Ser. A 162 (2019), 34-54]. We prove that a connected
graph is weakly norming if and only if it is step Sidorenko, and that if a
graph is norming then it is step forcing.
  Further, we study convexity properties of cut distance identifying graphon
parameters, and find a way to identify cut distance limits using spectra of
graphons. We also show that continuous cut distance identifying graphon
parameters have the {\guillemotleft}pumping property{\guillemotright}, and thus
can be used in the proof of the Frieze-Kannan regularity lemma.
