  We investigate the relative computability of exchangeable binary relational
data when presented in terms of the distribution of an invariant measure on
graphs, or as a graphon in either $L^1$ or the cut distance. We establish basic
computable equivalences, and show that $L^1$ representations contain
fundamentally more computable information than the other representations, but
that $0'$ suffices to move between computable such representations. We show
that $0'$ is necessary in general, but that in the case of random-free
graphons, no oracle is necessary. We also provide an example of an
$L^1$-computable random-free graphon that is not weakly isomorphic to any
graphon with an a.e. continuous version.
