On the computability of graphons

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.