We consider countable relational structures which can be generated randomly – that is, we begin with a countable set and choose which relations hold at each tuple by making independent random choices. There are various equivalent schemes for describing this situation – as exchangeable arrays, as graphons and their generalizations, as samples from ultraproducts, and as probability measures on the space of structures.

Ackerman-Freer-Patel showed that a structure can be generated randomly exactly when it has trivial group-theoretic definable closure. We wish to consider those structures which can be generated by random schemes which are themselves “quasirandom”. We observe that the characterization is more complicated than expected, and argue that by handling asymmetric relations in a slightly different way we obtain a more natural characterization.