  Our main results are quantitative bounds in the multivariate normal
approximation of centred subgraph counts in random graphs generated by a
general graphon and independent vertex labels. We are interested in these
statistics because they are key to understanding fluctuations of regular
subgraph counts -- a cornerstone of dense graph limit theory. We also identify
the resulting limiting Gaussian stochastic measures by means of the theory of
generalised $U$-statistics and Gaussian Hilbert spaces, which we think is a
suitable framework to describe and understand higher-order fluctuations in
dense random graph models. With this article, we believe we answer the question
"What is the central limit theorem of dense graph limit theory?". We complement
the theory with some statistical applications to illustrate the use of centred
subgraph counts in network modelling.
