  Network complexity has been studied for over half a century and has found a
wide range of applications. Many methods have been developed to characterize
and estimate the complexity of networks. However, there has been little
research with statistical guarantees. In this paper, we develop a statistical
theory of graph complexity in a general model of random graphs, the so-called
graphon model.
  Given a graphon, we endow the latent space of the nodes with the neighborhood
distance that measures the propensity of two nodes to be connected with similar
nodes. Our complexity index is then based on the covering number and the
Minkowski dimension of (a purified version of) this metric space. Although the
latent space is not identifiable, these indices turn out to be identifiable.
This notion of complexity has simple interpretations on popular examples of
random graphs: it matches the number of communities in stochastic block models;
the dimension of the Euclidean space in random geometric graphs; the regularity
of the link function in H\"older graphon models.
  From a single observation of the graph, we construct an estimator of the
neighborhood-distance and show universal non-asymptotic bounds for its risk,
matching minimax lower bounds. Based on this estimated distance, we compute the
corresponding covering number and Minkowski dimension and we provide optimal
non-asymptotic error bounds for these two plug-in estimators.
