  Random networks, constrained to reproduce specific features of networks, are
often used to represent and analyze network data as well as their mathematical
descriptions. Chief among them, the configuration model constrains random
networks by their degree distribution and is foundational to many areas of
network science. However, these representations are often selected based on
intuition or mathematical and computational simplicity rather than on
statistical evidence. To evaluate the quality of a network representation we
need to consider both the amount of information required by a random network
model as well as the probability of recovering the original data when using the
model as a generative process. To this end, we calculate the approximate size
of network ensembles generated by the popular configuration model and its
generalizations that include degree-correlations and centrality layers based on
the onion decomposition. We then apply minimum description length as a model
selection criterion and also introduce the Network Onion Divergence: model
selection and network comparison over a nested family of configuration models
with differing level of structural details. Using over 100 empirical sets of
network data, we find that a simple Layered Configuration Model offers the most
compact representation of the majority of real networks. We hope that our
results will continue to motivate the development of intricate random network
models that help capture network structure beyond the simple degree
distribution.
