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Random networks, constrained to reproduce specific features of networks, areoften used to represent and analyze network data as well as their mathematicaldescriptions. Chief among them, the configuration model constrains randomnetworks by their degree distribution and is foundational to many areas ofnetwork science. However, these representations are often selected based onintuition or mathematical and computational simplicity rather than onstatistical evidence. To evaluate the quality of a network representation weneed to consider both the amount of information required by a random networkmodel as well as the probability of recovering the original data when using themodel as a generative process. To this end, we calculate the approximate sizeof network ensembles generated by the popular configuration model and itsgeneralizations that include degree-correlations and centrality layers based onthe onion decomposition. We then apply minimum description length as a modelselection criterion and also introduce the Network Onion Divergence: modelselection and network comparison over a nested family of configuration modelswith differing level of structural details. Using over 100 empirical sets ofnetwork data, we find that a simple Layered Configuration Model offers the mostcompact representation of the majority of real networks. We hope that ourresults will continue to motivate the development of intricate random networkmodels that help capture network structure beyond the simple degreedistribution.

Network Onion Divergence: Network representation and comparison using  nested configuration models with fixed connectivity, correlation and  centrality patterns