  Represented as graphs, real networks are intricate combinations of order and
disorder. Fixing some of the structural properties of network models to their
values observed in real networks, many other properties appear as statistical
consequences of these fixed observables, plus randomness in other respects.
Here we employ the $dk$-series, a complete set of basic characteristics of the
network structure, to study the statistical dependencies between different
network properties. We consider six real networks---the Internet, US airport
network, human protein interactions, technosocial web of trust, English word
network, and an fMRI map of the human brain---and find that many important
local and global structural properties of these networks are closely reproduced
by $dk$-random graphs whose degree distributions, degree correlations, and
clustering are as in the corresponding real network. We discuss important
conceptual, methodological, and practical implications of this evaluation of
network randomness, and release software to generate $dk$-random graphs.
