  In order to detect patterns in real networks, randomized graph ensembles that
preserve only part of the topology of an observed network are systematically
used as fundamental null models. However, their generation is still
problematic. The existing approaches are either computationally demanding and
beyond analytic control, or analytically accessible but highly approximate.
Here we propose a solution to this long-standing problem by introducing an
exact and fast method that allows to obtain expectation values and standard
deviations of any topological property analytically, for any binary, weighted,
directed or undirected network. Remarkably, the time required to obtain the
expectation value of any property is as short as that required to compute the
same property on the single original network. Our method reveals that the null
behavior of various correlation properties is different from what previously
believed, and highly sensitive to the particular network considered. Moreover,
our approach shows that important structural properties (such as the modularity
used in community detection problems) are currently based on incorrect
expressions, and provides the exact quantities that should replace them.
