  The paper orders certain important issues related to both uncorrelated and
correlated networks with hidden variables. In particular, we show that networks
being uncorrelated at the hidden level are also lacking in correlations between
node degrees. The observation supported by the depoissonization idea allows to
extract distribution of hidden variables from a given node degree distribution.
It completes the algorithm for generating uncorrelated networks that was
suggested by other authors. In this paper we also carefully analyze the
interplay between hidden attributes and node degrees. We show how to extract
hidden correlations from degree correlations. Our derivations provide
mathematical background for the algorithm for generating correlated networks
that was proposed by Boguna and Pastor-Satorras.
