Networks with given two-point correlations: hidden correlations from degree correlations

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.