  In this paper we generalize the concept of random networks to describe
networks with non trivial features by a statistical mechanics approach. This
framework is able to describe ensembles of undirected, directed as well as
weighted networks. These networks might have not trivial community structure
or, in the case of networks embedded in a given space, non trivial distance
dependence of the link probability. These ensembles are characterized by their
entropy which evaluate the cardinality of networks in the ensemble. The general
framework we present in this paper is able to describe microcanonical ensemble
of networks as well as canonical or hidden variables network ensemble with
significant implication for the formulation of network constructing algorithms.
Moreover in the paper we define and and characterize in particular the
"structural entropy", i.e. the entropy of the ensembles of undirected
uncorrelated simple networks with given degree sequence. We discuss the
apparent paradox that scale-free degree distribution are characterized by
having small structural entropy but are so widely encountered in natural,
social and technological complex systems. We give the proof that while
scale-free networks ensembles have small structural entropy, they also
correspond to the most likely degree distribution with the corresponding value
of the structural entropy.
