Entropies of complex networks with hierarchically constrained topologies

The entropy of a hierarchical network topology in an ensemble of sparse random networks, with hidden variables associated with its nodes, is the log-likelihood that a given network topology is present in the chosen ensemble. We obtain a general formula for this entropy, which has a clear interpretation in some simple limiting cases. The results provide keys with which to solve the general problem of fitting a given network with an appropriate ensemble of random networks.