Gibbs entropy of network ensembles by cavity methods

The Gibbs entropy of a microcanonical network ensemble is the logarithm of the number of network configurations compatible with a set of hard constraints. This quantity characterizes the level of order and randomness encoded in features of a given real network. Here, we show how to relate this entropy to large deviations of conjugated canonical ensembles. We derive exact expression for this correspondence using the cavity methods for the configuration model, for the ensembles with constraint degree sequence and community structure and for the ensemble with constraint degree sequence and number of links at a given distance.