The entropy of randomized network ensembles

Randomized network ensembles are the null models of real networks and are extensively used to compare a real system to a null hypothesis. In this paper we study network ensembles with the same degree distribution, the same degree correlations and the same community structure of any given real network. We characterize these randomized network ensembles by their entropy, i.e. the normalized logarithm of the total number of networks which are part of these ensembles. We estimate the entropy of randomized ensembles starting from a large set of real directed and undirected networks. We propose entropy as an indicator to assess the role of each structural feature in a given real network. We observe that the ensembles with fixed scale-free degree distribution have smaller entropy than the ensembles with homogeneous degree distribution indicating a higher level of order in scale-free networks