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Any physical system can be viewed from the perspective that information isimplicitly represented in its state. However, the quantification of thisinformation when it comes to complex networks has remained largely elusive. Inthis work, we use techniques inspired by quantum statistical mechanics todefine an entropy measure for complex networks and to develop a set ofinformation-theoretic tools, based on network spectral properties, such asRenyi q-entropy, generalized Kullback-Leibler and Jensen-Shannon divergences,the latter allowing us to define a natural distance measure between complexnetworks. First we show that by minimizing the Kullback-Leibler divergencebetween an observed network and a parametric network model, inference of modelparameter(s) by means of maximum-likelihood estimation can be achieved andmodel selection can be performed appropriate information criteria. Second, weshow that the information-theoretic metric quantifies the distance betweenpairs of networks and we can use it, for instance, to cluster the layers of amultilayer system. By applying this framework to networks corresponding tosites of the human microbiome, we perform hierarchical cluster analysis andrecover with high accuracy existing community-based associations. Our resultsimply that spectral based statistical inference in complex networks results indemonstrably superior performance as well as a conceptual backbone, filling agap towards a network information theory.

Spectral entropies as information-theoretic tools for complex network comparison