Chapter 5 Information theory of spatial network ensembles
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This chapter provides a comprehensive and self-contained discussion of the most recent developments of information theory of networks. Maximum entropy models of networks are the least biased ensembles enforcing a set of constraints and are used in a number of applications to produce null model of networks. Here maximum entropy ensembles of networks are introduced by distinguishing between microcanonical and canonical ensembles revealing the non-equivalence of these two classes of ensembles in the case in which an extensive number of constraints is imposed. It is very common that network data includes also meta-data describing feature of the nodes such as their position in a real or in an abstract space. The features of the nodes can be treated as latent variables that determine the cost associated to each link. Maximum entropy network ensembles with latent variables include spatial networks and their generalisation. Here we cover the case of transportation networks including airport and rail networks. Maximum entropy network ensemble satisfy a given set of constraints. However traditional approaches do not provide any insight on the origin of such constraints. Here we use information theory principles to find the optimal distribution of latent variables in the framework of the classical information theory of networks. This theory explains the \blessing of non-uniformity" of data guaranteeing the efficiency of inference algorithms.

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