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Multilayer networks represent multiple types of connections between the same set of nodes. Clearly, a multilayer description of a system adds value only if the multiplex does not merely consist of independent layers, i.e. if the inter-layer overlap is nontrivial. On real-world multiplexes, it is expected that the observed overlap may partly result from spurious correlations arising from the heterogeneity of nodes and partly from true interdependencies. However, no rigorous way to disentangle these two effects has been developed. In this paper we introduce an unbiased maximum-entropy model of multiplexes with controllable node degrees and controllable overlap. The model can be mapped to a generalized Ising model where the combination of node heterogeneity and inter-layer coupling leads to the possibility of local phase transitions. In particular, we find that an increased heterogeneity in the network results in different critical points for different pairs of nodes, which in turn leads to local phase transitions that may ultimately increase the overlap. The model allows us to quantify how the overlap can be increased by either increasing the heterogeneity of the network (spurious correlation) or the strength of the inter-layer coupling (true correlation), thereby disentangling the two effects. As an application, we show that the empirical overlap in the International Trade Multiplex is not merely a spurious result of the correlation between node degrees across different layers, but requires a non-zero inter-layer coupling in its modeling.
Network robustness is a central point in network science, both from a theoretical and a practical point of view. In this paper, we show that layer degradation, understood as the continuous or discrete loss of links weight, triggers a structural trans
Universal spectral properties of multiplex networks allow us to assess the nature of the transition between disease-free and endemic phases in the SIS epidemic spreading model. In a multiplex network, depending on a coupling parameter, $p$, the inver
We study the robustness properties of multiplex networks consisting of multiple layers of distinct types of links, focusing on the role of correlations between degrees of a node in different layers. We use generating function formalism to address var
We explore the block nature of the matrix representation of multiplex networks, introducing a new formalism to deal with its spectral properties as a function of the inter-layer coupling parameter. This approach allows us to derive interesting result
Real data show that interdependent networks usually involve inter-similarity. Intersimilarity means that a pair of interdependent nodes have neighbors in both networks that are also interdependent (Parshani et al cite{PAR10B}). For example, the coupl