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Multilayer networks represent multiple types of connections between the sameset of nodes. Clearly, a multilayer description of a system adds value only ifthe multiplex does not merely consist of independent layers, i.e. if theinter-layer overlap is nontrivial. On real-world multiplexes, it is expectedthat the observed overlap may partly result from spurious correlations arisingfrom 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 multiplexeswith controllable node degrees and controllable overlap. The model can bemapped to a generalized Ising model where the combination of node heterogeneityand inter-layer coupling leads to the possibility of local phase transitions.In particular, we find that an increased heterogeneity in the network resultsin different critical points for different pairs of nodes, which in turn leadsto local phase transitions that may ultimately increase the overlap. The modelallows us to quantify how the overlap can be increased by either increasing theheterogeneity of the network (spurious correlation) or the strength of theinter-layer coupling (true correlation), thereby disentangling the two effects.As an application, we show that the empirical overlap in the InternationalTrade Multiplex is not merely a spurious result of the correlation between nodedegrees across different layers, but requires a non-zero inter-layer couplingin its modeling.

Local phase transitions in a model of multiplex networks with heterogeneous degrees and inter-layer coupling