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Several real-world systems can be represented as multi-layer complexnetworks, i.e. in terms of a superposition of various graphs, each related to adifferent mode of connection between nodes. Hence, the definition of propermathematical quantities aiming at capturing the level of complexity of thosesystems is required. Various attempts have been made to measure the empiricaldependencies between the layers of a multiplex, for both binary and weightednetworks. In the simplest case, such dependencies are measured viacorrelation-based metrics: we show that this is equivalent to the use ofcompletely homogeneous benchmarks specifying only global constraints, such asthe total number of links in each layer. However, these approaches do not takeinto account the heterogeneity in the degree and strength distributions, whichare instead a fundamental feature of real-world multiplexes. In this work, wecompare the observed dependencies between layers with the expected valuesobtained from reference models that appropriately control for the observedheterogeneity in the degree and strength distributions. This leads to novelmultiplexity measures that we test on different datasets, i.e. theInternational Trade Network (ITN) and the European Airport Network (EAN). Ourfindings confirm that the use of homogeneous benchmarks can lead to misleadingresults, and furthermore highlight the important role played by thedistribution of hubs across layers.

Multiplexity versus correlation: the role of local constraints in real multiplexes