Many complex systems can be represented as networks composed by distinct layers, interacting and depending on each others. For example, in biology, a good description of the full protein-protein interactome requires, for some organisms, up to seven distinct network layers, with thousands of protein-protein interactions each. A fundamental open question is then how much information is really necessary to accurately represent the structure of a multilayer complex system, and if and when some of the layers can indeed be aggregated. Here we introduce a method, based on information theory, to reduce the number of layers in multilayer networks, while minimizing information loss. We validate our approach on a set of synthetic benchmarks, and prove its applicability to an extended data set of protein-genetic interactions, showing cases where a strong reduction is possible and cases where it is not. Using this method we can describe complex systems with an optimal trade-off between accuracy and complexity. Network science has shown that characterising the topology of a complex system is fundamental when it comes to understanding its dynamical properties [1, 3, 21] . However, in most cases, the basic units of real-world systems are connected by different types of interactions occurring at comparable time scales. For instance, this is the case of social systems, in which the same set of people might have political or financial relationships [25] , or might be interacting using different platforms like e-mail, Twitter, Facebook, phone calls, etc. [18, 30] . Similarly in biological systems, basic constituents such as proteins can have physical, co-localization, genetic or many other types of interactions. Recently, it has been shown that retaining the whole multi-dimensional information [8] in the modeling of interdependent [6, 13] and multilayer systems [9, 17, 20, 22] leads to new non-trivial structural properties [2, 11, 24] and unexpected levels of dynamical complexity [10, 14, 15, 23, 26] . On the other hand, nothing is known about the inverse problem, that is under which circumstances a more compact aggregated representation of a system can give the same information of a fully multilayer representation. Inspired by quantum physics, where a similar question emerges to quantify the distance between mixed quantum states [19], we propose a method to aggregate some of the layers of a multilayer system without a sensible loss of information. Our procedure is based on the application of information theory to graphs, and allows to construct a reduced representation of a multilayer network which provides a good trade-off between accuracy and compactness. In quantum mechanics, there are pure states, describing the system by means of a single vector in the Hilbert space, and mixed states, arising from composite quantum systems described by a statistical ensemble of pure states. The most general quantum system can then be described by the so-called density operator ρ, a semidefinite positive matrix with eigenvalues summing up to 1, which encodes all the information about the statistical ensemble of pure states of the system [12] . A widely adopted descriptor to measure the mixedness of a quantum system is given by the Von Neumann entropy, the natural extension of Shannon information entropy to quantum operators, although other definitions, satisfying extensivity or nonextensivity paradigms, have been lately introduced and studied [28] . The Von Neumann entropy is defined for any density operator ρ. In particular, if the Von Neumann entropy is zero, then the system is in a pure state, otherwise it is in a mixed state. The quantum mechanics formalism can also be used to describe a complex system, if we imagine that each layer of a multilayer network represents one possible state of the system, so that the entire network is described by an ensemble of states. It has already been shown that the amount of information carried by a single-layer network can be quantified by the Von Neumann entropy of the graph [4], represented by a matrix which resembles quantum operators. Such a matrix can be obtained from the Laplacian associated to the graph, after a proper normalisation (see Appendix for details). The resulting number effectively summarises the complexity of the wiring patterns of a network. Here we propose to use the Von Neumann entropy to quantify the information gained or lost by aggregating some of the layers of a multiplex network in a single graph. This problem is surprisingly related to the separability of mixing states in quantum systems [7, 16, 27] , where, for instance, the entanglement of pure states is quantified by considering the minimum information loss due to a complete local measurement, in terms of the corresponding density operator ρ.
Although, techniques based on information theory have been proposed for multilayer protein-protein interactions REF , this is only for reducing the number of redundant layers through aggregation, but not as a generalized approach for composing different layers to represent the corresponding combination of features as proposed here.
Layer aggregation and reducibility of multilayer interconnected networks
Nature Communications 6, 6864 (2015)
Physics Computer Science