One of the most important challenges in network science is to quantify the information encoded in complex network structures. Disentangling randomness from organizational principles is even more demanding when networks have a multiplex nature. Multiplex networks are multilayer systems of nodes that can be linked in multiple interacting and co-evolving layers. In these networks, relevant information might not be captured if the single layers were analyzed separately. Here we demonstrate that such partial analysis of layers fails to capture significant correlations between weights and topology of complex multiplex networks. To this end, we study two weighted multiplex co-authorship and citation networks involving the authors included in the American Physical Society. We show that in these networks weights are strongly correlated with multiplex structure, and provide empirical evidence in favor of the advantage of studying weighted measures of multiplex networks, such as multistrength and the inverse multiparticipation ratio. Finally, we introduce a theoretical framework based on the entropy of multiplex ensembles to quantify the information stored in multiplex networks that would remain undetected if the single layers were analyzed in isolation.

he maxim of Jean Anthelme Brillat-Savarin, "Dites-moi ce que vous mangez et je vais vous dire ce que vous êtes"-'you are what you eat'-remains as pertinent today, in the era of modern medicine, as it did in 1826. Indeed, the exceptional role of diet in health is well documented by decades of research in nutritional epidemiology, unveiling the role of nutrients and other dietary factors in cardiovascular disease, obesity, type 2 diabetes mellitus (T2DM) and other common diseases 1. Yet, the bulk of our current understanding of the way food affects health is anchored in the 150 nutritional components that the United States Department of Agriculture (USDA) and other national databases track 2,3 , and these nutritional components represent only a subset of the total pool of definable biochemicals in the food supply (see Supplementary Discussion 1).

Networks are mathematical structures that are universally used to describe a large variety of complex systems such as the brain or the Internet. Characterizing the geometrical properties of these networks has become increasingly relevant for routing problems, inference and data mining. In real growing networks, topological, structural and geometrical properties emerge spontaneously from their dynamical rules. Nevertheless we still miss a model in which networks develop an emergent complex geometry. Here we show that a single two parameter network model, the growing geometrical network, can generate complex network geometries with non-trivial distribution of curvatures, combining exponential growth and small-world properties with finite spectral dimensionality. In one limit, the non-equilibrium dynamical rules of these networks can generate scale-free networks with clustering and communities, in another limit planar random geometries with non-trivial modularity. Finally we find that these properties of the geometrical growing networks are present in a large set of real networks describing biological, social and technological systems.

The problem of controllability of the dynamical state of a network is central in network theory and has wide applications ranging from network medicine to financial markets. The driver nodes of the network are the nodes that can bring the network to the desired dynamical state if an external signal is applied to them. Using the framework of structural controllability, here we show that the density of nodes with in-degree and out-degree equal to 0, 1 and 2 determines the number of driver nodes of random networks. Moreover we show that networks with minimum in-degree and out-degree greater than 2, are always fully controllable by an infinitesimal fraction of driver nodes, regardless on the other properties of the degree distribution. Finally, based on these results, we propose an algorithm to improve the controllability of networks. The controllability of a network [1-10] is a fundamental problem with wide applications ranging from medicine and drug discovery [11], to the characterization of dynamical processes in the brain [12][13][14], or the evaluation of risk in financial markets [15]. While the interplay between the structure of the network [16][17][18][19] and the dynamical processes defined on them has been an active subject of complex network research for more than ten years [20,21], only recently the rich interplay between the controllability of a network and its structure has started to be investigated. A pivotal role in this respect has been played by a paper by Liu et al. [6], in which the problem of finding the minimal set of driver nodes necessary to control a network was mapped into a maximum matching problem. Using a well established statistical mechanics approach [22][23][24][25][26][27], Liu et al. [6] characterize in detail the set of driver nodes for real networks and for ensembles of networks with given in-degree and out-degree distribution. By analyzing scale-free networks with minimum in-degree and minimum out-degree equal to 1 they have found that the smaller is the power-law exponent γ of the degree distribution, the larger is the fraction of driver nodes in the network. This result has prompted the authors of [6] to say that the higher is the heterogeneity of the degree distribution, the less controllable is the network. Later, different papers have addressed questions related to controllability of networks with similar tools [7,28].In this Letter we consider the network controllability and its mapping to the maximum matching problem, exploring the role of low in-degree and low out-degree nodes in the network. We show that by changing the fraction of nodes with in-degree and out-degree less than 3, the number of driver nodes of a network can change in a dramatic way. In particular if the minimum in-degree and the minimum out-degree of a network are both greater than 2 then any network, independently on the level of heterogeneity of the degree distribution, is fully controllable by an infinitesimal fraction of nodes. Therefore we show that the heterogeneity of the network is not the only e...

Multiplex networks describe a large number of systems ranging from social networks to the brain. These multilayer structure encode information in their structure. This information can be extracted by measuring the correlations present in the multiplex networks structure, such as the overlap of the links in different layers. Many multiplex networks are also weighted, and the weights of the links can be strongly correlated with the structural properties of the multiplex network. For example, in multiplex network formed by the citation and collaboration networks between PRE scientists it was found that the statistical properties of citations to coauthors differ from the one of citations to noncoauthors, i.e., the weights depend on the overlap of the links. Here we present a theoretical framework for modeling multiplex weighted networks with different types of correlations between weights and overlap. To this end, we use the framework of canonical network ensembles, and the recently introduced concept of multilinks, showing that null models of a large variety of network structures can be constructed in this way. In order to provide a concrete example of how this framework apply to real data we consider a multiplex constructed from gene expression data of healthy and cancer tissues.

We approach here the problem of defining and estimating the nature of the metabolite-metabolite association network underlying the human individual metabolic phenotype in healthy subjects. We retrieved significant associations using an entropy-based approach and a multiplex network formalism. We defined a significantly over-represented network formed by biologically interpretable metabolite modules. The entropy of the individual metabolic phenotype is also introduced and discussed.

We characterize different cell states, related to cancer and ageing phenotypes, by a measure of entropy of network ensembles, integrating gene expression profiling values and protein interaction network topology. In our case studies, network entropy, that by definition estimates the number of possible network instances satisfying the given constraints, can be interpreted as a measure of the "parameter space" available to the cell. Network entropy was able to characterize specific pathological conditions: normal versus cancer cells, primary tumours that developed metastasis or relapsed, and extreme longevity samples. Moreover, this approach has been applied at different scales, from whole network to specific subnetworks (biological pathways defined on a priori biological knowledge) and single nodes (genes), allowing a deeper understanding of the cell processes involved.

The controllability of a network is a theoretical problem of relevance in a variety of contexts ranging from financial markets to the brain. Until now, network controllability has been characterized only on isolated networks, while the vast majority of complex systems are formed by multilayer networks. Here we build a theoretical framework for the linear controllability of multilayer networks by mapping the problem into a combinatorial matching problem. We found that correlating the external signals in the different layers can significantly reduce the multiplex network robustness to node removal, as it can be seen in conjunction with a hybrid phase transition occurring in interacting Poisson networks. Moreover we observe that multilayer networks can stabilize the fully controllable multiplex network configuration that can be stable also when the full controllability of the single network is not stable.