The governance of the political and economic world order builds on a complex architecture of international treaties at various geographical scales. In a historical phase of high institutional turbulence, assessing the stability of such architecture with respect to the unilateral defection of single countries and the breakdown of single treaties is important. We carry out this analysis on the whole global architecture and find that the countries with the highest disruption potential are mostly medium-small and micro countries. Political stability is highly dependent on many former colonial overseas territories that are today part of the global network of fiscal havens, as well as on emerging economies, mostly from South-East Asia. Economic stability depends on medium-sized European and African countries. Single global treaties have surprisingly less disruptive potential, with the major exception of the WTO. Our results suggest that the potential fragility of the world order seems to be more directly related to global inequality and fiscal injustice than commonly believed and that the legacy of the colonial world order is still strong in the current international relations scenario. In particular, vested interests related to tax avoidance seem to have a structural role in the political architecture of global governance.

Despite the numerous ways now available to quantify which parts or subsystems of a network are most important, there remains a lack of centrality measures that are related to the complexity of information flows and are derived directly from entropy measures. Here, we introduce a ranking of edges based on how each edge's removal would change a system's von Neumann entropy (VNE), which is a spectral-entropy measure that has been adapted from quantum information theory to quantify the complexity of information dynamics over networks. We show that a direct calculation of such rankings is computationally inefficient (or unfeasible) for large networks: e.g.\ the scaling is (N3) per edge for networks with N nodes. To overcome this limitation, we employ spectral perturbation theory to estimate VNE perturbations and derive an approximate edge-ranking algorithm that is accurate and fast to compute, scaling as (N) per edge. Focusing on a form of VNE that is associated with a transport operator e−βL, where L is a graph Laplacian matrix and β>0 is a diffusion timescale parameter, we apply this approach to diverse applications including a network encoding polarized voting patterns of the 117th U.S. Senate, a multimodal transportation system including roads and metro lines in London, and a multiplex brain network encoding correlated human brain activity. Our experiments highlight situations where the edges that are considered to be most important for information diffusion complexity can dramatically change as one considers short, intermediate and long timescales β for diffusion.

Real systems are characterized by complex patterns of interactions between their units, by dynamical processes on them, and by the interplay of the two. It is well known that particular structures affect dynamical processes at different scales. Sometimes richly connected units are connected by costly, long-range links. In the brain, hubs form rich clubs for integrating information between different brain regions, and many biological and social networks show this same structural organization. It remains, however, unclear whether this structural organization alone enables a rapid communication between highly connected nodes or whether a functional rich club may emerge as a combination of direct links and longer paths between rich nodes. Here, we identify functional rich clubs through the diffusion geometry, providing a perspective on rich-club phenomena in complex networks. We show that weak structural rich clubs may be functionally stronger, thanks to bridge nodes, while diffusion inside strong structural rich clubs may be damped in modular networks.

Networks are a powerful tool to model complex systems, and the definition of many Graph Neural Networks (GNN), Deep Learning algorithms that can handle networks, has opened a new way to approach many real-world problems that would be hardly or even untractable. In this paper, we propose mGNN, a framework meant to generalize GNNs to the case of multi-layer networks, i.e., networks that can model multiple kinds of interactions and relations between nodes. Our approach is general (i.e., not task specific) and has the advantage of extending any type of GNN without any computational overhead. We test the framework into three different tasks (node and network classification, link prediction) to validate it.

For decades, from design theory to urban planning and management, from social sciences to urban environmental science, cities have been probed and analyzed from the partial perspective of single disciplines. The digital era, with its unprecedented data availability, is allowing for testing old theories and developing new ones, ultimately challenging relatively partial models. Our community has been in the last years providing more and more compelling evidence that cities are complex systems with emergent phenomena characterized by the collective behavior of their citizens who are themselves complex systems. However, more recently, it has also been shown that such multiscale complexity alone is not enough to describe some salient features of urban systems. Multilayer network modeling, accounting for both multiplexity of relationships and interdependencies among the city’s subsystems, is indeed providing a novel integrated framework to study urban backbones, their resilience to unexpected perturbations due to internal or external factors, and their human flows. In this paper, we first offer an overview of the transdisciplinary efforts made to cope with the three dimensions of complexity of the city: the complexity of the urban environment, the complexity of human cognition about the city, and the complexity of city planning. In particular, we discuss how the most recent findings, for example, relating the health and wellbeing of communities to urban structure and function, from traffic congestion to distinct types of pollution, can be better understood considering a city as a multiscale and multilayer complex system. The new challenges posed by the postpandemic scenario give to this perspective an unprecedented relevance, with the necessity to address issues of reconstruction of the social fabric, recovery from prolonged psychological, social and economic stress with the ensuing mental health and wellbeing issues, and repurposing of urban organization as a consequence of new emerging practices such as massive remote working. By rethinking cities as large-scale active matter systems far from equilibrium which consume energy, process information, and adapt to the environment, we argue that enhancing social engagement, for example, involving citizens in codesigning the city and its changes in this critical postpandemic phase, can trigger widespread adoption of good practices leading to emergent effects with collective benefits which can be directly measured.

Chapter for the book Coronavirus Disease - COVID-19. Advances in Experimental Medicine and Biology, edited by N. Rezaei

Large-scale neurophysiological networks are often reconstructed from band-pass filtered time series derived from magnetoencephalography (MEG) data. Common practice is to reconstruct these networks separately for different frequency bands and to treat them independently. Recent evidence suggests that this separation may be inadequate, as there can be significant coupling between frequency bands (interlayer connectivity). A multilayer network approach offers a solution to analyze frequency-specific networks in one framework. We propose to use a recently developed network reconstruction method in conjunction with phase oscillator models to estimate interlayer connectivity that optimally fits the empirical data. This approach determines interlayer connectivity based on observed frequency-specific time series of the phase and a connectome derived from diffusion weighted imaging. The performance of this interlayer reconstruction method was evaluated in-silico. Our reconstruction of the underlying interlayer connectivity agreed to very high degree with the ground truth. Subsequently, we applied our method to empirical resting-state MEG data obtained from healthy subjects and reconstructed two-layered networks consisting of either alpha-to-beta or theta-to-gamma band connectivity. Our analysis revealed that interlayer connectivity is dominated by a multiplex structure, i.e. by one-to-one interactions for both alpha-to-beta band and theta-to-gamma band networks. For theta–gamma band networks, we also found a plenitude of interlayer connections between distant nodes, though weaker connectivity relative to the one-to-one connections. Our work is an stepping stone towards the identification of interdependencies across frequency-specific networks. Our results lay the ground for the use of the promising multilayer framework in this field with more-informed and justified interlayer connections.

Complex networks are characterized by latent geometries induced by their topology or by the dynamics on the top of them. In the latter case, different network-driven processes induce distinct geometric features that can be captured by adequate metrics. Random walks, a proxy for a broad spectrum of processes, from simple contagion to metastable synchronization and consensus, have been recently used in [Phys. Rev. Lett. 118, 168301 (2017)] to define the class of diffusion geometry and pinpoint the functional mesoscale organization of complex networks from a genuine geometric perspective. Here, we firstly extend this class to families of distinct random walk dynamics -- including local and non-local information -- on the top of multilayer networks -- a paradigm for biological, neural, social, transportation, biological and financial systems -- overcoming limitations such as the presence of isolated nodes and disconnected components, typical of real-world networks. Secondly, we characterize the multilayer diffusion geometry of synthetic and empirical systems, highlighting the role played by different random search dynamics in shaping the geometric features of the corresponding diffusion manifolds.

How components of the distributed brain networks that support cognition participate in typical functioning remains a largely unanswered question. An important subgroup of regions in the larger network are connector hubs, which are areas that are highly connected to several other functionally-specialized sets of regions, and are likely important for sensorimotor integration. The present study attempts to characterize connector hubs involved in typical expressive language functioning using a data-driven, multimodal, full multilayer MEG connectivity-based pipeline. Twelve adolescents, 16-18 years of age (5 male) participated in this study. Participants underwent MEG scanning during a verb generation task. MEG and structural connectivity were calculated at the whole-brain level. Amplitude-amplitude coupling (AAC) was used to compute functional connections both within and between discrete frequency bins. AAC values were then multiplied by a binary structural connectivity matrix, then entered into full multilayer network analysis. Initially, hubs were defined based on multilayer versatility and subsequently re-ranked by a novel measure called delta centrality on interconnectedness (DCI). DCI is defined as the percent change in inter-frequency interconnectedness after removal of a hub. We resolved regions that are important for between-frequency communication among other areas during expressive language, with several potential theoretical and clinical applications that can be generalized to other cognitive domains. Our multilayer, data-driven framework captures nonlinear connections that span across scales that are often missed in conventional analyses. The present study suggests that crucial hubs may be conduits for inter-frequency communication between action and perception systems that are crucial for typical functioning.

Multilayer systems are coupled networks characterized by different contexts (layers) of interaction and have gained much attention recently due to their suitability to describe a broad spectrum of empirical complex systems. They are very fragile to percolation and first-neighbor failure propagation, but little is known about how they respond to non-local disruptions, as it occurs in failures induced by flow redistribution, for example. Acknowledging that many socio-technical and biological systems sustain a flow of some physical quantity, such as energy or information, across the their components, it becomes crucial to understand when the flow redistribution can cause global cascades of failures in order to design robust systems, to increase their resilience or to learn how to efficiently dismantle them. In this paper we study the impact that different multiplex topological features have on the robustness of the system when subjected to non-local cascade propagation. We first numerically demonstrate that this dynamics has a critical value at which a small initial perturbation effectively dismantles the entire network, and that the transition appears abruptly. Then we identify that the excess of flow caused by a failure is, in general, more homogeneously distributed the networks in which the average distance between nodes is small. Using this information we find that aggregated versions of multiplex networks tend to overestimate robustness, even though to make the system more robust can be achieved by increasing the number of layers. Our predictions are confirmed by simulated cascading failures in a real multilayer system.

Units of complex systems -- such as neurons in the brain or individuals in societies -- must communicate efficiently to function properly: e.g., allowing electrochemical signals to travel quickly among functionally connected neuronal areas in the human brain, or allowing for fast navigation of humans and goods in complex transportation landscapes. The coexistence of different types of relationships among the units, entailing a multilayer represention in which types are considered as networks encoded by layers, plays an important role in the quality of information exchange among them. While altering the structure of such systems -- e.g., by physically adding (or removing) units, connections or layers -- might be costly, coupling the dynamics of subset(s) of layers in a way that reduces the number of redundant diffusion pathways across the multilayer system, can potentially accelerate the overall information flow. To this aim, we introduce a framework for functional reducibility which allow us to enhance transport phenomena in multilayer systems by coupling layers together with respect to dynamics rather than structure. Mathematically, the optimal configuration is obtained by maximizing the deviation of system's entropy from the limit of free and non-interacting layers. Our results provide a transparent procedure to reduce diffusion time and optimize non-compact search processes in empirical multilayer systems, without the cost of altering the underlying structure.

Identification of modules in molecular networks is at the core of many current analysis methods in biomedical research. However, how well different approaches identify disease-relevant modules in different types of networks remains poorly understood. We launched the "Disease Module Identification DREAM Challenge", an open competition to comprehensively assess module identification methods across diverse gene, protein and signaling networks. Predicted network modules were tested for association with complex traits and diseases using a unique collection of 180 genome-wide association studies (GWAS). While a number of approaches were successful in terms of discovering complementary trait-associated modules, consensus predictions derived from the challenge submissions performed best. We find that most of these modules correspond to core disease-relevant pathways, which often comprise therapeutic targets and correctly prioritize candidate disease genes. This community challenge establishes benchmarks, tools and guidelines for molecular network analysis to study human disease biology (https://synapse.org/modulechallenge).

Untangling the complex interplay between phenotype and genotype is crucial to the effective characterization and subtyping of diseases. Here we build and analyze the multiplex network of 779 human diseases, which consists of a genotype-based layer and a phenotype-based layer. We show that diseases with common genetic constituents tend to share symptoms, and uncover how phenotype information helps boost genotype information. Moreover, we offer a flexible classification of diseases that considers their molecular underpinnings alongside their clinical manifestations. We detect cohesive groups of diseases that have high intra-group similarity at both the molecular and the phenotypic level. Inspecting these disease classes, we demonstrate the underlying pathways that connect diseases mechanistically. We observe monogenic disorders grouped together with complex diseases for which they increase the risk factor. We propose potentially new disease associations that arise as a unique feature of the information flow within and across the two layers.

Overwhelming empirical evidence has shown that online social dynamics mirrors real-world events. Hence, understanding the mechanisms leading to social contagion in online ecosystems is fundamental for predicting, and even manouvering, human behavior. It has been shown that one of such mechanisms is based on fabricating armies of automated agents that are known as social bots. Using the recent Italian elections as an emblematic case study, here we provide evidence for the existence of a special class of highly influential users, that we name "augmented humans". They exploit bots for enhancing both their visibility and influence, generating deep information cascades to the same extent of news media and other broadcasters. Augmented humans uniformly infiltrate across the full range of identified clusters of accounts, the latter reflecting political parties and their electoral ranks.

Social media can deeply influence reality perception, affecting millions of people’s voting behavior. Hence, maneuvering opinion dynamics by disseminating forged content over online ecosystems is an effective pathway for social hacking. We propose a framework for discovering such a potentially dangerous behavior promoted by automatic users, also called “bots,” in online social networks. We provide evidence that social bots target mainly human influencers but generate semantic content depending on the polarized stance of their targets. During the 2017 Catalan referendum, used as a case study, social bots generated and promoted violent content aimed at Independentists, ultimately exacerbating social conflict online. Our results open challenges for detecting and controlling the influence of such content on society.

Biological systems, from a cell to the human brain, are inherently complex. A powerful representation of such systems, described by an intricate web of relationships across multiple scales, is provided by complex networks. Recently, several studies are highlighting how simple networks – obtained by aggregating or neglecting temporal or categorical description of biological data – are not able to account for the richness of information characterizing biological systems. More complex models, namely multilayer networks, are needed to account for interdependencies, often varying across time, of biological interacting units within a cell, a tissue or parts of an organism. Gosak et al. [1] review the most recent advances in the application of multilayer networks for modeling complex biological systems, from molecular interactions within a cell to neuronal connectivity of the human brain.

A variety of complex systems exhibit different types of relationships simultaneously that can be modeled by multiplex networks. A typical problem is to determine the community structure of such systems that, in general, depend on one or more parameters to be tuned. In this study we propose one measure, grounded on information theory, to find the optimal value of the relax rate characterizing Multiplex Infomap, the generalization of the Infomap algorithm to the realm of multilayer networks. We evaluate our methodology on synthetic networks, to show that the most representative community structure can be reliably identified when the most appropriate relax rate is used. Capitalizing on these results, we use this measure to identify the most reliable meso-scale functional organization in the human protein-protein interaction multiplex network and compare the observed clusters against a collection of independently annotated gene sets from the Molecular Signatures Database (MSigDB). Our analysis reveals that modules obtained with the optimal value of the relax rate are biologically significant and, remarkably, with higher functional content than the ones obtained from the aggregate representation of the human proteome. Our framework allows us to characterize the meso-scale structure of those multilayer systems whose layers are not explicitly interconnected each other -- as in the case of edge-colored models -- the ones describing most biological networks, from proteomes to connectomes.

Chapter for the book Multiplex and Multilevel Networks, edited by Stefano Battiston, Guido Caldarelli, Antonios Garas

We introduce distance entropy as a measure of homogeneity in the distribution of path lengths between a given node and its neighbours in a complex network. Distance entropy defines a new centrality measure whose properties are investigated for a variety of synthetic network models. By coupling distance entropy information with closeness centrality, we introduce a network cartography which allows one to reduce the degeneracy of ranking based on closeness alone. We apply this methodology to the empirical multiplex lexical network encoding the linguistic relationships known to English speaking toddlers. We show that the distance entropy cartography better predicts how children learn words compared to closeness centrality. Our results highlight the importance of distance entropy for gaining insights from distance patterns in complex networks.

Word similarities affect language acquisition and use in a multi-relational way barely accounted for in the literature. We propose a multiplex network representation of this mental lexicon of word similarities as a natural framework for investigating large-scale cognitive patterns. Our representation accounts for semantic, taxonomic, and phonological interactions and it identifies a cluster of words which are used with greater frequency, are identified, memorised, and learned more easily, and have more meanings than expected at random. This cluster emerges around age 7 through an explosive transition not reproduced by null models. We relate this explosive emergence to polysemy – redundancy in word meanings. Results indicate that the word cluster acts as a core for the lexicon, increasing both lexical navigability and robustness to linguistic degradation. Our findings provide quantitative confirmation of existing conjectures about core structure in the mental lexicon and the importance of integrating multi-relational word-word interactions in psycholinguistic frameworks.

Understanding how the human brain is structured, and how its architecture is related to the function, is of paramount importance for a variety of applications, including, but not limited to, new ways to prevent, deal with and cure brain diseases, such as Alzheimer’s or Parkinson’s, and psychiatric disorders, such as Schizophrenia. The recent advances in structural and functional neuroimaging, together with the increasing attitude to interdisciplinary approaches involving computer science, mathematics and physics, are fostering interesting results from computational neuroscience, that are quite often based on the analysis of complex network representation of human brain. In the last years, this representation experienced a theoretical and computational revolution that are breaching neuroscience, allowing to cope with the increasing complexity of human brain across multiple scales and in multiple dimensions, and to model structural and functional connectivity from new perspectives, often combined with each other. In this work, we will review the main achievements obtained from interdisciplinary research based on magnetic resonance imaging and establishing, de facto, the birth of multilayer network analysis and modeling of human brain.

Nowadays, scientific challenges usually require approaches that cross traditional boundaries between academic disciplines, driving many researchers towards interdisciplinarity. Despite its obvious importance, there is a lack of studies on how to quantify the influence of interdisciplinarity on the research impact, posing uncertainty in a proper evaluation for hiring and funding purposes. Here we propose a method based on the analysis of bipartite interconnected multilayer networks of citations and disciplines, to assess scholars, institutions and countries interdisciplinary importance. Using data about physics publications and US patents, we show that our method allows to reveal, using a quantitative approach, that being more interdisciplinary causes -- in the Granger sense -- benefits in scientific productivity and impact. The proposed method could be used by funding agencies, universities and scientific policy decision makers for hiring and funding purposes, and to complement existing methods to rank universities and countries.

Despite the success of traditional network analysis, standard networks provide a limited representation of complex systems, which often include different types of relationships (or ‘multiplexity’) between their components. Such structural complexity has a significant effect on both dynamics and function. Throwing away or aggregating available structural information can generate misleading results and be a major obstacle towards attempts to understand complex systems. The recent multilayer approach for modelling networked systems explicitly allows the incorporation of multiplexity and other features of realistic systems. It allows one to couple different structural relationships by encoding them in a convenient mathematical object. It also allows one to couple different dynamical processes on top of such interconnected structures. The resulting framework plays a crucial role in helping to achieve a thorough, accurate understanding of complex systems. The study of multilayer networks has also revealed new physical phenomena that remain hidden when using ordinary graphs, the traditional network representation. Here we survey progress towards attaining a deeper understanding of spreading processes on multilayer networks, and we highlight some of the physical phenomena related to spreading processes that emerge from multilayer structure.

Any physical system can be viewed from the perspective that information is implicitly represented in its state. However, the quantification of this information when it comes to complex networks has remained largely elusive. In this work, we use techniques inspired by quantum statistical mechanics to define an entropy measure for complex networks and to develop a set of information-theoretic tools, based on network spectral properties, such as Renyi q-entropy, generalized Kullback-Leibler and Jensen-Shannon divergences, the latter allowing us to define a natural distance measure between complex networks. First we show that by minimizing the Kullback-Leibler divergence between an observed network and a parametric network model, inference of model parameter(s) by means of maximum-likelihood estimation can be achieved and model selection can be performed appropriate information criteria. Second, we show that the information-theoretic metric quantifies the distance between pairs of networks and we can use it, for instance, to cluster the layers of a multilayer system. By applying this framework to networks corresponding to sites of the human microbiome, we perform hierarchical cluster analysis and recover with high accuracy existing community-based associations. Our results imply that spectral based statistical inference in complex networks results in demonstrably superior performance as well as a conceptual backbone, filling a gap towards a network information theory.

Network analysis provides a powerful tool to analyze complex influences of social and ecological structures on community and household dynamics. Most network studies of social–ecological systems use simple, undirected, unweighted networks. We analyze multiplex, directed, and weighted networks of subsistence food flows collected in three small indigenous communities in Arctic Alaska potentially facing substantial economic and ecological changes. Our analysis of plausible future scenarios suggests that changes to social relations and key households have greater effects on community robustness than changes to specific wild food resources

We find that winning European research money (Nature 530, 33; 2016) is, among other things, contingent on how well national governments manage to retain their own scientists and to attract others from abroad. We analysed statistical indicators of scientists' mobility in the European Union for 2007-14 (erc.europa.eu and go.nature.com/bpeylu) to determine the 'attractiveness' of different countries to scientists from abroad, as well as their 'stickiness' in preventing native researchers from moving away (known as brain drain). We quantified attractiveness and stickiness as the relative difference between incoming or remaining researchers, respectively, and outgoing ones. For both measures, we found that the higher the value, the better are that country's chances of securing European research funding. The United Kingdom and Sweden are high scorers in both; Italy is among the lowest (further details from M. de D. and A. A.; see go.nature.com/wyvtls). We conclude that there is a 'rich get richer' effect for countries that have high attractiveness and stickiness scores. Those nations also boast a high gross domestic product per capita, and tend to invest more in research and development. This means that they can lure and retain the best researchers by providing competitive salaries and a guaranteed future in research.

Typical brain networks consist of many peripheral regions and a few highly central ones, i.e. hubs, playing key functional roles in cerebral inter-regional interactions. Studies have shown that networks, obtained from the analysis of specific frequency components of brain activity, present peculiar architectures with unique profiles of region centrality. However, the identification of hubs in networks built from different frequency bands simultaneously is still a challenging problem, remaining largely unexplored. Here we identify each frequency component with one layer of a multiplex network and face this challenge by exploiting the recent advances in the analysis of multiplex topologies. First, we show that each frequency band carries unique topological information, fundamental to accurately model brain functional networks. We then demonstrate that hubs in the multiplex network, in general different from those ones obtained after discarding or aggregating the measured signals as usual, provide a more accurate map of brain's most important functional regions, allowing to distinguish between healthy and schizophrenic populations better than conventional network approaches.

Academic research is driven by several factors causing different disciplines to act as "sources" or "sinks" of knowledge. However, how the flow of authors' research interests -- a proxy of human knowledge -- evolved across time is still poorly understood. Here, we build a comprehensive map of such flows across one century, revealing fundamental periods in the raise of interest in areas of human knowledge. We identify and quantify the most attractive topics over time, when a relatively significant number of researchers moved from their original area to another one, causing what we call a "diaspora of the knowledge" towards sinks of scientific interest, and we relate these points to crucial historical and political events. Noticeably, only a few areas -- like Medicine, Physics or Chemistry -- mainly act as sources of the diaspora, whereas areas like Material Science, Chemical Engineering, Neuroscience, Immunology and Microbiology or Environmental Science behave like sinks.

Real-world complex systems exhibit multiple levels of relationships. In many cases they require to be modeled as interconnected multilayer networks, characterizing interactions of several types simultaneously. It is of crucial importance in many fields, from economics to biology and from urban planning to social sciences, to identify the most (or the less) influential nodes in a network using centrality measures. However, defining the centrality of actors in interconnected complex networks is not trivial. In this paper, we rely on the tensorial formalism recently proposed to characterize and investigate this kind of complex topologies, and extend two well known random walk centrality measures, the random walk betweenness and closeness centrality, to interconnected multilayer networks. For each of the measures we provide analytical expressions that completely agree with numerically results.

The coexistence of multiple types of interactions within social, technological and biological networks has moved the focus of the physics of complex systems towards a multiplex description of the interactions between their constituents. This novel approach has unveiled that the multiplex nature of complex systems has strong influence in the emergence of collective states and their critical properties. Here we address an important issue that is intrinsic to the coexistence of multiple means of interactions within a network: their competition. To this aim, we study a two-layer multiplex in which the activity of users can be localized in each of the layer or shared between them, favoring that neighboring nodes within a layer focus their activity on the same layer. This framework mimics the coexistence and competition of multiple communication channels, in a way that the prevalence of a particular communication platform emerges as a result of the localization of users activity in one single interaction layer. Our results indicate that there is a transition from localization (use of a preferred layer) to delocalization (combined usage of both layers) and that the prevalence of a particular layer (in the localized state) depends on their structural properties.

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.

Real-world complex systems exhibit multiple levels of relationships. In many cases, they require to be modeled by interconnected multilayer networks, characterizing interactions on several levels simultaneously. It is of crucial importance in many fields, from economics to biology, from urban planning to social sciences, to identify the most (or the less) influent nodes in a network. However, defining the centrality of actors in an interconnected structure is not trivial. In this paper, we capitalize on the tensorial formalism, recently proposed to characterize and investigate this kind of complex topologies, to show how several centrality measures -- well-known in the case of standard ("monoplex") networks -- can be extended naturally to the realm of interconnected multiplexes. We consider diagnostics widely used in different fields, e.g., computer science, biology, communication and social sciences, to cite only some of them. We show, both theoretically and numerically, that using the weighted monoplex obtained by aggregating the multilayer network leads, in general, to relevant differences in ranking the nodes by their importance.

Unveiling the community structure of networks is a powerful methodology to comprehend interconnected systems across the social and natural sciences. To identify different types of functional modules in interaction data aggregated in a single network layer, researchers have developed many powerful methods. For example, flow-based methods have proven useful for identifying modular dynamics in weighted and directed networks that capture constraints on flow in the systems they represent. However, many networked systems consist of agents or components that exhibit multiple layers of interactions. Inevitably, representing this intricate network of networks as a single aggregated network leads to information loss and may obscure the actual organization. Here we propose a method based on compression of network flows that can identify modular flows in non-aggregated multilayer networks. Our numerical experiments on synthetic networks show that the method can accurately identify modules that cannot be identified in aggregated networks or by analyzing the layers separately. We capitalize on our findings and reveal the community structure of two multilayer collaboration networks: scientists affiliated to the Pierre Auger Observatory and scientists publishing works on networks on the arXiv. Compared to conventional aggregated methods, the multilayer method reveals smaller modules with more overlap that better capture the actual organization.

Nowadays, millions of people interact on a daily basis on online social media like Facebook and Twitter, where they share and discuss information about a wide variety of topics. In this paper, we focus on a specific online social network, Twitter, and we analyze multiple datasets each one consisting of individuals' online activity before, during and after an exceptional event in terms of volume of the communications registered. We consider important events that occurred in different arenas that range from policy to culture or science. For each dataset, the users' online activities are modeled by a multilayer network in which each layer conveys a different kind of interaction, specifically: retweeting, mentioning and replying. This representation allows us to unveil that these distinct types of interaction produce networks with different statistical properties, in particular concerning the degree distribution and the clustering structure. These results suggests that models of online activity cannot discard the information carried by this multilayer representation of the system, and should account for the different processes generated by the different kinds of interactions. Secondly, our analysis unveils the presence of statistical regularities among the different events, suggesting that the non-trivial topological patterns that we observe may represent universal features of the social dynamics on online social networks during exceptional events.

Human mobility and social structure are at the basis of disease spreading. Disease containment strategies are usually devised from coarse-grained assumptions about human mobility. Cellular networks data, however, provides finer-grained information, not only about how people move, but also about how they communicate.
In this paper, using cellular network data, we analyze the behavior of a large number of individuals in Ivory Coast. We model mobility and communication between individual by means of an interconnected multiplex structure where each node represents the population in a geographic area (i.e. a \textit{sous-pr\'efecture}, a third-level administrative region). We present a model that describes how diseases circulate around the country as people move between regions. We extend the model with a concurrent process of relevant information spreading. This process corresponds to people disseminating disease prevention information, e.g. hygiene practises, vaccination campaign notices and other, within their social network. Thus, this process interferes with the epidemic. We then evaluate how restricting the mobility or using an adverse information spreading process affects the epidemic. We find that restricting mobility does not delay the occurrence of an endemic state and that an information campaign might be an effective countermeasure.

Recent advances in the study of networked systems have highlighted that our interconnected world is composed of networks that are coupled to each other through different layers€™ that each represent one of many possible subsystems or types of interactions. Nevertheless, it is traditional to aggregate multilayer networks into a single weighted network in order to take advantage of existing tools. This is admittedly convenient, but it is also extremely problematic, as important information can be lost as a result. It is therefore important to develop multilayer generalizations of network concepts. In this paper, we analyze triadic relations and generalize the idea of transitivity to multiplex networks. By focusing on triadic relations, which yield the simplest type of transitivity, we generalize the concept and computation of clustering coefficients to multiplex networks. We show how the layered structure of such networks introduces a new degree of freedom that has a fundamental effect on transitivity. We compute multiplex clustering coefficients for several real multiplex networks and illustrate why one must take great care when generalizing standard network concepts to multiplex networks. We also derive analytical expressions for our clustering coefficients for ensemble averages of networks in a family of random multiplex networks. Our analysis illustrates that social networks have a strong tendency to promote redundancy by closing triads at every layer and that they thereby have a different type of multiplex transitivity from transportation networks, which do not exhibit such a tendency. These insights are invisible if one only studies aggregated networks.

Human mobility in a city represents a fascinating complex system that combines social interactions, daily constraints and random explorations. New collections of data that capture human mobility not only help us to understand their underlying patterns but also to design intelligent systems. Bringing us the opportunity to reduce traffic and to develop other applications that make cities more adaptable to human needs. In this paper, we propose an adaptive routing strategy which accounts for individual constraints to recommend personalized routes and, at the same time, for constraints imposed by the collectivity as a whole. Using big data sets recently released during the Telecom Italia Big Data Challenge, we show that our algorithm allows us to reduce the overall traffic in a smart city thanks to synergetic effects, with the participation of individuals in the system, playing a crucial role.

The vertiginous increase of e-platforms for social communication has boosted the ways people use to interact each other. Micro-blogging and decentralized posts are used indistinctly for social interaction, usually by the same individuals acting simultaneously in the different platforms. Multiplex networks are the natural abstraction representation of such "layered" relationships and others, like co-authorship. Here, we re-define the betweenness centrality measure to account for the inherent structure of multiplex networks and propose an algorithm to compute it in an efficient way. To show the necessity and the advantage of the proposed definition, we analyze the obtained centralities for two real multiplex networks, a social multiplex of two layers obtained from Twitter and Instagram and a co-authorship network of four layers obtained from arXiv. Results show that the proposed definition provides more accurate results than the current approach of evaluating the classical betweenness centrality on the aggregated network, in particular for the middle ranked nodes. We also analyze the computational cost of the presented algorithm.

Assessing the navigability of interconnected networks (transporting information, people, or goods) under eventual random failures is of utmost importance to design and protect critical infrastructures. Random walks are a good proxy to determine this navigability, specifically the coverage time of random walks, which is a measure of the dynamical functionality of the network. Here, we introduce the theoretical tools required to describe random walks in interconnected networks accounting for structure and dynamics inherent to real systems. We develop an analytical approach for the covering time of random walks in interconnected networks and compare it with extensive Monte Carlo simulations. Generally speaking, interconnected networks are more resilient to random failures than their individual layers per se, and we are able to quantify this effect. As an application, which we illustrate by considering the public transport of London, we show how the efficiency in exploring the multiplex critically depends on layers’ topology, interconnection strengths, and walk strategy. Our findings are corroborated by data-driven simulations, where the empirical distribution of check-ins and checks-out is considered and passengers travel along fastest paths in a network affected by real disruptions. These findings are fundamental for further development of searching and navigability strategies in real interconnected systems.

Multiplex networks are receiving increasing interests because they allow to model relationships between networked agents on several layers simultaneously. In this letter, we extend well-known random walks to multiplexes and we introduce a new type of walk that can exist only in multiplexes. We derive exact expressions for vertex occupation time and the coverage. Finally, we show how the efficiency in exploring the multiplex critically depends on the underlying topology of layers, the weight of their inter-connections and the strategy adopted to walk.

Multilayer relationships among entities and information about entities must be accompanied by the means to analyse, visualize and obtain insights from such data. We present open-source software (muxViz) that contains a collection of algorithms for the analysis of multilayer networks, which are an important way to represent a large variety of complex systems throughout science and engineering. We demonstrate the ability of muxViz to analyse and interactively visualize multilayer data using empirical genetic, neuronal and transportation networks. Our software is available at https://github.com/manlius/muxViz.

A network representation is useful for describing the structure of a large variety of complex systems. However, most real and engineered systems have multiple subsystems and layers of connectivity, and the data produced by such systems is very rich. Achieving a deep understanding of such systems necessitates generalizing "traditional" network theory, and the newfound deluge of data now makes it possible to test increasingly general frameworks for the study of networks. In particular, although adjacency matrices are useful to describe traditional single-layer networks, such a representation is insufficient for the analysis and description of multiplex and time-dependent networks. One must therefore develop a more general mathematical framework to cope with the challenges posed by multi-layer complex systems. In this paper, we introduce a tensorial framework to study multi-layer networks, and we discuss the generalization of several important network descriptors and dynamical processes ---including degree centrality, clustering coefficients, eigenvector centrality, modularity, Von Neumann entropy, and diffusion--- for this framework. We examine the impact of different choices in constructing these generalizations, and we illustrate how to obtain known results for the special cases of single-layer and multiplex networks. Our tensorial approach will be helpful for tackling pressing problems in multi-layer complex systems, such as inferring who is influencing whom (and by which media) in multichannel social networks and developing routing techniques for multimodal transportation systems.

One of the more challenging tasks in the understanding of dynamical properties of models on top of complex networks is to capture the precise role of multiplex topologies. In a recent paper, Gomez et al. [Phys. Rev. Lett. 101, 028701 (2013)] proposed a framework for the study of diffusion processes in such networks. Here, we extend the previous framework to deal with general configurations in several layers of networks, and analyze the behavior of the spectrum of the Laplacian of the full multiplex. We derive an interesting decoupling of the problem that allow us to unravel the role played by the interconnections of the multiplex in the dynamical processes on top of them. Capitalizing on this decoupling we perform an asymptotic analysis that allow us to derive analytical expressions for the full spectrum of eigenvalues. This spectrum is used to gain insight into physical phenomena on top of multiplex, specifically, diffusion processes and synchronizability.

Human mobility is one of the key factors at the basis of the spreading of diseases in a population. Containment strategies are usually devised on movement scenarios based on coarse-grained assumptions. Mobility phone data provide a unique opportunity for building models and defining strategies based on very precise information about the movement of people in a region or in a country. Another very important aspect is the underlying social structure of a population, which might play a fundamental role in devising information campaigns to promote vaccination and preventive measures, especially in countries with a strong family (or tribal) structure. In this paper we analyze a large-scale dataset describing the mobility and the call patterns of a large number of individuals in Ivory Coast. We present a model that describes how diseases spread across the country by exploiting mobility patterns of people extracted from the available data. Then, we simulate several epidemics scenarios and we evaluate mechanisms to contain the epidemic spreading of diseases, based on the information about people mobility and social ties, also gathered from the phone call data. More specifically, we find that restricting mobility does not delay the occurrence of an endemic state and that an information campaign based on one-to-one phone conversations among members of social groups might be an effective countermeasure.