As the coronavirus disease 2019 (COVID-19) spread globally, emerging variants such as B.1.1.529 quickly became dominant worldwide. Sustained community transmission favors the proliferation of mutated sub-lineages with pandemic potential, due to cross-national mobility flows, which are responsible for consecutive cases surge worldwide. We show that, in the early stages of an emerging variant, integrating data from national genomic surveillance and global human mobility with large-scale epidemic modeling allows to quantify its pandemic potential, providing quantifiable indicators for pro-active policy interventions. We validate our framework on worldwide spreading variants and gain insights about the pandemic potential of BA.5 and BA.2.75 sub-lineages. Country-level epidemic intelligence is not enough to contrast the pandemic of respiratory pathogens such as SARS-CoV-2 and a scalable integrated approach, i.e. pandemic intelligence, is required to enhance global preparedness.

We analyze social media activity during one of the largest protest mobilizations in US history to examine ideological asymmetries in the posting of news content. Using an unprecedented combination of four datasets (tracking offline protests, social media activity, web browsing, and the reliability of news sources), we show that there is no evidence of unreliable sources having any prominent visibility during the protest period, but we do identify asymmetries in the ideological slant of the sources shared on social media, with a clear bias towards right-leaning domains. These results support the “amplification of the right” thesis, which points to the structural conditions (social and technological) that lead to higher visibility of content with a partisan bent towards the right. Our findings provide evidence that right-leaning sources gain more visibility on social media and reveal that ideological asymmetries manifest themselves even in the context of movements with progressive goals.

When a large number of similar entities interact among each other and with their environment at a low scale, unexpected outcomes at higher spatio-temporal scales might spontaneously arise. This nontrivial phenomenon, known as emergence, characterizes a broad range of distinct complex systems -- from physical to biological and social ones -- and is often related to collective behavior. It is ubiquitous, from non-living entities such as oscillators that under specific conditions synchronize, to living ones, such as birds flocking or fish schooling. Despite the ample phenomenological evidence of the existence of systems' emergent properties, central theoretical questions to the study of emergence remain still unanswered, such as the lack of a widely accepted, rigorous definition of the phenomenon or the identification of the essential physical conditions that favour emergence. We offer here a general overview of the phenomenon of emergence and sketch current and future challenges on the topic. Our short review also serves as an introduction to the Theme Issue "Emergent phenomena in complex physical and socio-technical systems: from cells to societies", where we provide a synthesis of the contents tackled in the Issue and outline how they relate to these challenges, spanning from current advances in our understanding on the origin of life to the large-scale propagation of infectious diseases.

Localized perturbations in a real-world network have the potential to trigger cascade failures at the whole system level, hindering its operations and functions. Standard approaches analytically tackling this problem are mostly based either on static descriptions, such as percolation, or on models where the failure evolves through first-neighbor connections, crucially failing to capture the nonlocal behavior typical of real cascades. We introduce a dynamical model that maps the failure propagation across the network to a self-avoiding random walk that, at each step, has a probability to perform nonlocal jumps toward operational systems' units. Despite the inherent non-Markovian nature of the process, we are able to characterize the critical behavior of the system out of equilibrium, as well as the stopping time distribution of the cascades. Our numerical experiments on synthetic and empirical biological and transportation networks are in excellent agreement with theoretical expectation, demonstrating the ability of our framework to quantify the vulnerability to nonlocal cascade failures of complex systems with interconnected structure.

During the COVID-19 epidemic, draconian countermeasures forbidding nonessential human activities have been adopted in several countries worldwide, providing an unprecedented setup for testing and quantifying the current impact of humankind on climate and for driving potential sustainability policies in the postpandemic era from a perspective of complex systems. In this study, we consider heterogeneous sources of environmental and human activity observables, considered as components of a complex socioenvironmental system, and apply information theory, network science, and Bayesian inference to analyze their structural relations and nonlinear dynamics between January 2019 and August 2020 in northern Italy, i.e., before, during, and after the national lockdown. The topological structure of a complex system strongly impacts its collective behavior; therefore, mapping this structure is essential to fully understand the functions of the system as a whole and its fragility to unexpected disruptions or shocks. To this aim, we unravel the causal relationships between the 16 environmental conditions and human activity variables, mapping the backbone of the complex interplay between intervening physical observables—such as NO2 emissions, energy consumption, intervening climate variables, and different flavors of human mobility flows—to a causal network model. To identify a tipping point during the period of observation, denoting the presence of a regime shift between distinct network states (i.e., before and during the shock), we introduce a novel information-theoretic method based on statistical divergence widely used in statistical physics. We find that despite a measurable decrease in NO2 concentration, due to an overall decrease in human activities, locking down a region as a climate change mitigation is an insufficient remedy to reduce emissions. Our results provide a functional characterization of socioenvironmental interdependent systems, and our analytical framework can be used, more generally, to characterize environmental changes and their interdependencies using statistical physics.

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 to 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 not major world powers (with the exception of Germany and France) but 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. However, single global treaties have surprisingly less disruptive potential, with the major exception of the WTO. These apparently counter-intuitive results highlight the importance to a nonlinear approach to international relations where the complex multilayered architecture of global governance is analyzed using advanced network science techniques. Our results suggest that the potential fragility of the world order seem to be much more directly related to global inequality and fiscal injustice than it is commonly believed, and that the legacy of the colonial world order is still very 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.

After the leak of 11.5 million documents from the Panamanian corporation Mossack Fonseca, an intricate network of offshore business entities has been revealed. The emerging picture is that of legal entities, either individuals or companies, involved in offshore activities and transactions with several tax havens simultaneously which establish, indirectly, an effective network of countries acting on tax evasion. The analysis of this network quantitatively uncovers a strongly connected core (a rich-club) of countries whose indirect interactions, mediated by legal entities, form the skeleton for tax evasion worldwide. Intriguingly, the rich-club mainly consists of well-known tax havens such as British Virgin Islands and Hong Kong, and major global powers such as China, Russia, United Kingdom and United States of America. The analysis provides a new way to rank tax havens because of the role they play in this network, and the results call for an international coordination on taxation policies that take into account the complex interconnected structure of tax evaders in a globalized economy.

Microscopic structural damage, such as lesions in neural systems or disruptions in urban transportation networks, can impair the dynamics crucial for systems' functionality, such as electrochemical signals or human flows, or any other type of information exchange, respectively, at larger topological scales. Damage is usually modeled by progressive removal of components or connections and, consequently, systems' robustness is assessed in terms of how fast their structure fragments into disconnected sub-systems. Yet, this approach fails to capture how damage hinders the propagation of information across scales, since system function can be degraded even in absence of fragmentation -- e.g., pathological yet structurally integrated human brain. Here, we probe the response to damage of dynamical processes on the top of complex networks, to study how such an information flow is affected. We find that removal of nodes central for network connectivity might have insignificant effects, challenging the traditional assumption that structural metrics alone are sufficient to gain insights about how complex systems operate. Using a damaging protocol explicitly accounting for flow dynamics, we analyze synthetic and empirical systems, from biological to infrastructural ones, and show that it is possible to drive the system towards functional fragmentation before full structural disintegration.

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.

The dynamics of a spreading disease and individual behavioral changes are entangled processes that have to be addressed together in order to effectively manage an outbreak. Here, we relate individual risk perception to the adoption of a specific set of control measures, as obtained from an extensive large-scale survey performed via Facebook—involving more than 500,000 respondents from 64 countries—showing that there is a “one-to-one” relationship between perceived epidemic risk and compliance with a set of mitigation rules. We then develop a mathematical model for the spreading of a disease—sharing epidemiological features with COVID-19—that explicitly takes into account non-compliant individual behaviors and evaluates the impact of a population fraction of infectious risk-deniers on the epidemic dynamics. Our modeling study grounds on a wide set of structures, including both synthetic and more than 180 real-world contact patterns, to evaluate, in realistic scenarios, how network features typical of human interaction patterns impact the spread of a disease. In both synthetic and real contact patterns we find that epidemic spreading is hindered for decreasing population fractions of risk-denier individuals. From empirical contact patterns we demonstrate that connectivity heterogeneity and group structure significantly affect the peak of hospitalized population: higher modularity and heterogeneity of social contacts are linked to lower peaks at a fixed fraction of risk-denier individuals while, at the same time, such features increase the relative impact on hospitalizations with respect to the case where everyone correctly perceive the risks.

In the last two decades, network science has proven to be an invaluable tool for the analysis of empirical systems across a wide spectrum of disciplines, with applications to data structures admitting a representation in terms of complex networks. On the one hand, especially in the last decade, an increasing number of applications based on geometric deep learning have been developed to exploit, at the same time, the rich information content of a complex network and the learning power of deep architectures, highlighting the potential of techniques at the edge between applied math and computer science. On the other hand, studies at the edge of network science and quantum physics are gaining increasing attention, e.g., because of the potential applications to quantum networks for communications, such as the quantum Internet. In this work, we briefly review a novel framework grounded on statistical physics and techniques inspired by quantum statistical mechanics which have been successfully used for the analysis of a variety of complex systems. The advantage of this framework is that it allows one to define a set of information-theoretic tools which find widely used counterparts in machine learning and quantum information science, while providing a grounded physical interpretation in terms of a statistical field theory of information dynamics. We discuss the most salient theoretical features of this framework and selected applications to protein–protein interaction networks, neuronal systems, social and transportation networks, as well as potential novel applications for quantum network science and machine learning.

An infodemic—an outpouring of information, including misleading and also fake news—is accompanying the current pandemic caused by SARS-CoV-2. In the absence of valid therapeutic approaches, behavioral responses may seriously affect the social dynamics of contagion, so the infodemic may cause confusion and disorientation in the public, leading to possible individually and socially harmful choices. This new phenomenon requires specific modeling efforts to better understand the complex intertwining of the epidemic and infodemic components of a pandemic crisis, with a view to building an integrative public health approach. We propose three models, from epidemiology to game theory, as potential candidates for the onset of the infodemics and statistically assess their accuracy in reproducing real infodemic waves observed in a data set of 390 million tweets collected worldwide. Our results show that evolutionary game-theory models are the most suitable ones to reproduce the observed infodemic modulations around the onset of the local epidemic wave. Furthermore, we find that the number of confirmed COVID-19 reported cases in each country and worldwide are driving the modeling dynamics with opposite effects.

The infection caused by SARS-CoV-2, responsible for the COVID-19 pandemic, is characterized by an infectious period with either asymptomatic or pre-symptomatic phases, leading to a rapid surge of mild and severe cases putting national health systems under serious stress. To avoid their collapse, and in the absence of pharmacological treatments, during the early pandemic phase countries worldwide were forced to adopt strategies, from elimination to mitigation, based on non-pharmacological interventions which, in turn, overloaded social, educational and economic systems. To date, the heterogeneity and incompleteness of data sources does not allow to quantify the multifaceted impact of the pandemic at country level and, consequently, to compare the effectiveness of country responses. Here, we tackle this challenge from a complex systems perspective, proposing a model to evaluate the impact of systemic failures in response to the pandemic shock. We use health, behavioral and economic indicators for 44 countries to build a shock index quantifying responses in terms of robustness and resilience, highlighting the crucial advantage of proactive policy and decision making styles over reactive ones.

Online platforms play a relevant role in creation and diffusion of false or misleading news. Concerningly, the COVID-19 pandemic is shaping a communication network - barely considered in the literature - which reflects the emergence of collective attention towards a topic that rapidly gained universal interest. Here, we characterize the dynamics of this network on Twitter, analyzing how unreliable content distributes among its users. We find that a minority of accounts is responsible for the majority of the misinformation circulating online, and identify two categories of users: a few active ones, playing the role of "creators", and a majority playing the role of "consumers". The relative proportion of these groups (≈14% creators - 86% consumers) appears stable over time: Consumers are mostly exposed to the opinions of a vocal minority of creators, that could be mistakenly understood as of representative of the majority of users. The corresponding pressure from a perceived majority is identified as a potential driver of the ongoing COVID-19 infodemic.

A look at the Nobel laurate’s ground-breaking early work, and the academic environment that made it possible.

Complex biological systems consist of large numbers of interconnected units, characterized by emergent properties such as collective computation. In spite of all the progress in the last decade, we still lack a deep understanding of how these properties arise from the coupling between the structure and dynamics. Here, we introduce the multiscale emergent functional state, which can be represented as a network where links encode the flow exchange between the nodes, calculated using diffusion processes on top of the network. We analyze the emergent functional state to study the distribution of the flow among components of 92 fungal networks, identifying their functional modules at different scales and, more importantly, demonstrating the importance of functional modules for information content of networks, quantified in terms of network spectral entropy. Our results suggest that the topological complexity of fungal networks guarantees the existence of functional modules at different scales keeping the information entropy, and functional diversity, high.

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.

From physics to engineering, biology and social science, natural and artificial systems are characterized by interconnected topologies whose features – e.g., heterogeneous connectivity, mesoscale organization, hierarchy – affect their robustness to external perturbations, such as targeted attacks to their units. Identifying the minimal set of units to attack to disintegrate a complex network, i.e. network dismantling, is a computationally challenging (NP-hard) problem which is usually attacked with heuristics. Here, we show that a machine trained to dismantle relatively small systems is able to identify higher-order topological patterns, allowing to disintegrate large-scale social, infrastructural and technological networks more efficiently than human-based heuristics. Remarkably, the machine assesses the probability that next attacks will disintegrate the system, providing a quantitative method to quantify systemic risk and detect early-warning signals of system’s collapse. This demonstrates that machine-assisted analysis can be effectively used for policy and decision-making to better quantify the fragility of complex systems and their response to shocks.

The public health research agenda for infodemic management has five workstreams: (1) measuring and continuously monitoring the impact of infodemics during health emergencies; (2) detecting signals and understanding the spread and risk of infodemics; (3) responding and deploying interventions that mitigate and protect against infodemics and their harmful effects; (4) evaluating infodemic interventions and strengthening the resilience of individuals and communities to infodemics; and (5) promoting the development, adaptation, and application of interventions and toolkits for infodemic management. Each workstream identifies research questions and highlights 49 high priority research questions.

As COVID-19 spreads worldwide, an infodemic – i.e., an over-abundance of information, reliable or not – spreads across the physical and the digital worlds, triggering behavioral responses which cause public health concern. We study 200 million interactions captured from Twitter during the early stage of the pandemic, from January to April 2020, to understand its socio-informational structure on a global scale. The COVID-19 global communication network is characterized by knowledge groups, hierarchically organized in sub-groups with well-defined geo-political and ideological characteristics. Communication is mostly segregated within groups and driven by a small number of subjects: 0.1% of users account for up to 45% and 10% of activities and news shared, respectively, centralizing the information flow. Contradicting the idea that digital social media favor active participation and co-creation of online content, our results imply that public health policy strategies to counter the effects of the infodemic must not only focus on information content, but also on the social articulation of its diffusion mechanisms, as a given community tends to be relatively impermeable to news generated by non-aligned sources.

Interconnected systems have to route information to function properly: At the lowest scale neural cells exchange electrochemical signals to communicate, while at larger scales animals and humans move between distinct spatial patches and machines exchange information via the Internet through communication protocols. Nontrivial patterns emerge from the analysis of information flows, which are not captured either by broadcasting, such as in random walks, or by geodesic routing, such as shortest paths. In fact, alternative models between those extreme protocols are still eluding us. Here we propose a class of stochastic processes, based on biased random walks, where agents are driven by a physical potential pervading the underlying network topology. By considering a generalized Coulomb dependence on the distance on destination(s), we show that it is possible to interpolate between random walk and geodesic routing in a simple and effective way. We demonstrate that it is not possible to find a one-size-fit-all solution to efficient navigation and that network heterogeneity or modularity has measurable effects. We illustrate how our framework can describe the movements of animals and humans, capturing with a stylized model some measurable features of the latter. From a methodological perspective, our potential-driven random walks open the doors to a broad spectrum of analytical tools, ranging from random-walk centralities to geometry induced by potential-driven network processes.

With hindsight, the main weakness behind the ineffective response to the coronavirus disease 2019 (COVID-19) pandemic in some countries has been the failure to understand, and take account of, the multilayered systemic interdependencies that spread the effects of the pandemic across social, technological, economic and health-care dimensions. For example, to respond to the COVID-19 pandemic, all people were required to rapidly adjust to social distancing and travel restrictions. Such a complex behavioural response entails adaptation to achieve a full recovery from the systemic shock. To capitalize on the positive effects of disruption to the status quo, much more complex socioeconomic modelling needs to be considered when designing and evaluating possible public health interventions that have major behavioural implications. We provide a simple example of how this reasoning may highlight generally unacknowledged connections and interdependencies and guide the construction of scenarios that can inform policy decisions to enhance the resilience of society and tackle existing societal challenges.

This special theme issue of Big Data & Society presents leading-edge, interdisciplinary research that focuses on examining how health-related (mis-)information is circulating on social media. In particular, we are focusing on how computational and Big Data approaches can help to provide a better understanding of the ongoing COVID-19 infodemic (overexposure to both accurate and misleading information on a health topic) and to develop effective strategies to combat it.

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

Information exchange in the human brain is crucial for vital tasks and to drive diseases. Neuroimaging techniques allow for the indirect measurement of information flows among brain areas and, consequently, for reconstructing connectomes analyzed through the lens of network science. However, standard analyses usually focus on a small set of network indicators and their joint probability distribution. Here, we propose an information-theoretic approach for the analysis of synthetic brain networks (based on generative models) and empirical brain networks, and to assess connectome's information capacity at different stages of dementia. Remarkably, our framework accounts for the whole network state, overcoming limitations due to limited sets of descriptors, and is used to probe human connectomes at different scales. We find that the spectral entropy of empirical data lies between two generative models, indicating an interpolation between modular and geometry-driven structural features. In fact, we show that the mesoscale is suitable for characterizing the differences between brain networks and their generative models. Finally, from the analysis of connectomes obtained from healthy and unhealthy subjects, we demonstrate that significant differences between healthy individuals and the ones affected by Alzheimer’s disease arise at the microscale (max. posterior probability smaller than 1%) and at the mesoscale (max. posterior probability smaller than 10%).

Hypergraphs offer an explicit formalism to describe multibody interactions in complex systems. To connect dynamics and function in systems with these higher-order interactions, network scientists have generalised random-walk models to hypergraphs and studied the multibody effects on flow-based centrality measures. Mapping the large-scale structure of those flows requires effective community detection methods applied to cogent network representations. For different hypergraph data and research questions, which combination of random-walk model and network representation is best? We define unipartite, bipartite, and multilayer network representations of hypergraph flows and explore how they and the underlying random-walk model change the number, size, depth, and overlap of identified multilevel communities. These results help researchers choose the appropriate modelling approach when mapping flows on hypergraphs.

The spread of an infectious disease is well approximated by metapopulation networks connected by human mobility flow and upon which an epidemiological model is defined. In order to account for travel restrictions or cancellation we introduce a model with a parameter that explicitly indicates the ratio between the time scales of the intervening processes. We study the critical properties of the epidemic process and its dependence on such a parameter. We find that the critical threshold separating the absorbing state from the active state depends on the scale parameter and exhibits a critical behavior itself: a metacritical point – a critical value in the curve of critical points – reflected in the behavior of the attack rate measured for a wide range of empirical metapopulation systems. Our results have potential policy implications, since they establish a non-trivial critical behavior between temporal scales of reaction (epidemic spread) and diffusion (human mobility) processes.

Complex systems are large collections of entities that organize themselves into non-trivial structures, represented as networks. One of their key emergent properties is robustness against random failures or targeted attacks —i.e., the networks maintain their integrity under removal of nodes or links. Here, we introduce network entanglement to study network robustness through a multiscale lens, encoded by the time required for information diffusion through the system. Our measure’s foundation lies upon a recently developed statistical field theory for information dynamics within interconnected systems. We show that at the smallest temporal scales, the node-network entanglement reduces to degree, whereas at extremely large scales, it measures the direct role played by each node in keeping the network connected. At the meso-scale, entanglement plays a more important role, measuring the importance of nodes for the transport properties of the system. We use entanglement as a centrality measure capturing the role played by nodes in keeping the overall diversity of the information flow. As an application, we study the disintegration of empirical social, biological and transportation systems, showing that the nodes central for information dynamics are also responsible for keeping the network integrated.

Network science enables the effective analysis of real interconnected systems, characterized by a complex interplay between topology and network flows. It is well-known that the topology of a network affects its resilience to failures or attacks, as well as its functions. Many real systems—such as the Internet, transportation networks and the brain—exchange information, so it is crucial to quantify how efficiently system’s units communicate. Measures of parallel communication efficiency for weighted networks rely on the identification of an ideal version of the system, which currently lacks a universal definition. Consequently, an inattentive choice might hinder a rigorous comparison of network flows across scales or might lead to a descriptor not robust to fluctuations in the topology or the flows. We propose a physically-grounded estimator of flow efficiency valid for any weighted network, regardless of scale, nature of weights and (missing) metadata, allowing for comparison across disparate systems. Our estimator captures the effect of flows heterogeneity along with topological differences of both synthetic and empirical systems. We also show that cutting the heaviest connections may increase the average efficiency of the system and hence, counterintuively, a sparser network is not necessarily less efficient.

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.

Percolation is an emblematic model to assess the robustness of interconnected systems when some of their components are corrupted. It is usually investigated in simple scenarios, such as the removal of the system’s units in random order, or sequentially ordered by specific topological descriptors. However, in the vast majority of empirical applications, it is required to dismantle the network following more sophisticated protocols, for instance, by combining topological properties and non-topological node metadata. We propose a novel mathematical framework to fill this gap: networks are enriched with features and their nodes are removed according to the importance in the feature space. We consider features of different nature, from ones related to the network construction to ones related to dynamical processes such as epidemic spreading. Our framework not only provides a natural generalization of percolation but, more importantly, offers an accurate way to test the robustness of networks in realistic scenarios.

Protein–protein interaction networks have been used to investigate the influence of SARS-CoV-2 viral proteins on the function of human cells, laying out a deeper understanding of COVID–19 and providing ground for applications, such as drug repurposing. Characterizing molecular (dis)similarities between SARS-CoV-2 and other viral agents allows one to exploit existing information about the alteration of key biological processes due to known viruses for predicting the potential effects of this new virus. Here, we compare the novel coronavirus network against 92 known viruses, from the perspective of statistical physics and computational biology. We show that regulatory spreading patterns, physical features and enriched biological pathways in targeted proteins lead, overall, to meaningful clusters of viruses which, across scales, provide complementary perspectives to better characterize SARS-CoV-2 and its effects on humans. Our results indicate that the virus responsible for COVID–19 exhibits expected similarities, such as to Influenza A and Human Respiratory Syncytial viruses, and unexpected ones with different infection types and from distant viral families, like HIV1 and Human Herpes virus. Taken together, our findings indicate that COVID–19 is a systemic disease with potential effects on the function of multiple organs and human body sub-systems.

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.

Networks are finite metric spaces, with distances defined by the shortest paths between nodes. However, this is not the only form of network geometry: two others are the geometry of latent spaces underlying many networks and the effective geometry induced by dynamical processes in networks. These three approaches to network geometry are intimately related, and all three of them have been found to be exceptionally efficient in discovering fractality, scale invariance, self-similarity and other forms of fundamental symmetries in networks. Network geometry is also of great use in a variety of practical applications, from understanding how the brain works to routing in the Internet. We review the most important theoretical and practical developments dealing with these approaches to network geometry and offer perspectives on future research directions and challenges in this frontier in the study of complexity.

Information manipulation is widespread in today’s media environment. Online networks have disrupted the gatekeeping role of traditional media by allowing various actors to influence the public agenda; they have also allowed automated accounts (or bots) to blend with human activity in the flow of information. Here, we assess the impact that bots had on the dissemination of content during two contentious political events that evolved in real time on social media. We focus on events of heightened political tension because they are particularly susceptible to information campaigns designed to mislead or exacerbate conflict. We compare the visibility of bots with human accounts, verified accounts, and mainstream news outlets. Our analyses combine millions of posts from a popular microblogging platform with web-tracking data collected from two different countries and timeframes. We employ tools from network science, natural language processing, and machine learning to analyze the diffusion structure, the content of the messages diffused, and the actors behind those messages as the political events unfolded. We show that verified accounts are significantly more visible than unverified bots in the coverage of the events but also that bots attract more attention than human accounts. Our findings highlight that social media and the web are very different news ecosystems in terms of prevalent news sources and that both humans and bots contribute to generate discrepancy in news visibility with their activity.

A volunteer effort by Artificial Intelligence (AI) researchers has shown it can deliver significant research outcomes rapidly to help tackle COVID-19. Within two months, CLAIRE’s self-organising volunteers delivered the World’s first comprehensive curated repository of COVID-19-related datasets useful for drug-repurposing, drafted review papers on the role CT/X-ray scan analysis and robotics could play, and progressed research in other areas. Given the pace required and nature of voluntary efforts, the teams faced a number of challenges. These offer insights in how better to prepare for future volunteer scientific efforts and large scale, data-dependent AI collaborations in general. We offer seven recommendations on how to best leverage such efforts and collaborations in the context of managing future crises.

Revealing the structural features of a complex system from the observed collective dynamics is a fundamental problem in network science. To compute the various topological descriptors commonly used to characterize the structure of a complex system (e.g., the degree, the clustering coefficient, etc.), it is usually necessary to completely reconstruct the network of relations between the subsystems. Several methods are available to detect the existence of interactions between the nodes of a network. By observing some physical quantities through time, the structural relationships are inferred using various discriminating statistics (e.g., correlations, mutual information, etc.). In this setting, the uncertainty about the existence of the edges is reflected in the uncertainty about the topological descriptors. In this study, we propose a methodological framework to evaluate this uncertainty, replacing the topological descriptors, even at the level of a single node, with appropriate probability distributions, eluding the reconstruction phase. Our theoretical framework agrees with the numerical experiments performed on a large set of synthetic and real-world networks. Our results provide a grounded framework for the analysis and the interpretation of widely used topological descriptors, such as degree centrality, clustering, and clusters, in scenarios in which the existence of network connectivity is statistically inferred or when the probabilities of existence pij of the edges are known. To this purpose, we also provide a simple and mathematically grounded process to transform the discriminating statistics into the probabilities pij.

Increasing evidence suggests that cities are complex systems, with structural and dynamical features responsible for a broad spectrum of emerging phenomena. Here we use a unique data set of human flows and couple it with information on the underlying street network to study, simultaneously, the structural and functional organisation of 10 world megacities. We quantify the efficiency of flow exchange between areas of a city in terms of integration and segregation using well defined measures. Results reveal unexpected complex patterns that shed new light on urban organisation. Large cities tend to be more segregated and less integrated, while their overall topological organisation resembles that of small world networks. At the same time, the heterogeneity of flows distribution might act as a catalyst for further integrating a city. Our analysis unravels how human behaviour influences, and is influenced by, the urban environment, suggesting quantitative indicators to control integration and segregation of human flows that can be used, among others, for restriction policies to adopt during emergencies and, as an interesting byproduct, allows us to characterise functional (dis)similarities of different metropolitan areas, countries, and cultures.

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.

Political and environmental factors—e.g., regional conflicts and global warming—increase large-scale migrations, posing extraordinary societal challenges to policymakers of destination countries. A common concern is that such a massive arrival of people—often from a country with a disrupted healthcare system—can increase the risk of vaccine-preventable disease outbreaks like measles. We analyze human flows of 3.5 million (M) Syrian refugees in Turkey inferred from massive mobile-phone data to verify this concern. We use multilayer modeling of interdependent social and epidemic dynamics to demonstrate that the risk of disease reemergence in Turkey, the main host country, can be dramatically reduced by 75 to 90% when the mixing of Turkish and Syrian populations is high. Our results suggest that maximizing the dispersal of refugees in the recipient population contributes to impede the spread of sustained measles epidemics, rather than favoring it. Targeted vaccination campaigns and policies enhancing social integration of refugees are the most effective strategies to reduce epidemic risks for all citizens.

Introduction: We introduce in this study CovMulNet19, a comprehensive COVID-19 network containing all available known interactions involving SARS-CoV-2 proteins, interacting-human proteins, diseases and symptoms that are related to these human proteins, and compounds that can potentially target them. Materials and Methods: Extensive network analysis methods, based on a bootstrap approach, allow us to prioritize a list of diseases that display a high similarity to COVID-19 and a list of drugs that could potentially be beneficial to treat patients. As a key feature of CovMulNet19, the inclusion of symptoms allows a deeper characterization of the disease pathology, representing a useful proxy for COVID-19-related molecular processes. Results: We recapitulate many of the known symptoms of the disease and we find the most similar diseases to COVID-19 reflect conditions that are risk factors in patients. In particular, the comparison between CovMulNet19 and randomized networks recovers many of the known associated comorbidities that are important risk factors for COVID-19 patients, through identified similarities with intestinal, hepatic, and neurological diseases as well as with respiratory conditions, in line with reported comorbidities. Conclusion: CovMulNet19 can be suitably used for network medicine analysis, as a valuable tool for exploring drug repurposing while accounting for the intervening multidimensional factors, from molecular interactions to symptoms.

During COVID-19, governments and the public are fighting not only a pandemic but also a co-evolving infodemic—the rapid and far-reaching spread of information of questionable quality. We analysed more than 100 million Twitter messages posted worldwide during the early stages of epidemic spread across countries (from 22 January to 10 March 2020) and classified the reliability of the news being circulated. We developed an Infodemic Risk Index to capture the magnitude of exposure to unreliable news across countries. We found that measurable waves of potentially unreliable information preceded the rise of COVID-19 infections, exposing entire countries to falsehoods that pose a serious threat to public health. As infections started to rise, reliable information quickly became more dominant, and Twitter content shifted towards more credible informational sources. Infodemic early-warning signals provide important cues for misinformation mitigation by means of adequate communication strategies.

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.

The constituents of a complex system exchange information to function properly. Their signaling dynamics often leads to the appearance of emergent phenomena, such as phase transitions and collective behaviors. While information exchange has been widely modeled by means of distinct spreading processes—such as continuous-time diffusion, random walks, synchronization and consensus—on top of complex networks, a unified and physically grounded framework to study information dynamics and gain insights about the macroscopic effects of microscopic interactions is still eluding us. In this paper, we present this framework in terms of a statistical field theory of information dynamics, unifying a range of dynamical processes governing the evolution of information on top of static or time-varying structures. We show that information operators form a meaningful statistical ensemble and their superposition defines a density matrix that can be used for the analysis of complex dynamics. As a direct application, we show that the von Neumann entropy of the ensemble can be a measure of the functional diversity of complex systems, defined in terms of the functional differentiation of higher-order interactions among their components. Our results suggest that modularity and hierarchy, two key features of empirical complex systems—from the human brain to social and urban networks—play a key role to guarantee functional diversity and, consequently, are favored.

Online social networks are the perfect test bed to better understand large-scale human behavior in interacting contexts. Although they are broadly used and studied, little is known about how their terms of service and posting rules affect the way users interact and information spreads. Acknowledging the relation between network connectivity and functionality, we compare the robustness of two different online social platforms, Twitter and Gab, with respect to banning, or dismantling, strategies based on the recursive censor of users characterized by social prominence (degree) or intensity of inflammatory content (sentiment). We find that the moderated (Twitter) vs. unmoderated (Gab) character of the network is not a discriminating factor for intervention effectiveness. We find, however, that more complex strategies based upon the combination of topological and content features may be effective for network dismantling. Our results provide useful indications to design better strategies for countervailing the production and dissemination of anti-social content in online social platforms.

Peer review is a cornerstone of modern scientific endeavor. However, there is growing consensus that several limitations of the current peer review system, from lack of incentives to reviewers to lack of transparency, risks to undermine its benefits. Here, we introduce the PRINCIPIA (http://www.principia.network/) framework for peer-review of scientific outputs (e.g., papers, grant proposals or patents). The framework allows key players of the scientific ecosystem -- including existing publishing groups -- to create and manage peer-reviewed journals, by building a free market for reviews and publications. PRINCIPIA's referees are transparently rewarded according to their efforts and the quality of their reviews. PRINCIPIA also naturally allows to recognize the prestige of users and journals, with an intrinsic reputation system that does not depend on third-parties. PRINCIPIA re-balances the power between researchers and publishers, stimulates valuable assessments from referees, favors a fair competition between journals, and reduces the costs to access research output and to publish.

In this RISJ Factsheet, we assess the volume and patterns of toxic conversations on social media during the Covid-19 pandemic. We specifically analyse worldwide conversations on Twitter targeting the World Health Organization (WHO), a central actor during the pandemic. Our analysis contributes to the current research on the health of online debates amid the increasing role of social media as a critical entrance to information and mediator of public opinion building.

Equations governing the nonlinear dynamics of complex systems are usually unknown, and indirect methods are used to reconstruct their manifolds. In turn, they depend on embedding parameters requiring other methods and long temporal sequences to be accurate. In this paper, we show that an optimal reconstruction can be achieved by lossless compression of system’s time course, providing a self-consistent analysis of its dynamics and a measure of its complexity, even for short sequences. Our measure of complexity detects system’s state changes such as weak synchronization phenomena, characterizing many systems, in one step, integrating results from Lyapunov and fractal analysis

The analysis team distilled the suggestions into a set of 50 proposed actions for a framework for managing infodemics in health emergencies. The consultation revealed six policy implications to consider. First, interventions and messages must be based on science and evidence, and must reach citizens and enable them to make informed decisions on how to protect themselves and their communities in a health emergency. Second, knowledge should be translated into actionable behavior-change messages, presented in ways that are understood by and accessible to all individuals in all parts of all societies. Third, governments should reach out to key communities to ensure their concerns and information needs are understood, tailoring advice and messages to address the audiences they represent. Fourth, to strengthen the analysis and amplification of information impact, strategic partnerships should be formed across all sectors, including but not limited to the social media and technology sectors, academia, and civil society. Fifth, health authorities should ensure that these actions are informed by reliable information that helps them understand the circulating narratives and changes in the flow of information, questions, and misinformation in communities. Sixth, following experiences to date in responding to the COVID-19 infodemic and the lessons from other disease outbreaks, infodemic management approaches should be further developed to support preparedness and response, and to inform risk mitigation, and be enhanced through data science and sociobehavioral and other research.

In the era of social media, every day billions of individuals produce content in socio-technical systems resulting in a deluge of information. However, human attention is a limited resource and it is increasingly challenging to consume the most suitable content for one's interests. In fact, the complex interplay between individual and social activities in social systems overwhelmed by information results in bursty activity of collective attention which are still poorly understood. Here, we tackle this challenge by analyzing the online activity of millions of users in a popular microblogging platform during exceptional events, from NBA Finals to the elections of Pope Francis and the discovery of gravitational waves. We observe extreme fluctuations in collective attention that we are able to characterize and explain by considering the co-occurrence of two fundamental factors: the heterogeneity of social interactions and the preferential attention towards influential users. Our findings demonstrate how combining simple mechanisms provides a route towards complex social phenomena.

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.

In collective decision-making, a group of independent experts propose individual choices to reach a common decision. This is the case of competitive events such as Olympics, international Prizes or grant evaluation, where groups of experts evaluate individual performances to assign resources, e.g. scores, recognitions, or funding. However, there are systems where evaluating individual’s performance is difficult: in those cases, other factors play a relevant role, leading to unexpected emergent phenomena from micro-scale interactions. The Nobel assignment procedure, rooted on recommendations, is one of these systems. Here we unveil its network, reconstructed from official data and metadata about nominators, nominees and awardees between 1901 and 1965, consisting of almost 12,000 individuals and 17,000 nominations. We quantify the role of homophily, academic reputation of nominators and their prestige neighborhood, showing that nominees endorsed by central actors – who are part of the system’s core because of their prestigious reputation – are more likely to become laureate within a finite time scale than nominees endorsed by nominators in the periphery of the network. We propose a mechanistic model which reproduces all the salient observations and allows to design possible countermeasures to mitigate observed effects..

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.

Centrality descriptors are widely used to rank nodes according to specific concept(s) of importance. Despite the large number of centrality measures available nowadays, it is still poorly understood how to identify the node which can be considered as the `centre' of a complex network. In fact, this problem corresponds to finding the median of a complex network. The median is a non-parametric and robust estimator of the location parameter of a probability distribution. In this work, we present the most natural generalisation of the concept of median to the realm of complex networks, discussing its advantages for defining the centre of the system and percentiles around that centre. To this aim, we introduce a new statistical data depth and we apply it to networks embedded in a geometric space induced by different metrics. The application of our framework to empirical networks allows us to identify median nodes which are socially or biologically relevant.

Recent progress in applying complex network theory to problems faced in quantum information and computation has resulted in a beneficial crossover between two fields. Complex network methods have successfully been used to characterize quantum walk and transport models, entangled communication networks, graph-theoretic models of emergent space-time and in detecting mesoscale structure in quantum systems. Information physics is setting the stage for a theory of complex and networked systems with quantum information-inspired methods appearing in complex network science, including information-theoretic distance and correlation measures for network characterization. Novel quantum induced effects have been predicted in random graphs---where edges represent entangled links---and quantum computer algorithms have recently been proposed to offer super-polynomial enhancement for several network and graph theoretic problems. Here we review the results at the cutting edge, pinpointing the similarities and reconciling the differences found in the series of results at the intersection of these two fields.

With its fifth edition, the Mediterranean School of Complex Networks is one of the oldest international summer schools, run without interruptions, which is fully devoted to teach the fundamentals of Network Science and its current trends to early-career researchers.

The blockchain technology promises to transform finance, money and even governments. However, analyses of blockchain applicability and robustness typically focus on isolated systems whose actors contribute mainly by running the consensus algorithm. Here, we highlight the importance of considering trustless platforms within the broader ecosystem that includes social and communication networks. As an example, we analyse the flash-crash observed on 21st June 2017 in the Ethereum platform and show that a major phenomenon of social coordination led to a catastrophic cascade of events across several interconnected systems. We propose the concept of “emergent centralisation” to describe situations where a single system becomes critically important for the functioning of the whole ecosystem, and argue that such situations are likely to become more and more frequent in interconnected socio-technical systems. We anticipate that the systemic approach we propose will have implications for future assessments of trustless systems and call for the attention of policy-makers on the fragility of our interconnected and rapidly changing world.

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.

Collective phenomena emerge from the interaction of natural or artificial units with a complex organization. The interplay between structural patterns and dynamics might induce functional clusters that, in general, are different from topological ones. In biological systems, like the human brain, the overall functionality is often favored by the interplay between connectivity and synchronization dynamics, with functional clusters that do not coincide with anatomical modules in most cases. In social, sociotechnical, and engineering systems, the quest for consensus favors the emergence of clusters. Despite the unquestionable evidence for mesoscale organization of many complex systems and the heterogeneity of their interconnectivity, a way to predict and identify the emergence of functional modules in collective phenomena continues to elude us. Here, we propose an approach based on random walk dynamics to define the diffusion distance between any pair of units in a networked system. Such a metric allows us to exploit the underlying diffusion geometry to provide a unifying framework for the intimate relationship between metastable synchronization, consensus, and random search dynamics in complex networks, pinpointing the functional mesoscale organization of synthetic and biological systems.

Publishing reproducible analyses is a long-standing and widespread challenge for the scientific community, funding bodies and publishers. Although a definitive solution is still elusive, the problem is recognized to affect all disciplines and lead to a critical system inefficiency. Here, we propose a blockchain-based approach to enhance scientific reproducibility, with a focus on life science studies and precision medicine. While the interest of encoding permanently into an immutable ledger all the study key information-including endpoints, data and metadata, protocols, analytical methods and all findings-has been already highlighted, here we apply the blockchain approach to solve the issue of rewarding time and expertise of scientists that commit to verify reproducibility. Our mechanism builds a trustless ecosystem of researchers, funding bodies and publishers cooperating to guarantee digital and permanent access to information and reproducible results. As a natural byproduct, a procedure to quantify scientists and institutions reputation for ranking purposes is obtained.

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.

While the statistical and resilience properties of the Internet are no more changing significantly across time, the Darknet, a network devoted to keep anonymous its traffic, still experiences rapid changes to improve the security of its users. Here, we study the structure of the Darknet and we find that its topology is rather peculiar, being characterized by non-homogenous distribution of connections -- typical of scale-free networks --, very short path lengths and high clustering -- typical of small-world networks -- and lack of a core of highly connected nodes. We propose a model to reproduce such features, demonstrating that the mechanisms used to improve cyber-security are responsible for the observed topology. Unexpectedly, we reveal that its peculiar structure makes the Darknet much more resilient than the Internet -- used as a benchmark for comparison at a descriptive level -- to random failures, targeted attacks and cascade failures, as a result of adaptive changes in response to the attempts of dismantling the network across time.

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.

Understanding how people move within a geographic area, e.g. a city, a country or the whole world, is fundamental in several applications, from predicting the spatio-temporal evolution of an epidemics to inferring migration patterns. Mobile phone records provide an excellent proxy of human mobility, showing that movements exhibit a high level of memory. However, the precise role of memory in widely adopted proxies of mobility, as mobile phone records, is unknown. Here we use 560 millions of call detail records from Senegal to show that standard Markovian approaches, including higher-order ones, fail in capturing real mobility patterns and introduce spurious movements never observed in reality. We introduce an adaptive memory-driven approach to overcome such issues. At variance with Markovian models, it is able to realistically model conditional waiting times, i.e. the probability to stay in a specific area depending on individual's historical movements. Our results demonstrate that in standard mobility models the individuals tend to diffuse faster than what observed in reality, whereas the predictions of the adaptive memory approach significantly agree with observations. We show that, as a consequence, the incidence and the geographic spread of a disease could be inadequately estimated when standard approaches are used, with crucial implications on resources deployment and policy making during an epidemic outbreak.

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.

The analysis of the activity of neuronal cultures is considered to be a good proxy of the functional connectivity of in vivo neuronal tissues. Thus, the functional complex network inferred from activity patterns is a promising way to unravel the interplay between structure and functionality of neuronal systems. Here, we monitor the spontaneous self-sustained dynamics in neuronal cultures formed by interconnected aggregates of neurons (clusters). Dynamics is characterized by the fast activation of groups of clusters in sequences termed bursts. The analysis of the time delays between clusters' activations within the bursts allows the reconstruction of the directed functional connectivity of the network. We propose a method to statistically infer this connectivity and analyze the resulting properties of the associated complex networks. Surprisingly enough, in contrast to what has been reported for many biological networks, the clustered neuronal cultures present assortative mixing connectivity values, meaning that there is a preference for clusters to link to other clusters that share similar functional connectivity, as well as a rich-club core, which shapes a ‘connectivity backbone’ in the network. These results point out that the grouping of neurons and the assortative connectivity between clusters are intrinsic survival mechanisms of the culture.

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.

The announcement of the discovery of a Higgs boson-like particle at CERN will be remembered as one of the milestones of the scientific endeavor of the 21st century. In this paper we present a study of information spreading processes on Twitter before, during and after the announcement of the discovery of a new particle with the features of the elusive Higgs boson on 4th July 2012. We report evidence for non-trivial spatio-temporal patterns in user activities at individual and global level, such as tweeting, re-tweeting and replying to existing tweets. We provide a possible explanation for the observed time-varying dynamics of user activities during the spreading of this scientific "rumor". We model the information spreading in the corresponding network of individuals who posted a tweet related to the Higgs boson discovery. Finally, we show that we are able to reproduce the global behavior of about 500,000 individuals with remarkable accuracy.

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.

We present a novel way to characterize the structure of complex networks by studying the statistical properties of the trajectories of random walks over them. We consider time series corresponding to different properties of the nodes visited by the walkers. We show that the analysis of the fluctuations of these time series allows to define a set of characteristic exponents which capture the local and global organization of a network. This approach provides a way of solving two classical problems in network science, namely the systematic classification of networks, and the identification of the salient properties of growing networks. The results contribute to the construction of a unifying framework for the investigation of the structure and dynamics of complex systems.

Previous studies have shown that human movement is predictable to a certain extent at different geographic scales. Existing prediction techniques exploit only the past history of the person taken into consideration as input of the predictors. In this paper, we show that by means of multivariate nonlinear time series prediction techniques it is possible to increase the accuracy of movement forecasting by considering movements of friends or people with correlated mobility patterns (i.e., characterised by high mutual information) as input of the predictor. Finally, we evaluate the proposed techniques on the Nokia Mobile Data Challenge and Cabspotting datasets.

The study of the interdependence of human movement and social ties of individuals is one of the most interesting re- search areas in computational social science. Previous stud- ies have shown that human movement is predictable to a certain extent at different geographic scales. One of the open problems is how to improve the prediction exploiting additional available information. In particular, one of the key questions is how to characterise and exploit the corre- lation between movements of friends and acquaintances to increase the accuracy of the forecasting algorithms.

Prediction of sea-level is an important task for navigation, coastal engineering and geodetic applications, as well as recreational activities. This study presents a comparison of Chaos theory and Auto-Regressive Integrated Moving Average (ARIMA) techniques for sea level modelling for daily, weekly, 10-day and monthly time scale at the Cocos (Keeling) islands from 1992 to 2001. The state space reconstruction of the unknown underlying process is directly employed from time series data, through Takens delay embedding method: opti- mal embedding dimension and delay time are obtained from false nearest neighbours and average mutual information techniques, respectively. Optimal values are then used for the estimation of the correlation dimension and the largest Lyapunov exponent, for inspecting possible signatures of chaotic dynamics. We find a positive Lyapunov exponent an evident feature of chaos. Indeed, the nonlinear prediction of sea level, in the period ranging from January 2001 to December 2001, is in an excellent agreement with the data for the same period, evidencing the nonlinear nature of the process. ARIMA method is also used for sea level modelling, for the same time scales; the performances of the two models are compared using such statistical indices as the root mean square error (RMSE) and correlation coefficient (CC). The comparative analyses show that the chaos theory model has a slight edge over ARIMA while both models are in principal acceptable.

We investigate flow dynamics in rivers characterized by basin areas and daily mean discharge spanning different orders of magnitude. We show that the delayed increments evaluated at time scales ranging from days to months can be opportunely rescaled to the same non-Gaussian probability density function. Such a scaling breaks up above a certain critical horizon, where a behavior typical of thermodynamic systems at the critical point emerges. We finally show that both the scaling behavior and the break up of the scaling are universal features of river flow dynamics.

We introduce a statistical method to detect nonlinearity and nonstationarity in time series, that works even for short sequences and in presence of noise. The method has a discrimination power similar to that of the most advanced estimators on the market, yet it depends only on one parameter, is easier to implement and faster. Applications to real data sets reject the null hypothesis of an underlying stationary linear stochastic process with a higher confidence interval than the best known nonlinear discriminators up to date.

Adequate knowledge of the nature of river flow process is crucial for proper planning and management of our water resources and environment. This study attempts to detect the salient characteristics of flow dynamics of the Karoon River in Iran. Daily discharge series observed over a period of six years (1999-2004) is analyzed to examine the chaotic and scaling characteristics of the flow dynamics. The presence of chaos is investigated through the correlation dimension and Lyapunov exponent methods, while the Hurst exponent and Renyi dimension analyses are performed to explore the scaling characteristics. The low correlation dimension (2.60 +- 0.07) and the positive largest Lyapunov exponent (0.014 +- 0.001) suggest the presence of low-dimensional chaos; they also imply that the flow dynamics are dominantly governed by three variables and can be reliably predicted up to 48 days (i.e. prediction horizon). Results from the Hurst exponent and Renyi dimension analyses reveal the multifractal character of the flow dynamics, with persistent and anti-persistent behaviors observed at different time scales.

A linear and a nonlinear time series analysis on the experimental rate of SD events normalized to the sensitive detector area, as described in [1], has been done. The same analysis have been performed on simulated time series with annual periodic behavior. An evident annual period has been found in the P. Auger data due to a deterministic ciclostationary physical effect which a recent paper from the P. Auger Collaboration attributes to atmospheric effects, such as pressure, temperature and density, on extensive air showers [2] [3]. Our results are in good agreement with this interpretation. Another deterministic stationary aperiodic (or quasi-periodic) or nonstationary (deterministic or stochastic) behavior has been found in the P. Auger data, probably due to some physical unknown effect which needs further investigations in the next years, when a larger amount of data will be available.

We present the results about a deep and detailed nonlinear time series anal- ysis of the UHECR arrival times series observed at the P. Auger Observatory, Malargue, Argentina. The analysis has been developed over a known chaotic time series (derived from a Lorenz standard model) in order to test that our algorithms and our software work fine. We analysed in addition 2 arrival times series obtained from a chaotic system and a stochastic one, in order to get our conclusion by a direct comparison among the UHECR system’s dynamical properties and the 2 known ones. We tried to reconstruct the phase space of each dynamical system from a non-linear analysis of its own time arrival series by using a special multidimensional space, called embedding, with the same dynamical invariants of the original phase space like the fractal dimension, the Renyi dimension, the Lyapunov exponents and the time reversal asymmetry. We performed a statistical analysis based on the surrogate data method and nonparametric statistics with about 2400 numerical analysis to test the null hypothesis of white noise, linear gaussian model, nonlinear gaussian model. From the analysis of the Lyapunov spectrums and the time reversal asymmetry we deduced that there are time correlation in UHECR with energy above 1019eV and that, at this preliminary step, it is possible that the dynamical system, which they depend to, can be considered deterministic and low-dimensional chaotic with a time-projected attractor of fractal dimension D2 = 1.14. Important implications on the nature of their cosmological origin and propagation need further investigations.

Arrival time distributions of the UHECRs should contain relevant information about specific feature of the source. To distinguish these ’source features’ is not an easy task. We suggest to use nonlinear time series analysis. The approach has two main goals: on one side we will try to investigate the astrophysical nature of the process (origin, propagation feature), while on the other side we would like to investigate the deterministic, low-dimensional chaotic or stochastic natute of the dynamical system they belong to. We present preliminary results about the nonlinearity and the nonstationarity of UHECR with E > 10EeV detected from surface detector at PAO with zenith angle θ < 60◦, for both T5 and no T5 cuts. Preliminary results for UHECR with E > 40EeV are also included.