Publications -- Networks

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  Centralities of nodes and influences of layers in large multiplex networks

Journal of Complex Networks, cnx050 - doi: 10.1103/comnet/cnx050 - 2017

C. Rahmede, J. Iacovacci, A. Arenas and G. Bianconi

We formulate and propose an algorithm (MultiRank) for the weighted ranking of nodes and layers in large multiplex networks. MultiRank takes into account the full multiplex network structure of the data and exploits the dual nature of the network in terms of nodes and layers. The proposed centrality of the layers (influences) and the centrality of the nodes are determined by a coupled set of equations. The basic idea consist on assigning more centrality to nodes that are linked to central nodes in highly influential layers. The layers are more influential if highly central nodes are active in them. The algorithm applies to directed/undirected as well as to weighted/unweighted multiplex networks. We discuss the application of MultiRank to three major examples of multiplex network data sets: the European Air Transportation Multiplex Network, the Pierre Auger Multiplex Collaboration Network and the FAO Multiplex Trade Network.

  Modeling structure and resilience of the Dark Network

Physical Review E 95, 022313 - doi: 10.1103/PhysRevE.022313 - 2017

M. De Domenico and A. Arenas

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 Đ typical of scale-free networks Đ, very short path lengths and high clustering Đ typical of small-world networks Đ 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 Đ used as a benchmark for comparison at a descriptive level Đ 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.

  Influence of trust in the spreading of information

Physical Review E 95, 012301 - doi: 10.1103/PhysRevE.95.012301 - 2017

H. Wu, A. Arenas, S. Gómez

The understanding and prediction of information diffusion processes on networks is a major challenge in network theory with many implications in social sciences. Many theoretical advances occurred due to stochastic spreading models. Nevertheless, these stochastic models overlooked the influence of rational decisions on the outcome of the process. For instance, different levels of trust in acquaintances do play a role in information spreading, and actors may change their spreading decisions during the information diffusion process accordingly. Here, we study an information-spreading model in which the decision to transmit or not is based on trust. We explore the interplay between the propagation of information and the trust dynamics happening on a two-layer multiplex network. ActorsŐ trustable or untrustable states are defined as accumulated cooperation or defection behaviors, respectively, in a PrisonerŐs Dilemma setup, and they are controlled by a memory span. The propagation of information is abstracted as a threshold model on the information-spreading layer, where the threshold depends on the trustability of agents. The analysis of the model is performed using a tree approximation and validated on homogeneous and heterogeneous networks. The results show that the memory of previous actions has a significant effect on the spreading of information. For example, the less memory that is considered, the higher is the diffusion. Information is highly promoted by the emergence of trustable acquaintances. These results provide insight into the effect of plausible biases on spreading dynamics in a multilevel networked system.

  The physics of spreading processes in multilayer networks

Nature Physics - doi:10.1038/nphys3865 - 2016

M. De Domenico, C. Granell, M. A. Porter, A. Arenas

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.

  Functional multiplex pagerank

Europhysics Letters 116, 28004 - 10.1209/0295-5075/116/28004 - 2016

J. Iacovacci, C. Rahmede, A. Arenas and G. Bianconi

Recently it has been recognized that many complex social, technological and biological networks have a multilayer nature and can be described by multiplex networks. Multiplex networks are formed by a set of nodes connected by links having different connotations forming the different layers of the multiplex. Characterizing the centrality of the nodes in a multiplex network is a challenging task since the centrality of the node naturally depends on the importance associated to links of a certain type. Here we propose to assign to each node of a multiplex network a centrality called Functional Multiplex PageRank that is a function of the weights given to every different pattern of connections (multilinks) existent in the multiplex network between any two nodes. Since multilinks distinguish all the possible ways in which the links in different layers can overlap, the Functional Multiplex PageRank can describe important non-linear effects when large relevance or small relevance is assigned to multilinks with overlap. Here we apply the Functional Page Rank to the multiplex airport networks, to the neuronal network of the nematode c.elegans, and to social collaboration and citation networks between scientists. This analysis reveals important differences existing between the most central nodes of these networks, and the correlations between their so called "pattern to success".

  Untangling the role of diverse social dimensions in the diffusion of microfinance

Applied Network Science 1:14 - 10.1007/s41109-016-0016-x - 2016

E. Omodei and A. Arenas

Ties between individuals on a social network can represent different dimensions of interactions, and the spreading of information and innovations on these networks could potentially be driven by some dimensions more than by others. In this paper we investigate this issue by studying the diffusion of microfinance within rural India villages and accounting for the whole multilayer structure of the underlying social networks. We define a new measure of node centrality, diffusion versatility, and show that this is a better predictor of microfinance participation rate than previously introduced measures defined on aggregated single-layer social networks. Moreover, we untangle the role played by each social dimension and find that the most prominent role is played by the nodes that are central on layers concerned with trust, shedding new light on the key triggers of the diffusion of microfinance.

  Multiplex social ecological network analysis reveals how social changes affect community robustness more than resource depletion

PNAS USA 113, 13708 - - 2016

J.A. Baggio, S.B. BurnSilver, A. Arenas, J.S. Magdanzd, G.P. Kofinasd, M. De Domenico

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.

  On controlling networks of limit-cycle oscillators

Chaos: An Interdisciplinary Journal of Nonlinear Science 26, 094812 - 10.1063/1.4954273 - 2016

PS. Skardal and A. Arenas

The control of network-coupled nonlinear dynamical systems is an active area of research in the nonlinear science community. Coupled oscillator networks represent a particularly important family of nonlinear systems, with applications ranging from the power grid to cardiac excitation. Here we study the control of network-coupled limit cycle oscillators, extending previous work that focused on phase oscillators. Based on stabilizing a target fixed point, our method aims to attain complete frequency synchronization, i.e., consensus, by applying control to as few oscillators as possible. We develop two types of control. The first type directs oscillators towards to larger amplitudes, while the second does not. We present numerical examples of both control types and comment on the potential failures of the method.

  Spectral entropies as information-theoretic tools for complex network comparison

Phys. Rev. X 6, 041062 - 10.1103/PhysRevX.6.041062 - 2016

M. De Domenico, J. Biamonte

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 RŽnyi 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 with 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.

  Collective frequency variation in network synchronization and reverse PageRank

Physical Review E 93, 042314 - - 2016

PS. Skardal, D. Taylor, J. Sun, and A. Arenas

A wide range of natural and engineered phenomena rely on large networks of interacting units to reach a dynamical consensus state where the system collectively operates. Here we study the dynamics of self-organizing systems and show that for generic directed networks the collective frequency of the ensemble is not the same as the mean of the individuals' natural frequencies. Specifically, we show that the collective frequency equals a weighted average of the natural frequencies, where the weights are given by an outflow centrality measure that is equivalent to a reverse PageRank centrality. Our findings uncover an intricate dependence of the collective frequency on both the structural directedness and dynamical heterogeneity of the network, and also reveal an unexplored connection between synchronization and PageRank, which opens the possibility of applying PageRank optimization to synchronization. Finally, we demonstrate the presence of collective frequency variation in real-world networks by considering the UK and Scandinavian power grids.

  A model to identify urban traffic congestion hotspots in complex networks

Royal Society Open Science 3, 160098 - 10.1098/rsos.160098 - 2016

Albert Solé-Ribalta, Sergio Gómez, Alex Arenas

The rapid growth of population in urban areas is jeopardizing the mobility and air quality worldwide. One of the most notable problems arising is that of traffic congestion. With the advent of technologies able to sense real-time data about cities, and its public distribution for analysis, we are in place to forecast scenarios valuable for improvement and control. Here, we propose an idealized model, based on the critical phenomena arising in complex networks, that allows to analytically predict congestion hotspots in urban environments. Results on real citiesŐ road networks, considering, in some experiments, real traffic data, show that the proposed model is capable of identifying susceptible junctions that might become hotspots if mobility demand increases.

  Erosion of synchronization: Coupling heterogeneity and network structure

Physica D: Nonlinear Phenomena 323, 40-48 - - 2016

PS. Skardal, D. Taylor, J. Sun, and A. Arenas

We study the dynamics of network-coupled phase oscillators in the presence of coupling frustration. It was recently demonstrated that in heterogeneous network topologies, the presence of coupling frustration causes perfect phase synchronization to become unattainable even in the limit of infinite coupling strength. Here, we consider the important case of heterogeneous coupling functions and extend previous results by deriving analytical predictions for the total erosion of synchronization. Our analytical results are given in terms of basic quantities related to the network structure and coupling frustration. In addition to fully heterogeneous coupling, where each individual interaction is allowed to be distinct, we also consider partially heterogeneous coupling and homogeneous coupling in which the coupling functions are either unique to each oscillator or identical for all network interactions, respectively. We demonstrate the validity of our theory with numerical simulations of multiple network models, and highlight the interesting effects that various coupling choices and network models have on the total erosion of synchronization. Finally, we consider some special network structures with well-known spectral properties, which allows us to derive further analytical results.

  Nonlinear Dynamics on Interconnected Networks

Physica D; Volumes 323–324 - - 2016

M. De Domenico, A. Arenas

Networks of dynamical interacting units can represent many complex systems, from the human brain to transportation systems and societies. The study of these complex networks, when accounting for different types of interactions has become a subject of interest in the last few years, especially because its representational power in the description of users’ interactions in diverse online social platforms (Facebook, Twitter, Instagram, etc.) [1], or in representing different transportation modes in urban networks [2,3]. The general name coined for these networks is multilayer networks, where each layer accounts for a type of interaction (see Fig. 1). In the last years, it has been a boosted interest in the analysis of the structure and dynamics on multilayer networks [4–7], essentially because the well-known theory of complex networks developed for the study of a single layer network has to be revisited when analyzing multilayer networks. Moreover, the outcome of the analysis has revealed that new emergent physical phenomena can appear as a direct consequence of the multilayer structure. In particular, the analysis of the robustness of the structure in the presence of perturbations or defects, as well as the cascade propagation of failures has focussed the analysis of important contributions in the field [4,8–13]. From the physics point of view, the study of simple diffusion processes (or more complex like epidemic spreading, etc.) has driven the understanding of the interplay between dynamical processes and structure in the subject [11,14–16] From the mathematical point of view, the new level of complexity required the definition of a novel mathematical framework [17], based on tensorial algebra, for their representation and their structural reduction to simpler subsets of networks [18]. The outgrowth of these mathematical approaches is the development of new structural descriptors, from centrality measures [19–24] to partitions in communities to describe the me

  Bond percolation on multiplex networks

Physical Review X 6, 021002 - - 2016

A. Hackett, D. Cellai, S. Gómez, A. Arenas and J.P. Gleeson

We present an analytical approach for bond percolation on multiplex networks and use it to determine the expected size of the giant connected component and the value of the critical bond occupation probability in these networks. We advocate the relevance of these tools to the modeling of multilayer robustness and contribute to the debate on whether any benefit is to be yielded from studying a full multiplex structure as opposed to its monoplex projection, especially in the seemingly irrelevant case of a bond occupation probability that does not depend on the layer. Although we find that in many cases the predictions of our theory for multiplex networks coincide with previously derived results for monoplex networks, we also uncover the remarkable result that for a certain class of multiplex networks, well described by our theory, new critical phenomena occur as multiple percolation phase transitions are present. We provide an instance of this phenomenon in a multiplex network constructed from London rail and European air transportation data sets.

  The dynamics of information-driven coordination phenomena: A transfer entropy analysis

Science Advances 2(4) e1501158 - 10.1126/sciadv.1501158 - 2016

J. Borge-Holthoefer, N. Perra, B. Gonçalves, S. Gonzalez-Bailon, A. Arenas, Y. Moreno and A. Vespignani

Data from social media are providing unprecedented opportunities to investigate the processes that rule the dynamics of collective social phenomena. Here, we consider an information theoretical approach to define and measure the temporal and structural signatures typical of collective social events as they arise and gain prominence. We use the symbolic transfer entropy analysis of micro-blogging time series to extract directed networks of influence among geolocalized sub-units in social systems. This methodology captures the emergence of system-level dynamics close to the onset of socially relevant collective phenomena. The framework is validated against a detailed empirical analysis of five case studies. In particular, we identify a change in the characteristic time-scale of the information transfer that flags the onset of information-driven collective phenomena. Furthermore, our approach identifies an order-disorder transition in the directed network of influence between social sub-units. In the absence of a clear exogenous driving, social collective phenomena can be represented as endogenously-driven structural transitions of the information transfer network. This study provides results that can help define models and predictive algorithms for the analysis of societal events based on open source data.

  Mapping Multiplex Hubs in Human Functional Brain Networks

Frontiers in Neuroscience 10, 326 - - 2016

M. De Domenico,S. Sasai, A. Arenas

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.

  Evaluating the impact of interdisciplinary research: a multilayer network approach

Network Science - - 2016

E. Omodei, M. De Domenico, A. Arenas

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.

  Quantifying the Diaspora of Knowledge in the Last Century

Applied Network Science 1, 15 - - 2016

M. De Domenico, E. Omodei, A. Arenas

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.

  Researcher incentives: EU cash goes to the sticky and attractive

Nature 531, 580 - doi: 10.1038/531580c - 2016

M. De Domenico, A. Arenas

Winning European research money does not depend only on a well-funded research base. We find that it is also contingent on national governments' ability to retain their own scientists ('stickiness') and to attract others from abroad ('attractiveness'). We analysed statistical indicators of EU scientists' mobility for 2007–14 to determine the stickiness and attractiveness of different countries. We quantified attractiveness and stickiness as the relative difference between the numbers of incoming or remaining researchers, respectively, and of outgoing ones. For both measures, we found that the higher the value, the better were that country's chances of securing European research funding. The United Kingdom and Sweden are examples of high scorers in both; Italy is among the lowest . 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.

  Assessing reliable human mobility patterns from higher-order memory in mobile communications

J. Roy. Soc. Inter. 13, 20160203 - DOI: 10.1098/rsif.2016.0203 - 2016

M. De Domenico, J. T. Matamalas, A. Arenas

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.

  Congestion Induced by the Structure of Multiplex Networks

Physical Review Lett. 116, 108701 - DOI: 10.1103/PhysRevLett.116.108701 - 2016

A. Solé-Ribalta, S. Gómez, A. Arenas

Multiplex networks are representations of multilayer interconnected complex networks where the nodes are the same at every layer. They turn out to be good abstractions of the intricate connectivity of multimodal transportation networks, among other types of complex systems. One of the most important critical phenomena arising in such networks is the emergence of congestion in transportation flows. Here, we prove analytically that the structure of multiplex networks can induce congestion for flows that otherwise would be decongested if the individual layers were not interconnected. We provide explicit equations for the onset of congestion and approximations that allow us to compute this onset from individual descriptors of the individual layers. The observed cooperative phenomenon is reminiscent of Braess’ paradox in which adding extra capacity to a network when the moving entities selfishly choose their route can in some cases reduce overall performance. Similarly, in the multiplex structure, the efficiency in transportation can unbalance the transportation loads resulting in unexpected congestion.

   Random walk centrality in interconnected multilayer networks

Physica D 323-324, 73 - - 2016

A. Sole-Ribalta, M. De Domenico, S. Gómez, A. Arenas

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.

  Enhancing the stability of the synchronization of multivariable coupled oscillators

Physical Review E 92, 032804 - DOI: 10.1103/PhysRevE.92.032804 - 2015

R. Sevilla-Escoboza, R. Gutierrez, G. Huerta-Cuellar, S. Boccaletti, J. Gómez-Gardeñes, A. Arenas and J. M. Buldú

Synchronization processes in populations of identical networked oscillators are the focus of intense studies in physical, biological, technological, and social systems. Here we analyze the stability of the synchronization of a network of oscillators coupled through different variables. Under the assumption of an equal topology of connections for all variables, the master stability function formalism allows assessing and quantifying the stability properties of the synchronization manifold when the coupling is transferred from one variable to another. We report on the existence of an optimal coupling transference that maximizes the stability of the synchronous state in a network of Rossler-like oscillators. Finally, we design an experimental implementation (using nonlinear electronic circuits) which grounds the robustness of the theoretical predictions against parameter mismatches, as well as against intrinsic noise of the system.

  Layer-layer competition in multiplex complex networks

Phil. Trans. R. Soc. A 373: 20150117 - DOI: 10.1098/rsta.2015.0117 - 2015

J. Gómez-Gardeñes, M. de Domenico, G. Gutiérrez, A. Arenas, S. Gómez

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 layers or shared between them, favouring that neighbouring 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 user 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 the structural properties.

  Control of coupled oscillator networks with application to microgrid technologies

Science Advances 1, 7 - DOI: 10.1126/sciadv.1500339 - 2015

P.S. Skardal and A. Arenas

The control of complex systems and network-coupled dynamical systems is a topic of vital theoretical importance in mathematics and physics with a wide range of applications in engineering and various other sciences. Motivated by recent research into smart grid technologies, we study the control of synchronization and consider the important case of networks of coupled phase oscillators with nonlinear interactions—a paradigmatic example that has guided our understanding of self-organization for decades. We develop a method for control based on identifying and stabilizing problematic oscillators, resulting in a stable spectrum of eigenvalues, and in turn a linearly stable synchronized state. The amount of control, that is, number of oscillators, required to stabilize the network is primarily dictated by the coupling strength, dynamical heterogeneity, and mean degree of the network, and depends little on the structural heterogeneity of the network itself.

  Ranking nodes in interconnected multilayer networks reveals their versatility

Nature Communications 6, 6868 - DOI: 10.1038/ncomms7868 - 2015

M. De Domenico, A. Sole-Ribalta, E. Omodei, S. Gomez, A. Arenas

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.

  Information transfer in community structured multiplex networks

Frontiers in Physics 3, 61 - - 2015

A. Sole-Ribalta, C. Granell, S. GĂłmez and A. Arenas

The study of complex networks that account for different types of interactions has become a subject of interest in the last few years, specially because its representational power in the description of users interactions in diverse online social platforms (Facebook, Twitter, Instagram, etc.). The mathematical description of these interacting networks has been coined under the name of multilayer networks, where each layer accounts for a type of interaction. It has been shown that diffusive processes on top of these networks present a phenomenology that cannot be explained by the naive superposition of single layer diffusive phenomena but require the whole structure of interconnected layers. Nevertheless, the description of diffusive phenomena on multilayer networks has obviated the fact that social networks have strong mesoscopic structure represented by different communities of individuals driven by common interests, or any other social aspect. In this work, we study the transfer of information in multilayer networks with community structure. The final goal is to understand and quantify, if the existence of well-defined community structure at the level of individual layers, together with the multilayer structure of the whole network, enhances or deteriorates the diffusion of packets of information.

   Characterizing interactions in online social networks during exceptional events

Frontiers in Physics 3, 59 - - 2015

E. Omodei, M. De Domenico, A. Arenas

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.

  A benchmark model to assess community structure in evolving networks

Physical Review E 012805 - - 2015

C. Granell, R. K. Darst, A. Arenas, S. Fortunato and S. GĂłmez

Detecting the time evolution of the community structure of networks is crucial to identify major changes in the internal organization of many complex systems, which may undergo important endogenous or exogenous events. This analysis can be done in two ways: considering each snapshot as an independent community detection problem or taking into account the whole evolution of the network. In the first case, one can apply static methods on the temporal snapshots, which correspond to configurations of the system in short time windows, and match afterward the communities across layers. Alternatively, one can develop dedicated dynamic procedures so that multiple snapshots are simultaneously taken into account while detecting communities, which allows us to keep memory of the flow. To check how well a method of any kind could capture the evolution of communities, suitable benchmarks are needed. Here we propose a model for generating simple dynamic benchmark graphs, based on stochastic block models. In them, the time evolution consists of a periodic oscillation of the system's structure between configurations with built-in community structure. We also propose the extension of quality comparison indices to the dynamic scenario.

  Structural reducibility of multilayer networks

Nature Communications 6, 6864 - DOI: 10.1038/ncomms7864 - 2015

M. De Domenico, V. Nicosia, A. Arenas, V. Latora

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.

  Identifying modular flows on multilayer networks reveals highly overlapping organization in interconnected systems

Physical Review X 5, 011027 - DOI: 10.1103/PhysRevX.5.011027 - 2015

M. De Domenico, A. Lancichinetti, A. Arenas, M. Rosvall

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.

  Strategical incoherence regulates cooperation in social dilemmas on multiplex networks

Scientific Reports 5, 9519 - doi:10.1038/srep09519 - 2015

J. Matamalas, J. Poncela-Casasnovas, S. GĂłmez, and Alexandre Arenas

Cooperation is a very common, yet not fully-understood phenomenon in natural and human systems. The introduction of a network structure within the population is known to affect the outcome of cooperative dynamics, as described by the Game Theory paradigm, allowing for the survival of cooperation in adverse scenarios. Recently, the introduction of multilayered networks has yet again modified the expectations for the outcome of the Prisoner’s Dilemma game, compared to the monoplex case. However, much remains to be studied regarding other games in the plane of social dilemmas on multiplex, as well as the unexplored microscopic underpinnings of it. In this paper, we systematically and carefully study the evolution and outcome of all four games in the S − T plane (Prisoner’s Dilemma, Stag-Hung, Snow Drift and Harmony) on multiplex, as a function of the number of layers. More importantly, we find some remarkable and previously unknown features in the microscopic organization of the strategies, that are at the root of the important differences between cooperative dynamics in monoplex and multiplex. Specifically, we find that in the stationary state, there are individuals that play the same strategy in all layers (coherent), and others that don’t (incoherent). This second group of players is responsible for the surprising fact of a non full-cooperation in the Harmony Game on multiplex, never observed before, as well as a higher-than-expected survival of cooperation in some regions of the other three social dilemmas.

  Quantifying sudden changes in dynamical systems using symbolic networks

New Journal of Physics 17, 023068 - - 2015

C. Masoller, Y.Y. Hong, S. Ayad, F. Gustave, S. Barland, A. Pons, S. GĂłmez and A. Arenas

We characterize the evolution of a dynamical system by combining two well-known complex systems' tools, namely, symbolic ordinal analysis and networks. From the ordinal representation of a time series we construct a network in which every node weight represents the probability of an ordinal pattern (OP) to appear in the symbolic sequence and each edge's weight represents the probability of transitions between two consecutive OPs. Several network-based diagnostics are then proposed to characterize the dynamics of different systems: logistic, tent, and circle maps. We show that these diagnostics are able to capture changes produced in the dynamics as a control parameter is varied. We also apply our new measures to empirical data from semiconductor lasers and show that they are able to anticipate the polarization switchings, thus providing early warning signals of abrupt transitions.

  Personalized Routing for Multitudes in Smart Cities

EPJ Data Science 4, 1 - - 2015

M. De Domenico, A. Lima, M. Gonzalez, A. Arenas

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.

  Disease Containment Strategies based on Mobility and Information Dissemination

Scientific Reports 5, 10650 - DOI:10.1038/srep10650 - 2015

A. Lima, M. De Domenico, V. Pejovic, M. Musolesi

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.

  MuxViz: a tool for multilayer analysis and visualization of networks

Journal of Complex Networks 3, 159 - doi:10.1093/comnet/cnu038 - 2015

M. De Domenico, Mason A. Porter, A. Arenas

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

   Structure of triadic relations in multiplex networks

New Journal of Physics 17, 073029 - - 2015

E. Cozzo, M. Kivela, M. De Domenico, A. Sole-Ribalta, A. Arenas, S. Gomez, M. Porter and Y. Moreno

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.

  Centrality Rankings in Multiplex Networks

Proceedings of the 2014 ACM conference on Web science, 149-155 - - 2014

A. Sole-Ribalta, M. De Domenico, S. Gomez, A. Arenas

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.

  Navigability of interconnected networks under random failures

PNAS 11, 8351 - DOI: 10.1073/pnas.1318469111 - 2014

M. De Domenico, A. Sole-Ribalta, S. Gomez, A. Arenas

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.

  Emergence of assortative mixing between clusters of cultured neurons

PLOS Comput. Biol. 10(9), e1003796 - - 2014

S. Teller, C. Granell, M. De Domenico, J. Soriano, S. Gomez, A. Arenas

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.

  Multilayer networks

Journal of Complex Networks, Vol. 2, No. 3: 203-271 - DOI: 10.1093/comnet/cnu016 - 2014

M. Kivela, A. Arenas, M Barthelemy, J.P. Gleeson, Y. Moreno and M. Porter

In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such ‘multilayer’ features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize ‘traditional’ network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion. The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science. In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks. To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary of terminology to relate the numerous existing concepts to each other and provide a thorough discussion that compares, contrasts and translates between related notions such as multilayer networks, multiplex networks, interdependent networks, networks of networks and many others. We also survey and discuss existing data sets that can be represented as multilayer networks. We review attempts to generalize single-layer-network diagnostics to multilayer networks. We also discuss the rapidly expanding research on multilayer-network models and notions like community structure, connected components, tensor decompositions and various types of dynamical processes on multilayer networks. We conclude with a summary and an outlook.

  Competing spreading processes on multiplex networks: Awareness and epidemics

Physical Review E 90, 012808 - DOI: - 2014

C. Granell, S. Gomez and A. Arenas

Epidemic-like spreading processes on top of multilayered interconnected complex networks reveal a rich phase diagram of intertwined competition effects. A recent study by the authors [Granell et al. Phys. Rev. Lett. 111, 128701 (2013)] presented the analysis of the interrelation between two processes accounting for the spreading of an epidemics, and the spreading of information awareness to prevent its infection, on top of multiplex networks. The results in the case in which awareness implies total immunization to the disease, revealed the existence of a metacritical point at which the critical onset of the epidemics starts depending on the reaching of the awareness process. Here we present a full analysis of these critical properties in the more general scenario where the awareness spreading does not imply total immunization, and where infection does not imply immediate awareness of it. We find the critical relation between both competing processes for a wide spectrum of parameters representing the interaction between them. We also analyze the consequences of a massive broadcast of awareness (mass media) on the final outcome of the epidemic incidence. Importantly enough, the mass media makes the metacritical point to disappear. The results reveal that the main finding i.e. existence of a metacritical point, is rooted on the competition principle and holds for a large set of scenarios.

  Disorder induces explosive synchronization

Physical Review E 89, 062811 - DOI: - 2014

P. S. Skardal and A. Arenas

We study explosive synchronization, a phenomenon characterized by first-order phase transitions between incoherent and synchronized states in networks of coupled oscillators. While explosive synchronization has been the subject of many recent studies, in each case strong conditions on either the heterogeneity of the network, its link weights, or its initial construction are imposed to engineer a first-order phase transition. This raises the question of how robust explosive synchronization is in view of more realistic structural and dynamical properties. Here we show that explosive synchronization can be induced in mildly heterogeneous networks by the addition of quenched disorder to the oscillators’ frequencies, demonstrating that it is not only robust to, but moreover promoted by, this natural mechanism. We support these findings with numerical and analytical results, presenting simulations of a real neural network as well as a self-consistency theory used to study synthetic networks.

  Atapuerca: evolution of scientific collaboration in an emergent large-scale research infrastructure

Scientometrics, 98, 1505 - DOI 10.1007/s11192-013-1162-x - 2014

S. Lozano, X-P. Rodriguez and A. Arenas

We study the evolution of scientific collaboration at Atapuerca’s archaeological complex along its emergence as a large-scale research infrastructure (LSRI). Using bibliometric and fieldwork data, we build and analyze co-authorship networks corresponding to the period 1992–2011. The analysis of such structures reveals a stable core of scholars with a long experience in Atapuerca’s fieldwork, which would control coauthorship-related information flows, and a tree-like periphery mostly populated by ‘external’ researchers. Interestingly, this scenario corresponds to the idea of a Equipo de Investigacio´n de Atapuerca, originally envisioned by Atapuerca’s first director 30 years ago. These results have important systemic implications, both in terms of resilience of co-authorship structures and of ‘oriented’ or ‘guided’ self-organized network growth. Taking into account the scientific relevance of LSRIs, we expect a growing number of quantitative studies addressing collaboration among scholars in this sort of facilities in general and, particularly, emergent phenomena like the Atapuerca case.

  Mathematical Formulation of Multi-Layer Networks

Phys. Rev. X 3, 041022 - DOI: 10.1103/PhysRevX.3.041022 - 2013

M. De Domenico, A. Sole-Ribalta, E. Cozzo, M. Kivela, Y. Moreno, M. A. Porter, S. Gomez, A. Arenas

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.

  Spectral properties of the Laplacian of multiplex networks

Phys. Rev. E 88, 032807 - DOI: 10.1103/PhysRevE.88.032807 - 2013

A. Sole-Ribalta, M. De Domenico, N. E. Kouvaris, A. Diaz-Guilera, S. Gomez, A. Arenas

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.

  Abrupt transition in the structural formation of interconnected networks

Nature Physics, 9, 717 - doi:10.1038/nphys2761 - 2013

F. Radicchi and A. Arenas

Our current world is linked by a complex mesh of networks where information, people and goods flow. These networks are interdependent each other, and present structural and dynamical features different from those observed in isolated networks. While examples of such “dissimilar” properties are becoming more abundant, for example diffusion, robustness and competition, it is not yet clear where these differences are rooted in. Here we show that the composition of independent networks into an interconnected network of networks undergoes a structurally sharp transition as the interconnections are formed. Depending of the relative importance of inter and intra-layer connections, we find that the entire interdependent system can be tuned between two regimes: in one regime, the various layers are structurally decoupled and they act as independent entities; in the other regime, network layers are indistinguishable and the whole system behave as a single-level network. We analytically show that the transition between the two regimes is discontinuous even for finite size networks. Thus, any real-world interconnected system is potentially at risk of abrupt changes in its structure that may reflect in new dynamical properties.

  Modeling self-sustained activity cascades in socio-technical networks

Europhysics Letters, 104, 48004 - - 2013

P. Piedrahita, J. Borge-Holthoefer, Y. Moreno and A. Arenas

The ability to understand and eventually predict the emergence of information and activation cascades in social networks is core to complex socio-technical systems research. However, the complexity of social interactions makes this a challenging enterprise. Previous works on cascade models assume that the emergence of this collective phenomenon is related to the activity observed in the local neighborhood of individuals, but do not consider what determines the willingness to spread information in a time-varying process. Here we present a mechanistic model that accounts for the temporal evolution of the individual state in a simplified setup. We model the activity of the individuals as a complex network of interacting integrate-and-fire oscillators. The model reproduces the statistical characteristics of the cascades in real systems, and provides a framework to study the time evolution of cascades in a state-dependent activity scenario.

  Dynamical interplay between awareness and epidemic spreading in multiplex networks

Physical Review Letters, 111, 128701 - DOI: 10.1103/PhysRevLett.111.128701 - 2013

C. Granell, S. Gomez and A. Arenas

We present the analysis of the interrelation between two processes accounting for the spreading of an epidemic, and the information awareness to prevent its infection, on top of multiplex networks. This scenario is representative of an epidemic process spreading on a network of persistent real contacts, and a cyclic information awareness process diffusing in the network of virtual social contacts between the same individuals. The topology corresponds to a multiplex network where two diffusive processes are interacting affecting each other. The analysis using a microscopic Markov chain approach reveals the phase diagram of the incidence of the epidemics and allows us to capture the evolution of the epidemic threshold depending on the topological structure of the multiplex and the interrelation with the awareness process. Interestingly, the critical point for the onset of the epidemics has a critical value (metacritical point) defined by the awareness dynamics and the topology of the virtual network, from which the onset increases and the epidemics incidence decreases

  Structural patterns in complex systems using multidendrograms

Entropy, 15(12), 5464-5474 - doi:10.3390/e15125464 - 2013

S. Gomez, A. Fernandez, C. Granell and A. Arenas

Complex systems are usually represented as an intricate set of relations between their components forming a complex graph or network. The understanding of their functioning and emergent properties are strongly related to their structural properties. The finding of structural patterns is of utmost importance to reduce the problem of understanding the structure–function relationships. Here we propose the analysis of similarity measures between nodes using hierarchical clustering methods. The discrete nature of the networks usually leads to a small set of different similarity values, making standard hierarchical clustering algorithms ambiguous. We propose the use of multidendrograms, an algorithm that computes agglomerative hierarchical clusterings implementing a variable-group technique that solves the non-uniqueness problem found in the standard pair-group algorithm. This problem arises when there are more than two clusters separated by the same maximum similarity (or minimum distance) during the agglomerative process. Forcing binary trees in this case means breaking ties in some way, thus giving rise to different output clusterings depending on the criterion used. Multidendrograms solves this problem by grouping more than two clusters at the same time when ties occur.

  Degree of intervality of food webs: From body-size data to models

Journal of Theoretical Biology, 334, 35-44 - DOI: 10.1016/j.jtbi.2013.06.004 - 2013

J.A. Capitan, A. Arenas and R. Guimera

In food webs, the degree of intervality of consumers' diets is an indicator of the number of dimensions that are necessary to determine the niche of a species. Previous studies modeling food-web structure have shown that real networks are compatible with a high degree of diet contiguity. However, current models are also compatible with the opposite, namely that species' diets have relatively low contiguity. This is particularly true when one takes species' body size as a proxy for niche value, in which case the indeterminacy of diet contiguities provided by current models can be large. We propose a model that enables us to narrow down the range of possible values of diet contiguity. According to this model, we find that diet contiguity not only can be high, but must be high when species are ranked in ascending order of body size.

  Diffusion dynamics on multiplex networks

Physical Review Letters, 110, 028701 - DOI: - 2013

S. Gomez, A. Diaz-Guilera, J. Gomez-Gardenes, C.J. Perez-Vicente, Y. Moreno and A. Arenas

We study the time scales associated with diffusion processes that take place on multiplex networks, i.e., on a set of networks linked through interconnected layers. To this end, we propose the construction of a supra-Laplacian matrix, which consists of a dimensional lifting of the Laplacian matrix of each layer of the multiplex network. We use perturbative analysis to reveal analytically the structure of eigenvectors and eigenvalues of the complete network in terms of the spectral properties of the individual layers. The spectrum of the supra-Laplacian allows us to understand the physics of diffusionlike processes on top of multiplex networks.

  On the Routability of the Internet

Dynamics On and Of Complex Networks, Volume 2, A. Mukherjee, M. Choudhury, F. Peruani, N. Ganguly and B. Mitra (eds.), Modeling and Simulation in Science, Engineering and Technology, 41-54 - - 2013

P. Erola, S. Gomez and A. Arenas

The Internet is increasingly changing the way we do everyday tasks at work, at home, and how we communicate with one another. In its entrails, the Internet is structured as a network of networks. From a bottom-up perspective, the Internet is made up of networks of routers, each one under the control of a single technical administration. These networks are called Autonomous Systems (AS). An AS can use an exterior gateway protocol to route packets to other ASes [35] forming one of the largest synthetic complex system ever built. The Internet1 comprises a decentralized collection of more than 30,000 computer networks from all around the world. Two ASes are connected if and only if they establish a business relationship (customer-provider or peer-topeer relationships), making the Internet a “living” self-organized system