Publications -- Methods

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   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.

  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. Gómez 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. Gómez

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. Gómez, 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. Gómez 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.

  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.

  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.

  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.

  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.

  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