Multiplex Networks

In many complex systems the interactions among the elementary components can be of qualitatively different nature. Such systems are therefore naturally described and represented in terms of multiplex or multi-layer networks, i.e. networks where each layer stands for a different type of interaction between the same set of nodes. We have proposed basic metrics [PRE14] for multiplex networks, with particular attention to correlations [PRE15]. Together with M. Barthelemy (CNRS, Paris) and G. Bianconi (QMUL, London) we have introduced some LINEAR [PRL13] AND NONLINEAR GROWTH MODELS [PRE14] to construct multiplex networks. With A. Arenas and M. De Domenico (Alephsys Lab, Terragona), we have introduced a method to REDUCE the number of layers in multilayer networks while minimizing information loss, which allows to describe the structure of a multilayer complex system with an optimal tradeoff between accuracy and complexity [NatComm15]. We have also studied dynamical processes, such as biased random walks on multilayer networks, and even the case of two different types of INTERTWINED DYNAMICAL PROCESSES, namely synchronization and energy transport, taking place on interconnected networks [arXiv14]. We are now working to an extension of the Axelroad model for the dissemination of cultures in multiplex networks, and to the study of Turing patterns in networks with two layers. Together with L. Lacasa (QMUL, London) we have introduced a method to study multivariate time series by transforming them into multiplex networks [NatSR15]. More on our work can be found at the website of the EU project LASAGNE, on multi-LAyer SpAtiotemporal Generalized NEtworks, of which I am the Scientific Coordinator.

Temporal Networks

Real-world networks are inherently dynamic, with the links fluctuating and changing over time. E.g., human contacts or relationships change over time because individuals lose old acquaintances, acquire new ones, or move over geographic space. Despite this fact, most of the classic studies on complex networks are based on the analysis of static (aggregated) graphs, as if the links were all concurrent in time. In order to capture the real dynamic behavior and time correlations of temporal networks, we describe them as time-varying graphs, i.e. ensembles of time-ordered graphs. We have extended to time-varying graphs various metrics and models originally developed for static graphs. In collaboration with C. Mascolo (Univ. of Cambridge), M. Musolesi (Univ. of Birmingham) and G. Russo (Univ. of Catania) we have introduced the concepts of TEMPORAL SHORTEST PATHS [PRE10] and CONNECTED COMPONENTS [Chaos12] in time-varying graphs, and we have defined as TEMPORAL SMALL WORLD a time-varying graph in which the links are highly clustered in time, yet the nodes are at small average temporal distances [PRE10]. We have explored the small-world behavior in synthetic time-varying networks of mobile agents, and in real social and biological time-varying systems, and we have also shown how to exploit TEMPORAL CENTRALITY MEASURES to contain mobile phone viruses that spread via Bluetooth contacts [arXiv10] (MIT Tech Review). More on our work can be found in two Chapters we have recently wrote for a new book on temporal networks [Spr13] [Spr13_2].

Random Walks on Complex Networks

Random walks are the simplest way to explore a graph. In [PRL08], in collaboration with J. Gomez-Gardenes and Y. Moreno (Bifi, Zaragoza) we have studied traffic in a complex network by modeling packets of information as non-interacting random walkers. The model is amenable to analytical solution and, notwithstanding its simplicity, is able to capture the essential ingredients determining the SCALING OF FLUCTUATIONS empirically observed for traffic flow in the Internet and in other real networks. In [EPL14] we have shown how to characterize the structure of a network by studying the properties of the trajectories of unbiased random walks over it. In [PRE08] we have introduced the DYNAMICAL ENTROPY of the trajectory of DEGREE-BIASED RANDOM WALKS. In such walkers, the probability of moving to a node depends on some power of the degree of the target node, and depending on whether the exponent is positive or negative, this can give rise to walks that favor or disfavor high-degree vertices [PRE14]. In collaboration with R. Sinatra (Northeastern University, USA) and R. Lambiotte (Univ. Namur) we have shown that, by opportunely tuning the value of the exponent, we can get diffusion processes with local rules and maximal entropy rate [PRE11], and we can even induce the emergence of synchronization in mobile Kuramoto oscillators [PRE13]. Turning on the interactions, in collaboration with F. Bagnoli (Univ. Florence) we have introduced and studied a model of interacting random walks competing for the nodes of a complex network [EPL11], while in collaboration with S. Thurner (Univ. Vienna) we have studied and modeled the motion on a network of a real socio-economic system [NatSR12].

Synchronization in Complex Networks

As another example of interesting dynamic process, we have studied the emergence of collective synchronized dynamics in NETWORKS OF COUPLED DYNAMICAL SYSTEMS, with the main focus on the interplay between the network topology and the features of the network dynamics. In particular, with S. Boccaletti (CNR, IT) we have addressed the problem of understanding the variable abundance of different MOTIFS by means of the MASTER STABILITY FUNCTION, an analytic method to measure the stability of a synchronous state on a graph. We have found that, for undirected graphs, the stability of the synchronous state is positively correlated with the relative motif abundance, while in directed graphs the correlation exists only for some specific motifs [EPL07]. Furthermore, in collaboration with A. Rapisarda (Univ. of Catania) we have introduced a method for the detection and identification of COMMUNITY STRUCTURES in complex networks, based on the formation of synchronized groups of dynamical units [PRE07], and we have tested the method on computer generated and real-world networks whose modular structure is already known or has been studied by means of other methods [EPJ08]. In collaboration with J. Gomez-Gardenes (Univ. Zaragoza, SP), in [NatSR11] and [PRL11], we have studied synchronization in ADAPTIVE NETWORKS, i.e. in networks whose structure coevolves with the dynamic state of the nodes (based on the two principles of homophily and homeostasis). More recently, with A. Diaz-Guilera (Univ. Barcelona, SP), and M. Chavez (Lab. de Neuroscience, CNRS, Paris), we have discovered a novel phenomenon of REMOTE SYNCHRONIZATION, where pairs of nodes with the same network symmetry are fully synchronized, despite their distance on the graph [PRL13].

APPLICATIONS 1: Complex Networks and the Brain

Recent developments in neuroimaging, including structural and functional magnetic resonance imaging (MRI), magnetoencephalography (MEG), and electroencephalography (EEG) have provided the possibility to study human brain at a global scale as a complex network. In collaboration with the experimental group of F. Babiloni (Univ. Roma La Sapienza) we have studied the topological properties of functional connectivity patterns among different cortical areas of the human brain. The networks were obtained from high-resolution EEG recordings in a set of SPINAL CORD INJURED PATIENTS during the preparation of a limb movement [IJBC09], [JPA08], and in couples of individuals playing an Iterated Prisoner's Dilemma game [PLoS10]. In particular, in the latter paper we showed how it is possible to predict human cooperative behaviors from the analysis of the so-called HYPERBRAIN NETWORKS, i.e. networks representing the connections among the areas of two distinct brains. In collaboration with M. Chavez (Lab. de Neurosciences Cognitives and Imagerie Cerebrale, CNRS, Paris) we have found that the modular structure of weighted brain networks extracted from MEG signals of EPILEPTIC PATIENTS recorded at rest, and far from the absence seizures, is intrinsically different from that of healthy subjects [PRL10] (On the cover page of PRL) . Neural systems can also be studied at the level of physical connections between neurons. With E. Bullmore and P. Vertes (Univ. of Cambridge) we have studied the nervous system of the nematode worm CAENORHABDITIS ELEGANS, together with information on the birth times of neurons and on their spatial locations, finding one of the first evidence of a PHASE TRANSITION IN THE GROWTH OF A SPATIAL NETWORK [PNAS13].

APPLICATIONS 2: The Science of Cities

Everyone knows that a place which is central has some special features to offer in many ways to those who live or work in cities: it is more visible, more accessible from the immediate surroundings as well as from far away, it is more popular in terms of people walking around and potential customers, it has a greater probability to develop as an urban landmark and a social catalyst, or offer first level functions like theatres or office headquarters as well as a larger diversity of opportunities and goods. In a long-term joint project with the Urban Design group of S. Porta (Univ. of Strathclyde), our aim is the development of new tools for the network analysis of urban spatial systems. One of such tools, named the MULTIPLE CENTRALITY ASSESSMENT (MCA), allows for mapping centrality in urban spaces [EPB06], [PhysA06], and establishing correlations with relevant dynamics such as land-use [EPB09], vehicular or pedestrian flows and crime rates. MCA has also been used for the statistical characterization of different types of urban fabrics taken from the history of cities, in order to infer relationships between them in an urban evolutionary perspective [PRE06], [PRE06]. Our latest research is mainly oriented to the definition of procedures, attitudes and tools for sustainable/human/adaptive urban analysis and design, ranging from GIS-based space analysis to sustainable community design [UDI08], transportation planning and traffic calming techniques [EPJB10], to strategies for safety and live-ability in the public domain. Much more on our work can be found at the website of the Urban Design Study Unit, now based at the Department of Architecture of Strathclyde University, Glasgow. A recent study in collaboration with M. Barthelemy (CNRS, Paris) on the ELEMENTARY PROCESSES GOVERNING URBANIZATION [NatSR12] has received press coverage in Nature, Scientific American and Le Scienze, and wide diffusion in forums on urban design.

Damage and Disease Spreading in Complex Networks

In collaboration with M. Marchiori (W3C MIT and University of Padova) we have first studied the resistance of scale-free networks to the removal of a certain percentage of their nodes [PA03]. Then, we have developed one of the first MODELS FOR CASCADING FAILURES in complex networks based on the dynamical redistribution of flows on the network, showing that the breakdown of a single node is sufficient to collapse the efficiency of the entire system if the node is initially among the ones with largest load [PRE04]. Together with R. Albert (Penn State Univ, USA) we have then applied our model of cascading failure to predict BLACKOUTS in the North American power grid [EPJB05]. With J. Gomez-Gardenes and Y. Moreno (Bifi, Zaragoza) we have studied the spreading of SEXUALLY TRANSMITTED DISEASES on bipartite scale-free graphs, representing heterosexual contact networks, deriving analytically the expression for the epidemic threshold and its dependence with the system size in finite populations [PNAS08]. With J. Gomez-Gardenes and L. Fortuna and M. Frasca we have studied the effect of movements on disease spreading [EPL08].

Basic Structural Properties of Complex Networks

The connection topology of many biological, technological and social networks is neither completely regular nor completely random. These networks, named small worlds, are in fact highly clustered, like regular lattices, yet having small characteristics path lengths, like random graphs. In collaboration with M. Marchiori (W3C MIT and University of Padova) we have proposed a new theory of the small-world behavior based on the concept of information transport over the network and also valid for weighted networks [PRL01] (Press coverage). Our formalism is based on the definition of the EFFICIENCY of a network, which measures how well the network nodes exchange information. By using this measure both at a a global and at a local scale, small-world networks result as systems that are both globally and locally efficient. We have performed precise quantitative analysis of neural networks such as the C. elegans nervous system, and two databases of cortico-cortical connections in the macaque and in the cat, and transportation systems [PhysA02], also studying the small-world behavior in connection to the COST of a network [EPJB03]. The concept of efficiency can be used to introduce a completely new idea of centrality, that we have named DELTA CENTRALITY, that applies to groups, as well as individual nodes and links [NJP07], and also allows to quantify the relevance of different mediators in the HUMAN IMMUNE CELL NETWORK [BIO05], to find the critical components of CRITICAL INFRASTRUCTURE NETWORKS [PRE05] [FNL05] and to develop strategies to protect from terrorist attacks [CSF04].