In the past years, network theory has successfully characterized the interaction among the constituents of a variety of complex systems, ranging from biological to technological, and social systems. However, up until recently, attention was almost exclusively given to networks in which all components were treated on equivalent footing, while neglecting all the extra information about the temporal-or context-related properties of the interactions under study. Only in the last years, taking advantage of the enhanced resolution in real data sets, network scientists have directed their interest to the multiplex character of real-world systems, and explicitly considered the time-varying and multilayer nature of networks. We offer here a comprehensive review on both structural and dynamical organization of graphs made of diverse relationships (layers) between its constituents, and cover several relevant issues, from a full redefinition of the basic structural measures, to understanding how the multilayer nature of the network affects processes and dynamics.

The rate at which nodes in a network increase their connectivity depends on their fitness to compete for links. For example, in social networks some individuals acquire more social links than others, or on the www some webpages attract considerably more links than others. We find that this competition for links translates into multiscaling, i.e. a fitness dependent dynamic exponent, allowing fitter nodes to overcome the more connected but less fit ones. Uncovering this fitter-gets-richer phenomena can help us understand in quantitative terms the evolution of many competitive systems in nature and society.PACS numbers: 5.65+b, 89. 89.75Fb, 89.75Hc. Typeset using REVT E X The complexity of many systems can be attributed to the interwoven web in which their constituents interact with each other. For example, the society is organized in a social web, whose nodes are individuals and links represent various social interactions, or the www forms a complex web whose nodes are documents and links are URLs. While for a long time these networks have been modeled as completely random [1,2], recently there is increasing evidence that they in fact have a number of generic non-random characteristics, obeying various scaling laws or displaying short length-scale clustering [3][4][5][6][7][8][9][10][11][12][13][14][15][16].A generic property of these complex systems is that they constantly evolve in time.This implies that the underlying networks are not static, but continuously change through the addition and/or removal of new nodes and links. Such evolving networks characterize the society, thanks to the birth and death of nodes and their constant acquisition of social links; or characterize the www, where the number of nodes increases exponentially and the links connecting them are constantly modified [3,6,7]. Consequently, in addressing these complex systems we have to uncover the dynamical forces that act at the level of individual nodes, whose cumulative effect determines the system's large-scale topology. A first step in this direction was the introduction of the scale-free model [8], that incorporates the fact that network evolution is driven by at least two coexisting mechanisms: (1) growth, implying that networks continuously expand by the addition of new nodes that attach to the nodes already present in the system; (2) preferential attachment, mimicking the fact that a new node links with higher probability to the nodes that already have a large number of links. With these two ingredients the scale-free model predicts the emergence of a power-law connectivity distribution, observed in many systems [3,[8][9][10], ranging from the Internet to citation networks. Furthermore, extensions of this model, including rewiring [11] or aging [12,13] have been able to account for more realistic aspects of the network evolution, such as the existence of various scaling exponents or cutoffs in the connectivity distribution.Despite its success in predicting the large-scale topology of real networks, the scalefree model neglects an ...

Brain function operates through the coordinated activation of neuronal assemblies. Graph theory predicts that scale-free topologies, which include "hubs" (superconnected nodes), are an effective design to orchestrate synchronization. Whether hubs are present in neuronal assemblies and coordinate network activity remains unknown. Using network dynamics imaging, online reconstruction of functional connectivity, and targeted whole-cell recordings in rats and mice, we found that developing hippocampal networks follow a scale-free topology, and we demonstrated the existence of functional hubs. Perturbation of a single hub influenced the entire network dynamics. Morphophysiological analysis revealed that hub cells are a subpopulation of gamma-aminobutyric acid-releasing (GABAergic) interneurons possessing widespread axonal arborizations. These findings establish a central role for GABAergic interneurons in shaping developing networks and help provide a conceptual framework for studying neuronal synchrony.

The evolution of many complex systems, including the World Wide Web, business, and citation networks, is encoded in the dynamic web describing the interactions between the system's constituents. Despite their irreversible and nonequilibrium nature these networks follow Bose statistics and can undergo Bose-Einstein condensation. Addressing the dynamical properties of these nonequilibrium systems within the framework of equilibrium quantum gases predicts that the "first-mover-advantage," "fit-get-rich," and "winner-takes-all" phenomena observed in competitive systems are thermodynamically distinct phases of the underlying evolving networks. DOI: 10.1103/PhysRevLett.86.5632 PACS numbers: 89.75.Hc, 03.75.Fi, 05.65.+b, 87.23.Ge Competition for links is a common feature of complex systems: on the World Wide Web (www) the sites compete for URLs to enhance their visibility [1], in the business world companies compete for links to consumers [2], and in the scientific community scientists and publications compete for citations, a measure of their impact on the field [3]. A common feature of these systems is that the nodes self-organize into a complex network, whose topology and evolution closely reflect the dynamics and outcome of this competition [1,[3][4][5][6]. Here we show that, despite their nonequilibrium and irreversible nature, evolving networks can be mapped into an equilibrium Bose gas [7], nodes corresponding to energy levels, and links representing particles. This mapping predicts that the common epithets used to characterize competitive systems, such as "winner takes all," "fit get rich" (FGR), or "first mover advantage," emerge naturally as thermodynamically and topologically distinct phases of the underlying complex evolving network. In particular, we predict that such networks can undergo Bose-Einstein (BE) condensation, in which a single node captures a macroscopic fraction of links.Fitness model [8]. -Consider a network that grows through the addition of new nodes such as the creation of new webpages, the emergence of new companies, or the publication of new papers. At each time step we add a new node, connecting it with m links to the nodes already present in the system. The rate at which nodes acquire links can vary widely as supported by measurements on the www [4], and by empirical evidence in citation [3] and economic networks. To incorporate the different ability of the nodes to compete for links we assign a fitness parameter to each node h, chosen from a distribution r͑h͒, accounting for the differences in the content of webpages, the quality of products and marketing of companies, or the importance of the findings reported in a publication. The probability P i that a new node connects one of its m links to a node i already present in the network depends on the number of links k i and on the fitness h i of node i, such thatEquation (1) incorporates in the simplest possible way the fact that new nodes link preferentially to nodes with higher k [9] (i.e., connecting to more visible websites, fav...

We introduce a general stochastic model for the spread of rumours, and derive meanfield equations that describe the dynamics of the model on complex social networks (in particular those mediated by the Internet). We use analytical and numerical solutions of these equations to examine the threshold behavior and dynamics of the model on several models of such networks: random graphs, uncorrelated scalefree networks and scale-free networks with assortative degree correlations. We show that in both homogeneous networks and random graphs the model exhibits a critical threshold in the rumour spreading rate below which a rumour cannot propagate in the system. In the case of scale-free networks, on the other hand, this threshold becomes vanishingly small in the limit of infinite system size. We find that the initial rate at which a rumour spreads is much higher in scale-free networks than in random graphs, and that the rate at which the spreading proceeds on scale-free networks is further increased when assortative degree correlations are introduced. The impact of degree correlations on the final fraction of nodes that ever hears a rumour, however, depends on the interplay between network topology and the rumour spreading rate. Our results show that scale-free social networks are prone to the spreading of rumours, just as they are to the spreading of infections. They are relevant to the spreading dynamics of chain emails, viral advertising and large-scale information dissemination algorithms on the Internet.

There is growing interest in multiplex networks where individual nodes take part in several layers of networks simultaneously. This is the case for example in social networks where each individual node has different kind of social ties or transportation systems where each location is connected to another location by different types of transport. Many of these multiplex are characterized by a significant overlap of the links in different layers. In this paper we introduce a statistical mechanics framework to describe multiplex ensembles. A multiplex is a system formed by N nodes and M layers of interactions where each node belongs to the M layers at the same time. Each layer α is formed by a network G α . Here we introduce the concept of correlated multiplex ensembles in which the existence of a link in one layer is correlated with the existence of a link in another layer. This implies that a typical multiplex of the ensemble can have a significant overlap of the links in the different layers. Moreover we characterize microcanonical and canonical multiplex ensembles satisfying respectively hard and soft constraints and we discuss how to construct multiplex in these ensembles. Finally we provide the expression for the entropy of these ensembles that can be useful to address different inference problems involving multiplexes.

The quantification of the complexity of networks is, today, a fundamental problem in the physics of complex systems. A possible roadmap to solve the problem is via extending key concepts of information theory to networks. In this paper we propose how to define the Shannon entropy of a network ensemble and how it relates to the Gibbs and von Neumann entropies of network ensembles. The quantities we introduce here will play a crucial role for the formulation of null models of networks through maximum-entropy arguments and will contribute to inference problems emerging in the field of complex networks.PACS numbers: 89.75. Hc, 89.75.Fb, 89.75.Da Complex networks [1,2,3,4] are found to characterize the underlying structure of many biological, social and technological systems. Following ten years of active research in the field of complex networks, the state of the art includes, a deep understanding of their evolution [1], an unveiling of the rich interplay between network topology and dynamics [3] and a description of networks through structural characteristics [2,4]. Nevertheless, we still lack the means to quantify, how complex is a complex network. In order to answer this question we need a new theory of information of complex networks. This new theory will contribute to solving many challenging inference problems in the field [4,5,6]. By providing an evaluation of the information encoded in complex networks, this will resolve one of the outstanding problems in the statistical mechanics of complex systems.In information theory [7] entropy measures play a key role. In fact, it is well known that the Shannon entropy and the von Neumann entropy are related to the information present in classical and quantum systems, respectively. Moreover, the afore mentioned measures also have statistical mechanics interpretations. Traditionally, in statistical mechanics, for configurations drawn from canonical ensembles, the Shannon entropy corresponds to the entropy for classical systems, while the von Neumann entropy provides the statistical description of quantum systems.In the context of complex networks a number of different entropy measures have been introduced [5,8,9,10,11,12,13]. In Ref.[9] the Gibbs entropy per node, in a network of N nodes, denoted Σ, was introduced for microcanonical network ensembles following a statistical mechanics paradigm. Microcanonical network ensembles are defined as those networks that satisfy a given set of constraints. Examples of some popular constraints include, fixed number of links per node, given degree sequence and community structure. The Gibbs entropy of these ensembles is given bywhere N indicates the cardinality of the ensemble, i.e., the total number of networks in the ensemble. As demonstrated further in [9] the statistical mechanics formalism enables us to develop canonical network ensembles where the structural constraints under consideration are satisfied, on average. In classical statistical mechanics the microcanonical ensemble is formed by configurations having constant en...

PACS 89.75-k -Complex systems PACS 89.75.Fb -Structure and organization in complex systems PACS 89.75.Hc -Networks and genealogical trees Abstract. -Randomized network ensembles are the null models of real networks and are extensively used to compare a real system to a null hypothesis. In this paper we study network ensembles with the same degree distribution, the same degree-correlations and the same community structure of any given real network. We characterize these randomized network ensembles by their entropy, i.e. the normalized logarithm of the total number of networks which are part of these ensembles. We estimate the entropy of randomized ensembles starting from a large set of real directed and undirected networks. We propose entropy as an indicator to assess the role of each structural feature in a given real network.We observe that the ensembles with fixed scale-free degree distribution have smaller entropy than the ensembles with homogeneous degree distribution indicating a higher level of order in scale-free networks.