Synchronization processes in populations of locally interacting elements are in the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understand synchronization phenomena in natural systems take now advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also overview the new emergent features coming out from the interplay between the structure and the function of the underlying pattern of connections. Extensive numerical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to different disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences.Comment: Final version published in Physics Reports. More information available at http://synchronets.googlepages.com

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We compare recent approaches to community structure identification in terms of sensitivity and computational cost. The recently proposed modularity measure is revisited and the performance of the methods as applied to ad hoc networks with known community structure, is compared. We find that the most accurate methods tend to be more computationally expensive, and that both aspects need to be considered when choosing a method for practical purposes. The work is intended as an introduction as well as a proposal for a standard benchmark test of community detection methods.

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.

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.

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.

We study the relationship between topological scales and dynamic time scales in complex networks. The analysis is based on the full dynamics towards synchronization of a system of coupled oscillators. In the synchronization process, modular structures corresponding to well-defined communities of nodes emerge in different time scales, ordered in a hierarchical way. The analysis also provides a useful connection between synchronization dynamics, complex networks topology, and spectral graph analysis.

We propose a procedure for analyzing and characterizing complex networks. We apply this to the social network as constructed from email communications within a medium sized university with about 1700 employees. Email networks provide an accurate and nonintrusive description of the flow of information within human organizations. Our results reveal the self-organization of the network into a state where the distribution of community sizes is self-similar. This suggests that a universal mechanism, responsible for emergence of scaling in other self-organized complex systems, as, for instance, river networks, could also be the underlying driving force in the formation and evolution of social networks. DOI: 10.1103/PhysRevE.68.065103 PACS number͑s͒: 89.75.Fb, 89.75.Da, 89.75.Hc Signatures of complex systems appear in disciplines as diverse as biology, chemistry, economy, and computer science, to name just a few. More specifically, the study of the complex networks of interactions in such systems has received a lot of attention from the statistical physics community ͓1-5͔. The structure of these complex networks is a reflection of the dynamics of their formation and evolution, and can be partially characterized using statistical observables such as the average distance between nodes ͓1͔, the clustering coefficient ͓1͔, and the degree distribution ͓2,3͔. Even though these measures are very useful in some situations, often they are not sufficient to describe key features of networks. In the specific field of social sciences, a more detailed description of human interactions is crucial to understand the formation and evolution of complex social networks.In this paper we describe a procedure to characterize the structure of networks, based on a recently proposed algorithm to identify communities in graphs ͓6͔. Our procedure allows one to study quantitatively the hierarchical structure of nested communities in networks. Moreover we apply the procedure to a real social network. We define and analyze the complex email network of an organization with about 1700 employees and determine its community structure. Our results reveal that this network self-organizes into a selfsimilar structure, suggesting that some universal mechanism could be the underlying driving force in the formation and evolution of social networks, as happens in other complex systems ͓7,8͔.Apart from work related reasons, ties between individuals in any organization arise, without external influence, due to personal, political, and cultural reasons, among others. The rapid development of electronic communications provides a powerful tool to analyze the informal self-organized social network arising as a result of the formation of such ties. Indeed, every time an email is sent, the addresses of the sender and the receiver are routinely registered in a server. Therefore, an email network can be built regarding each email address as a node and linking two nodes if there is an email communication between them. We take as a case study the email network of Uni...