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Sanz J.,University of Zaragoza | Xia C.-Y.,University of Zaragoza | Xia C.-Y.,Tianjin University of Technology | Meloni S.,University of Zaragoza | And 2 more authors.
Physical Review X

Current modeling of infectious diseases allows for the study of complex and realistic scenarios that go from the population to the individual level of description. However, most epidemic models assume that the spreading process takes place on a single level (be it a single population, a metapopulation system, or a network of contacts). In particular, interdependent contagion phenomena can be addressed only if we go beyond the scheme-one pathogen-one network. In this paper, we propose a framework that allows us to describe the spreading dynamics of two concurrent diseases. Specifically, we characterize analytically the epidemic thresholds of the two diseases for different scenarios and compute the temporal evolution characterizing the unfolding dynamics. Results show that there are regions of the parameter space in which the onset of a disease's outbreak is conditioned to the prevalence levels of the other disease. Moreover, we show, for the susceptible-infected-susceptible scheme, that under certain circumstances, finite and not vanishing epidemic thresholds are found even at the limit for scale-free networks. For the susceptibleinfected-removed scenario, the phenomenology is richer and additional interdependencies show up. We also find that the secondary thresholds for the susceptible-infected-susceptible and susceptibleinfected-removed models are different, which results directly from the interaction between both diseases. Our work thus solves an important problem and paves the way toward a more comprehensive description of the dynamics of interacting diseases. Source

De Domenico M.,Rovira i Virgili University | Sole-Ribalta A.,Rovira i Virgili University | Cozzo E.,University of Zaragoza | Kivela M.,Oxford Centre for Industrial and Applied Mathematics | And 5 more authors.
Physical Review X

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 are 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 multilayer complex systems. In this paper, we introduce a tensorial framework to study multilayer 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 multilayer 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. Source

Fortunato S.,Lagrange Systems
Physics Reports

The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i.e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such clusters, or communities, can be considered as fairly independent compartments of a graph, playing a similar role like, e.g., the tissues or the organs in the human body. Detecting communities is of great importance in sociology, biology and computer science, disciplines where systems are often represented as graphs. This problem is very hard and not yet satisfactorily solved, despite the huge effort of a large interdisciplinary community of scientists working on it over the past few years. We will attempt a thorough exposition of the topic, from the definition of the main elements of the problem, to the presentation of most methods developed, with a special focus on techniques designed by statistical physicists, from the discussion of crucial issues like the significance of clustering and how methods should be tested and compared against each other, to the description of applications to real networks. © 2009 Elsevier B.V. Source

Perc M.,University of Maribor | Gomez-Gardenes J.,University of Zaragoza | Szolnoki A.,Hungarian Academy of Sciences | Floria L.M.,University of Zaragoza | And 2 more authors.
Journal of the Royal Society Interface

Interactions among living organisms, from bacteria colonies to human societies, are inherently more complex than interactions among particles and non-living matter. Group interactions are a particularly important and widespread class, representative of which is the public goods game. In addition, methods of statistical physics have proved valuable for studying pattern formation, equilibrium selection and self-organization in evolutionary games. Here, we review recent advances in the study of evolutionary dynamics of group interactions on top of structured populations, including lattices, complex networks and coevolutionary models. We also compare these results with those obtained on well-mixed populations. The review particularly highlights that the study of the dynamics of group interactions, like several other important equilibrium and non-equilibrium dynamical processes in biological, economical and social sciences, benefits from the synergy between statistical physics, network science and evolutionary game theory. © 2013 The Authors. Source

Lancichinetti A.,Lagrange Systems | Lancichinetti A.,Polytechnic University of Turin | Fortunato S.,Lagrange Systems | Fortunato S.,Aalto University
Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Modularity maximization is the most popular technique for the detection of community structure in graphs. The resolution limit of the method is supposedly solvable with the introduction of modified versions of the measure, with tunable resolution parameters. We show that multiresolution modularity suffers from two opposite coexisting problems: the tendency to merge small subgraphs, which dominates when the resolution is low; the tendency to split large subgraphs, which dominates when the resolution is high. In benchmark networks with heterogeneous distributions of cluster sizes, the simultaneous elimination of both biases is not possible and multiresolution modularity is not capable to recover the planted community structure, not even when it is pronounced and easily detectable by other methods, for any value of the resolution parameter. This holds for other multiresolution techniques and it is likely to be a general problem of methods based on global optimization. © 2011 American Physical Society. Source

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