Leitura de Férias no Review of Modern Physics
Departamento de Física, Universidade de Aveiro, 3810-193 Aveiro, Portugal and A. F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia
J. F. F. MendesDepartamento de Física, Universidade de Aveiro, 3810-193 Aveiro, Portugal
Published 6 October 2008
The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, important steps have been made toward understanding the qualitatively new critical phenomena in complex networks. The results, concepts, and methods of this rapidly developing field are reviewed. Two closely related classes of these critical phenomena are considered, namely, structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. Systems where a network and interacting agents on it influence each other are also discussed. A wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, k-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks are mentioned. Strong finite-size effects in these systems and open problems and perspectives are also discussed.
Published 6 October 2008
The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, important steps have been made toward understanding the qualitatively new critical phenomena in complex networks. The results, concepts, and methods of this rapidly developing field are reviewed. Two closely related classes of these critical phenomena are considered, namely, structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. Systems where a network and interacting agents on it influence each other are also discussed. A wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, k-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks are mentioned. Strong finite-size effects in these systems and open problems and perspectives are also discussed.
Statistical physics of social dynamics
Claudio Castellano, Santo Fortunato and Vittorio Loreto
Statistical physics has proven to be a very fruitful framework to describe phenomena outside the realm of traditional physics. The last years have witnessed the attempt by physicists to study collective phenomena emerging from the interactions of individuals as elementary units in social structures. Here we review the state of the art by focusing on a wide list of topics ranging from opinion, cultural and language dynamics to crowd behavior, hierarchy formation, human dynamics, social spreading. We highlight the connections between these problems and other, more traditional, topics of statistical physics. We also emphasize the comparison of model results with empirical data from social systems.
Accepted Mon Nov 3, 2008
Claudio Castellano, Santo Fortunato and Vittorio Loreto
Statistical physics has proven to be a very fruitful framework to describe phenomena outside the realm of traditional physics. The last years have witnessed the attempt by physicists to study collective phenomena emerging from the interactions of individuals as elementary units in social structures. Here we review the state of the art by focusing on a wide list of topics ranging from opinion, cultural and language dynamics to crowd behavior, hierarchy formation, human dynamics, social spreading. We highlight the connections between these problems and other, more traditional, topics of statistical physics. We also emphasize the comparison of model results with empirical data from social systems.
Accepted Mon Nov 3, 2008
Fractal structures in nonlinear dynamics
Jacobo Aguirre, Ricardo L. Viana and Miguel A. F. Sanjuan
Besides the striking beauty inherent to their complex nature, fractals have become a fundamental ingredient of nonlinear dynamics and chaos theory since they were defined in the 1970s. Moreover, fractals have been detected in nature, in most fields of science, with even a certain influence in arts. Fractal structures appear naturally in dynamical systems, in particular associated to the phase space. The analysis of these structures is especially useful for obtaining information about the future behavior of complex systems, since they provide fundamental knowledge about their relation with uncertainty and indeterminism. Dynamical systems are divided in two main groups, Hamiltonian and dissipative systems. The concepts of attractor and basin of attraction are related to dissipative systems. In the case of open Hamiltonian systems, there are no attractors, but we have the analogous concepts of exit and exit basin. Therefore, basins formed by initial conditions can be computed both in Hamiltonian and dissipative systems, being some of them smooth and some of them fractal. This fact has fundamental consequences in our ability to predict the future of the system. The existence of this deterministic unpredictability, usually known as final state sensitivity, is typical of chaotic systems, and makes deterministic systems become, in practice, random processes where only a probabilistic approach is possible. The main types of fractal basins, their nature, and the numerical and experimental techniques used to obtain them both from mathematical models and real phenomena are described here, with special attention to their ubiquity in different fields of physics.
Accepted Fri Oct 3, 2008
Jacobo Aguirre, Ricardo L. Viana and Miguel A. F. Sanjuan
Besides the striking beauty inherent to their complex nature, fractals have become a fundamental ingredient of nonlinear dynamics and chaos theory since they were defined in the 1970s. Moreover, fractals have been detected in nature, in most fields of science, with even a certain influence in arts. Fractal structures appear naturally in dynamical systems, in particular associated to the phase space. The analysis of these structures is especially useful for obtaining information about the future behavior of complex systems, since they provide fundamental knowledge about their relation with uncertainty and indeterminism. Dynamical systems are divided in two main groups, Hamiltonian and dissipative systems. The concepts of attractor and basin of attraction are related to dissipative systems. In the case of open Hamiltonian systems, there are no attractors, but we have the analogous concepts of exit and exit basin. Therefore, basins formed by initial conditions can be computed both in Hamiltonian and dissipative systems, being some of them smooth and some of them fractal. This fact has fundamental consequences in our ability to predict the future of the system. The existence of this deterministic unpredictability, usually known as final state sensitivity, is typical of chaotic systems, and makes deterministic systems become, in practice, random processes where only a probabilistic approach is possible. The main types of fractal basins, their nature, and the numerical and experimental techniques used to obtain them both from mathematical models and real phenomena are described here, with special attention to their ubiquity in different fields of physics.
Accepted Fri Oct 3, 2008
Figura: The Internet is a scale-free network in that some sites have a seemingly unlimited number of connections to other sites. This map, made on February 6, 2003, traces the shortest routes from a test Web site to about 100,000 others, using like colors for similar Web addresses. (Image credit: Internet Mapping Project of Lumeta Corporation; Legend credit: Scientific American)
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