# Showing papers in "Physical Review E in 2002"

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TL;DR: This paper shows that a large class of standard epidemiological models, the so-called susceptible/infective/removed (SIR) models can be solved exactly on a wide variety of networks.

Abstract: The study of social networks, and in particular the spread of disease on networks, has attracted considerable recent attention in the physics community. In this paper, we show that a large class of standard epidemiological models, the so-called susceptible/infective/removed (SIR) models can be solved exactly on a wide variety of networks. In addition to the standard but unrealistic case of fixed infectiveness time and fixed and uncorrelated probability of transmission between all pairs of individuals, we solve cases in which times and probabilities are nonuniform and correlated. We also consider one simple case of an epidemic in a structured population, that of a sexually transmitted disease in a population divided into men and women. We confirm the correctness of our exact solutions with numerical simulations of SIR epidemics on networks.

2,872 citations

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TL;DR: It is shown that discrete lattice effects must be considered in the introduction of a force into the lattice Boltzmann equation, and a representation of the forcing term is proposed that derived the Navier-Stokes equation through the Chapman-Enskog expansion.

Abstract: We show that discrete lattice effects must be considered in the introduction of a force into the lattice Boltzmann equation. A representation of the forcing term is then proposed. With the representation, the Navier-Stokes equation is derived from the lattice Boltzmann equation through the Chapman-Enskog expansion. Several other existing force treatments are also examined.

1,518 citations

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TL;DR: It is found that the removals by the recalculated degrees and betweenness centralities are often more harmful than the attack strategies based on the initial network, suggesting that the network structure changes as important vertices or edges are removed.

Abstract: We study the response of complex networks subject to attacks on vertices and edges. Several existing complex network models as well as real-world networks of scientific collaborations and Internet traffic are numerically investigated, and the network performance is quantitatively measured by the average inverse geodesic length and the size of the largest connected subgraph. For each case of attacks on vertices and edges, four different attacking strategies are used: removals by the descending order of the degree and the betweenness centrality, calculated for either the initial network or the current network during the removal procedure. It is found that the removals by the recalculated degrees and betweenness centralities are often more harmful than the attack strategies based on the initial network, suggesting that the network structure changes as important vertices or edges are removed. Furthermore, the correlation between the betweenness centrality and the degree in complex networks is studied.

1,506 citations

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TL;DR: It is demonstrated that the heterogeneity of these networks makes them particularly vulnerable to attacks in that a large-scale cascade may be triggered by disabling a single key node.

Abstract: We live in a modern world supported by large, complex networks. Examples range from financial markets to communication and transportation systems. In many realistic situations the flow of physical quantities in the network, as characterized by the loads on nodes, is important. We show that for such networks where loads can redistribute among the nodes, intentional attacks can lead to a cascade of overload failures, which can in turn cause the entire or a substantial part of the network to collapse. This is relevant for real-world networks that possess a highly heterogeneous distribution of loads, such as the Internet and power grids. We demonstrate that the heterogeneity of these networks makes them particularly vulnerable to attacks in that a large-scale cascade may be triggered by disabling a single key node. This brings obvious concerns on the security of such systems.

1,428 citations

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TL;DR: The Hermiticity of the fractional Hamilton operator and the parity conservation law for fractional quantum mechanics are established and the energy spectra of a hydrogenlike atom and of a fractional oscillator in the semiclassical approximation are found.

Abstract: Some properties of the fractional Schrodinger equation are studied. We prove the Hermiticity of the fractional Hamilton operator and establish the parity conservation law for fractional quantum mechanics. As physical applications of the fractional Schrodinger equation we find the energy spectra of a hydrogenlike atom (fractional "Bohr atom") and of a fractional oscillator in the semiclassical approximation. An equation for the fractional probability current density is developed and discussed. We also discuss the relationships between the fractional and standard Schrodinger equations.

1,190 citations

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TL;DR: An analytical expression for the cluster coefficient is derived, which shows that the graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality are distinctly different from standard random graphs, even for infinite dimensionality.

Abstract: We analyze graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present. The critical connectivity is found numerically by examining the size of the largest cluster. We derive an analytical expression for the cluster coefficient, which shows that the graphs are distinctly different from standard random graphs, even for infinite dimensionality. Insights relevant for graph bipartitioning are included.

1,106 citations

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TL;DR: In this paper, the authors show that random uniform immunization of individuals does not lead to the eradication of infections in all complex networks, and that the absence of any critical immunization threshold is due to the unbounded connectivity fluctuations of scale-free networks.

Abstract: Complex networks such as the sexual partnership web or the Internet often show a high degree of redundancy and heterogeneity in their connectivity properties. This peculiar connectivity provides an ideal environment for the spreading of infective agents. Here we show that the random uniform immunization of individuals does not lead to the eradication of infections in all complex networks. Namely, networks with scale-free properties do not acquire global immunity from major epidemic outbreaks even in the presence of unrealistically high densities of randomly immunized individuals. The absence of any critical immunization threshold is due to the unbounded connectivity fluctuations of scale-free networks. Successful immunization strategies can be developed only by taking into account the inhomogeneous connectivity properties of scale-free networks. In particular, targeted immunization schemes, based on the nodes' connectivity hierarchy, sharply lower the network's vulnerability to epidemic attacks.

990 citations

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Umeå University

^{1}TL;DR: The standard scale-free network model is extended to include a "triad formation step" and the clustering coefficient is shown to be tunable simply by changing a control parameter---the average number of triad formation trials per time step.

Abstract: We extend the standard scale-free network model to include a "triad formation step." We analyze the geometric properties of networks generated by this algorithm both analytically and by numerical calculations, and find that our model possesses the same characteristics as the standard scale-free networks such as the power-law degree distribution and the small average geodesic length, but with the high clustering at the same time. In our model, the clustering coefficient is also shown to be tunable simply by changing a control parameter---the average number of triad formation trials per time step.

930 citations

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TL;DR: The resulting network exhibits a scale-free link distribution and pronounced small-world behavior, as observed in other social networks, implying that the spreading of e-mail viruses is greatly facilitated in real e- mail networks compared to random architectures.

Abstract: We study the topology of e-mail networks with e-mail addresses as nodes and e-mails as links using data from server log files. The resulting network exhibits a scale-free link distribution and pronounced small-world behavior, as observed in other social networks. These observations imply that the spreading of e-mail viruses is greatly facilitated in real e-mail networks compared to random architectures.

927 citations

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Boston College

^{1}, Boston University^{2}, Harvard University^{3}, Lund University^{4}, Max Planck Society^{5}TL;DR: A analysis of cross correlations between price fluctuations of different stocks using methods of random matrix theory finds that the largest eigenvalue corresponds to an influence common to all stocks, and discusses applications to the construction of portfolios of stocks that have a stable ratio of risk to return.

Abstract: We analyze cross correlations between price fluctuations of different stocks using methods of random matrix theory (RMT). Using two large databases, we calculate cross-correlation matrices C of returns constructed from (i) 30-min returns of 1000 US stocks for the 2-yr period 1994-1995, (ii) 30-min returns of 881 US stocks for the 2-yr period 1996-1997, and (iii) 1-day returns of 422 US stocks for the 35-yr period 1962-1996. We test the statistics of the eigenvalues lambda(i) of C against a "null hypothesis" - a random correlation matrix constructed from mutually uncorrelated time series. We find that a majority of the eigenvalues of C fall within the RMT bounds [lambda(-),lambda(+)] for the eigenvalues of random correlation matrices. We test the eigenvalues of C within the RMT bound for universal properties of random matrices and find good agreement with the results for the Gaussian orthogonal ensemble of random matrices-implying a large degree of randomness in the measured cross-correlation coefficients. Further, we find that the distribution of eigenvector components for the eigenvectors corresponding to the eigenvalues outside the RMT bound display systematic deviations from the RMT prediction. In addition, we find that these "deviating eigenvectors" are stable in time. We analyze the components of the deviating eigenvectors and find that the largest eigenvalue corresponds to an influence common to all stocks. Our analysis of the remaining deviating eigenvectors shows distinct groups, whose identities correspond to conventionally identified business sectors. Finally, we discuss applications to the construction of portfolios of stocks that have a stable ratio of risk to return. (Less)

846 citations

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TL;DR: In this article, the effects of three types of non-stationarities often encountered in real data were studied. And the authors compared the difference between the scaling results obtained for stationary correlated signals and correlated signals with these three types and showed how the characteristics of these crossovers depend on the fraction and size of the parts cut out from the signal, the concentration of spikes and their amplitudes.

Abstract: Detrended fluctuation analysis ~DFA! is a scaling analysis method used to quantify long-range power-law correlations in signals. Many physical and biological signals are ‘‘noisy,’’ heterogeneous, and exhibit different types of nonstationarities, which can affect the correlation properties of these signals. We systematically study the effects of three types of nonstationarities often encountered in real data. Specifically, we consider nonstationary sequences formed in three ways: ~i! stitching together segments of data obtained from discontinuous experimental recordings, or removing some noisy and unreliable parts from continuous recordings and stitching together the remaining parts—a ‘‘cutting’’ procedure commonly used in preparing data prior to signal analysis; ~ii! adding to a signal with known correlations a tunable concentration of random outliers or spikes with different amplitudes; and ~iii! generating a signal comprised of segments with different properties—e.g., different standard deviations or different correlation exponents. We compare the difference between the scaling results obtained for stationary correlated signals and correlated signals with these three types of nonstationarities. We find that introducing nonstationarities to stationary correlated signals leads to the appearance of crossovers in the scaling behavior and we study how the characteristics of these crossovers depend on ~a! the fraction and size of the parts cut out from the signal, ~b! the concentration of spikes and their amplitudes ~c! the proportion between segments with different standard deviations or different correlations and ~d! the correlation properties of the stationary signal. We show how to develop strategies for preprocessing ‘‘raw’’ data prior to analysis, which will minimize the effects of nonstationarities on the scaling properties of the data, and how to interpret the results of DFA for complex signals with different local characteristics.

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TL;DR: Applying measures of complexity based on vertical structures in recurrence plots and applying them to the logistic map as well as to heart-rate-variability data is able to detect and quantify the laminar phases before a life-threatening cardiac arrhythmia occurs thereby facilitating a prediction of such an event.

Abstract: The knowledge of transitions between regular, laminar or chaotic behaviors is essential to understand the underlying mechanisms behind complex systems. While several linear approaches are often insufficient to describe such processes, there are several nonlinear methods that, however, require rather long time observations. To overcome these difficulties, we propose measures of complexity based on vertical structures in recurrence plots and apply them to the logistic map as well as to heart-rate-variability data. For the logistic map these measures enable us not only to detect transitions between chaotic and periodic states, but also to identify laminar states, i.e., chaos-chaos transitions. The traditional recurrence quantification analysis fails to detect the latter transitions. Applying our measures to the heart-rate-variability data, we are able to detect and quantify the laminar phases before a life-threatening cardiac arrhythmia occurs thereby facilitating a prediction of such an event. Our findings could be of importance for the therapy of malignant cardiac arrhythmias.

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TL;DR: Empirically the structure of this network of connections between individuals over which the virus spreads is investigated using data drawn from a large computer installation, and the implications for the understanding and prevention of computer virus epidemics are discussed.

Abstract: Many computer viruses spread via electronic mail, making use of computer users' email address books as a source for email addresses of new victims. These address books form a directed social network of connections between individuals over which the virus spreads. Here we investigate empirically the structure of this network using data drawn from a large computer installation, and discuss the implications of this structure for the understanding and prevention of computer virus epidemics.

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TL;DR: In this article, the synchronization between left and right hemisphere rat electroencephalographic (EEG) channels by using various synchronization measures, namely nonlinear interdependences, phase synchronizations, mutual information, cross correlation, and the coherence function, was studied.

Abstract: We study the synchronization between left and right hemisphere rat electroencephalographic (EEG) channels by using various synchronization measures, namely nonlinear interdependences, phase synchronizations, mutual information, cross correlation, and the coherence function. In passing we show a close relation between two recently proposed phase synchronization measures and we extend the definition of one of them. In three typical examples we observe that except mutual information, all these measures give a useful quantification that is hard to be guessed beforehand from the raw data. Despite their differences, results are qualitatively the same. Therefore, we claim that the applied measures are valuable for the study of synchronization in real data. Moreover, in the particular case of EEG signals their use as complementary variables could be of clinical relevance.

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TL;DR: It is found that the connectivity structure of the Internet presents statistical distributions settled in a well-defined stationary state and the large-scale properties are characterized by a scale-free topology consistent with previous observations.

Abstract: We study the large-scale topological and dynamical properties of real Internet maps at the autonomous system level, collected in a 3-yr time interval. We find that the connectivity structure of the Internet presents statistical distributions settled in a well-defined stationary state. The large-scale properties are characterized by a scale-free topology consistent with previous observations. Correlation functions and clustering coefficients exhibit a remarkable structure due to the underlying hierarchical organization of the Internet. The study of the Internet time evolution shows a growth dynamics with aging features typical of recently proposed growing network models. We compare the properties of growing network models with the present real Internet data analysis.

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TL;DR: It is demonstrated that policies that discriminate between the nodes, curing mostly the highly connected nodes, can restore a finite epidemic threshold and potentially eradicate a virus.

Abstract: The vanishing epidemic threshold for viruses spreading on scale-free networks indicate that traditional methods, aiming to decrease a virus' spreading rate cannot succeed in eradicating an epidemic. We demonstrate that policies that discriminate between the nodes, curing mostly the highly connected nodes, can restore a finite epidemic threshold and potentially eradicate a virus. We find that the more biased a policy is towards the hubs, the more chance it has to bring the epidemic threshold above the virus' spreading rate. Furthermore, such biased policies are more cost effective, requiring less cures to eradicate the virus.

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TL;DR: A lattice Boltzmann model is proposed for isothermal incompressible flow in porous media and the key point is to include the porosity into the equilibrium distribution, and add a force term to the evolution equation to account for the linear and nonlinear drag forces of the medium.

Abstract: In this paper a lattice Boltzmann model is proposed for isothermal incompressible flow in porous media. The key point is to include the porosity into the equilibrium distribution, and add a force term to the evolution equation to account for the linear and nonlinear drag forces of the medium (the Darcy's term and the Forcheimer's term). Through the Chapman-Enskog procedure, the generalized Navier-Stokes equations for incompressible flow in porous media are derived from the present lattice Boltzmann model. The generalized two-dimensional Poiseuille flow, Couette flow, and lid-driven cavity flow are simulated using the present model. It is found the numerical results agree well with the analytical and/or the finite-difference solutions.

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TL;DR: In this article, the authors study epidemic dynamics in bounded scale-free networks with soft and hard connectivity cutoffs and show that the induced epidemic threshold is very small even at a relatively small cutoff, showing that the neglection of connectivity fluctuations in such networks leads to a strong overestimation of the epidemic threshold.

Abstract: Many real networks present a bounded scale-free behavior with a connectivity cutoff due to physical constraints or a finite network size. We study epidemic dynamics in bounded scale-free networks with soft and hard connectivity cutoffs. The finite size effects introduced by the cutoff induce an epidemic threshold that approaches zero at increasing sizes. The induced epidemic threshold is very small even at a relatively small cutoff, showing that the neglection of connectivity fluctuations in bounded scale-free networks leads to a strong overestimation of the epidemic threshold. We provide the expression for the infection prevalence and discuss its finite size corrections. The present paper shows that the highly heterogeneous nature of scale-free networks does not allow the use of homogeneous approximations even for systems of a relatively small number of nodes.

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TL;DR: It is demonstrated that optical discrete solitons are possible in appropriately oriented biased photorefractive crystals in optically induced periodic waveguide lattices that are created via plane-wave interference and paves the way towards the observation of entirely new families of discretesolitons.

Abstract: We demonstrate that optical discrete solitons are possible in appropriately oriented biased photorefractive crystals. This can be accomplished in optically induced periodic waveguide lattices that are created via plane-wave interference. Our method paves the way towards the observation of entirely new families of discrete solitons. These include, for example, discrete solitons in two-dimensional self-focusing and defocusing lattices of different group symmetries, incoherently coupled vector discrete solitons, discrete soliton states in optical diatomic chains, as well as their associated collision properties and interactions. We also present results concerning transport anomalies of discrete solitons that depend on their initial momentum within the Brillouin zone.

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TL;DR: A dynamical model of epidemic spreading on complex networks in which there are explicit correlations among the node's connectivities finds an epidemic threshold inversely proportional to the largest eigenvalue of the connectivity matrix that gives the average number of links.

Abstract: We study a dynamical model of epidemic spreading on complex networks in which there are explicit correlations among the node's connectivities. For the case of Markovian complex networks, showing only correlations between pairs of nodes, we find an epidemic threshold inversely proportional to the largest eigenvalue of the connectivity matrix that gives the average number of links, which from a node with connectivity k go to nodes with connectivity k('). Numerical simulations on a correlated growing network model provide support for our conclusions.

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TL;DR: It is found that scale-free random networks are excellently modeled by simple deterministic graphs and exactly and numerically with high precision all main characteristics of the graph are found.

Abstract: We find that scale-free random networks are excellently modeled by simple deterministic graphs. Our graph has a discrete degree distribution (degree is the number of connections of a vertex), which is characterized by a power law with exponent $\ensuremath{\gamma}=1+\mathrm{ln}3/\mathrm{ln}2.$ Properties of this compact structure are surprisingly close to those of growing random scale-free networks with \ensuremath{\gamma} in the most interesting region, between 2 and 3. We succeed to find exactly and numerically with high precision all main characteristics of the graph. In particular, we obtain the exact shortest-path-length distribution. For a large network $(\mathrm{ln}N\ensuremath{\gg}1)$ the distribution tends to a Gaussian of width $\ensuremath{\sim}\sqrt{\mathrm{ln}N}$ centered at $\mathcal{l}\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\sim}\mathrm{ln}N.$ We show that the eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail with exponent $2+\ensuremath{\gamma}.$

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TL;DR: It is shown that it is possible to smoothly connect generally applicable simulation schemes to more restricted loop algorithms that can be constructed only for a limited range of Hamiltonians (where backtracking can be avoided) and the "algorithmic discontinuities" between general and special points (or regions) in parameter space can be eliminated.

Abstract: We introduce the concept of directed loops in stochastic series expansion and path-integral quantum Monte Carlo methods. Using the detailed balance rules for directed loops, we show that it is possible to smoothly connect generally applicable simulation schemes (in which it is necessary to include backtracking processes in the loop construction) to more restricted loop algorithms that can be constructed only for a limited range of Hamiltonians (where backtracking can be avoided). The ``algorithmic discontinuities'' between general and special points (or regions) in parameter space can hence be eliminated. As a specific example, we consider the anisotropic $S=1/2$ Heisenberg antiferromagnet in an external magnetic field. We show that directed-loop simulations are very efficient for the full range of magnetic fields (zero to the saturation point) and anisotropies. In particular, for weak fields and anisotropies, the autocorrelations are significantly reduced relative to those of previous approaches. The back-tracking probability vanishes continuously as the isotropic Heisenberg point is approached. For the $\mathrm{XY}$ model, we show that back tracking can be avoided for all fields extending up to the saturation field. The method is hence particularly efficient in this case. We use directed-loop simulations to study the magnetization process in the two-dimensional Heisenberg model at very low temperatures. For $L\ifmmode\times\else\texttimes\fi{}L$ lattices with L up to $64,$ we utilize the step structure in the magnetization curve to extract gaps between different spin sectors. Finite-size scaling of the gaps gives an accurate estimate of the transverse susceptibility in the thermodynamic limit: ${\ensuremath{\chi}}_{\ensuremath{\perp}}=0.0659\ifmmode\pm\else\textpm\fi{}0.0002.$

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TL;DR: Molecular dynamic simulations of a medium consisting of disks in a periodically tilted box yield two dynamic modes differing considerably in the total potential and kinetic energies of the disks.

Abstract: Molecular dynamic simulations of a medium consisting of disks in a periodically tilted box yield two dynamic modes differing considerably in the total potential and kinetic energies of the disks. Depending on parameters, these modes display the following features: (i) hysteresis (coexistence of the two modes in phase space); (ii) intermingledlike basins of attraction (uncertainty exponent indistinguishable from zero); (iii) two-state on-off intermittency; and (iv) bimodal velocity distributions. Bifurcations are defined by a cross-shaped phase diagram.

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TL;DR: By considering a sharp boundary as a limit of anisotropically smoothed systems, this work is able to derive a correct first-order perturbation theory and mode-coupling constants, involving only surface integrals of the unperturbed fields over the perturbed interface.

Abstract: Perturbation theory permits the analytic study of small changes on known solutions, and is especially useful in electromagnetism for understanding weak interactions and imperfections Standard perturbation-theory techniques, however, have difficulties when applied to Maxwell's equations for small shifts in dielectric interfaces (especially in high-index-contrast, three-dimensional systems) due to the discontinous field boundary conditions---in fact, the usual methods fail even to predict the lowest-order behavior By considering a sharp boundary as a limit of anisotropically smoothed systems, we are able to derive a correct first-order perturbation theory and mode-coupling constants, involving only surface integrals of the unperturbed fields over the perturbed interface In addition, we discuss further considerations that arise for higher-order perturbative methods in electromagnetism

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TL;DR: The dynamics of an epidemiclike model for the spread of a rumor on a small-world network is studied and a quantitative characterization of the evolution in the two regimes is performed.

Abstract: We study the dynamics of an epidemiclike model for the spread of a rumor on a small-world network. It has been shown that this model exhibits a transition between regimes of localization and propagation at a finite value of the network randomness. Here, by numerical means, we perform a quantitative characterization of the evolution in the two regimes. The variant of dynamic small worlds, where the quenched disorder of small-world networks is replaced by randomly changing connections between individuals, is also analyzed in detail and compared with a mean-field approximation.

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TL;DR: In this paper, the authors studied the problem of satisfiability of randomly chosen clauses, each with K Boolean variables, using the cavity method at zero temperature, and showed the existence of an intermediate phase in the satisfiable region.

Abstract: We study the problem of satisfiability of randomly chosen clauses, each with K Boolean variables. Using the cavity method at zero temperature, we find the phase diagram for the K=3 case. We show the existence of an intermediate phase in the satisfiable region, where the proliferation of metastable states is at the origin of the slowdown of search algorithms. The fundamental order parameter introduced in the cavity method, which consists of surveys of local magnetic fields in the various possible states of the system, can be computed for one given sample. These surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems. One such algorithm is shown here for the K=3 satisfiability problem, with very good performances.

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TL;DR: Phase synchronization is observed to emerge in the presence of even a tiny fraction P of shortcuts and to display saturated behavior for P > or similar to 0.5, indicating that the same synchronizability as the random network (P=1) can be achieved with relatively small number of shortcuts.

Abstract: We investigate collective synchronization in a system of coupled oscillators on small-world networks. The order parameters that measure synchronization of phases and frequencies are introduced and analyzed by means of dynamic simulations and finite-size scaling. Phase synchronization is observed to emerge in the presence of even a tiny fraction P of shortcuts and to display saturated behavior for $P\ensuremath{\gtrsim}0.5.$ This indicates that the same synchronizability as the random network $(P=1)$ can be achieved with relatively small number of shortcuts. The transient behavior of the synchronization, obtained from the measurement of the relaxation time, is also discussed.

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TL;DR: A comprehensive solution to the technical problems that arise when two discretized surfaces come into contact is described and an efficient update of the particle velocities is described, taking into account the possibility that some of the differential equations are stiff.

Abstract: The lattice-Boltzmann method has been refined to take account of near-contact interactions between spherical particles. First, we describe a comprehensive solution to the technical problems that arise when two discretized surfaces come into contact. Second, we describe how to incorporate lubrication forces and torques into lattice-Boltzmann simulations, and test our method by calculating the forces and torques between a spherical particle and a plane wall. Third, we describe an efficient update of the particle velocities, taking into account the possibility that some of the differential equations are stiff.

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TL;DR: A nonlinear continuum model of large-scale brain electrical activity is used to analyze arousal states and their stability and nonlinear dynamics for physiologically realistic parameters and provides a single, powerful framework for quantitative understanding of a wide variety of brain phenomena.

Abstract: Links between electroencephalograms (EEGs) and underlying aspects of neurophysiology and anatomy are poorly understood. Here a nonlinear continuum model of large-scale brain electrical activity is used to analyze arousal states and their stability and nonlinear dynamics for physiologically realistic parameters. A simple ordered arousal sequence in a reduced parameter space is inferred and found to be consistent with experimentally determined parameters of waking states. Instabilities arise at spectral peaks of the major clinically observed EEG rhythms---mainly slow wave, delta, theta, alpha, and sleep spindle---with each instability zone lying near its most common experimental precursor arousal states in the reduced space. Theta, alpha, and spindle instabilities evolve toward low-dimensional nonlinear limit cycles that correspond closely to EEGs of petit mal seizures for theta instability, and grand mal seizures for the other types. Nonlinear stimulus-induced entrainment and seizures are also seen, EEG spectra and potentials evoked by stimuli are reproduced, and numerous other points of experimental agreement are found. Inverse modeling enables physiological parameters underlying observed EEGs to be determined by a new, noninvasive route. This model thus provides a single, powerful framework for quantitative understanding of a wide variety of brain phenomena.