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In this work, we study the synchronization in a generalized Kuramoto model with frequency-weighted coupling. In particular, we focus on the situations in which the frequency distributions of oscillators are asymmetric. For typical unimodal frequency distributions, such as Lorentzian, Gaussian, triangle, and even special Rayleigh, we find that the synchronization transition in the model generally converts from the first order to the second order as the central frequency shifts toward positive direction. We characterize two interesting coherent states in the system: In the former, two phase-locking clusters are formed, rotating with the same frequency. They correspond to those oscillators with relatively high frequencies while the oscillators with relatively small frequencies are not entrained. In the latter, two phase-locking clusters rotate with different frequencies, leading to the oscillation of the order parameter. We further conduct theoretical analysis to reveal the relation between the asymmetric frequency distribution and the conversion of synchronization type, and justify the coherent states observed in the system.

Evolutionary game theory is crucial to capturing the characteristic interaction patterns among selfish individuals. In a population of coordination games of two strategies, one of the central problems is to determine the fixation probability that the system reaches a state of networkwide of only one strategy, and the corresponding expectation times. The deterministic replicator equations predict the critical value of initial density of one strategy, which separates the two absorbing states of the system. However, numerical estimations of this separatrix show large deviations from the theory in finite populations. Here we provide a stochastic treatment of this dynamic process on complex networks of finite sizes as Markov processes, showing the evolutionary time explicitly. We describe analytically the effects of network structures on the intermediate fixations as observed in numerical simulations. Our theoretical predictions are validated by various simulations on both random and scale free networks. Therefore, our stochastic framework can be helpful in dealing with other networked game dynamics.

The suprachiasmatic nucleus (SCN) is a principal circadian clock in mammals, which controls physiological and behavioral daily rhythms. The SCN has two main features: Maintaining a rhythmic cycle of approximately 24 h in the absence of a light-dark cycle (free-running period) and the ability to entrain to external light-dark cycles. Both free-running period and range of entrainment vary from one species to another. To understand this phenomenon, we investigated the diversity of a free-running period by the distribution of coupling strengths in our previous work [Phys. Rev. E 80, 030904(R) (2009)]. In this paper we numerically found that the dispersion of intrinsic periods among SCN neurons influence the entrainment range of the SCN, but has little influence on the free-running periods under constant darkness. This indicates that the dispersion of coupling strengths determines the diversity in free-running periods, while the dispersion of intrinsic periods determines the diversity in the entrainment range. A theoretical analysis based on two coupled neurons is presented to explain the results of numerical simulations.

Explosive synchronization (ES) has recently attracted much attention, where its two necessary conditions are found to be a scale-free network topology and a positive correlation between the natural frequencies of the oscillators and their degrees. Here we present a framework for ES to be observed in a general complex network, where a positive correlation between coupling strengths of the oscillators and the absolute of their natural frequencies is assumed and the previous studies are included as specific cases. In the framework, the previous two necessary conditions are replaced by another one, thus fundamentally deepening the understanding of the microscopic mechanism toward synchronization. A rigorous analytical treatment by a mean field is provided to explain the mechanism of ES in this alternate framework.

The synchronization of biological activity with the alternation of day and night (circadian rhythm) is performed in the brain by a group of neurons, constituting the suprachiasmatic nucleus (SCN). The SCN is divided into two subgroups of oscillating cells: the ventrolateral (VL) neurons, which are exposed to light (photic signal), and the dorsomedial (DM) neurons, which are coupled to the VL cells. When the coupling between these neurons is strong enough, the system synchronizes with the photic period. Upon increasing the cell coupling, the entrainment of the DM cells has been recently shown to occur via a very sharp (jumping) transition when the period of the photic input is larger than the intrinsic period of the cells. Here, we characterize this transition with a simple realistic model. We show that two bifurcations possibly lead to the disappearance of the endogenous mode. Using a mean-field model, we show that the jumping transition results from a supercritical Hopf-like bifurcation. This finding implies that both the period and strength of the stimulating photic signal, and the relative fraction of cells in the VL and DM compartments, are crucial in determining the synchronization of the system.

Epidemic spreading has been well studied in the past decade, where the main concentration is focused on the influence of network topology but little attention is paid to the individual's crisis awareness. We here study how the crisis awareness, i.e., personal self-protection, influences the epidemic spreading by presenting a susceptible-infected-recovered model with information-driven vaccination. We introduce two parameters to quantitatively characterize the crisis awareness. One is the information creation rate λ and the other is the information sensitivity η. We find that the epidemic spreading can be significantly suppressed in both the homogeneous and heterogeneous networks when both λ and η are relatively large. More interesting is that the needed vaccine will be significantly reduced when the information is well spread, which is a good news for the poor countries and regions with limited resources.

Phase-disorder-induced resonance has been recently uncovered in an ensemble of coupled excitable neurons with weak external signal, where each neuron takes a constant initial signal phase [Phys. Rev. E 82, 010902(R) (2010)]. However, it is unclear how the initial phase disorder influences the behavior of a single or isolated neuron, which constitutes the ensemble. In order to answer this question, we here consider the case of a single neuron with phase noise originated from the time-varying initial signal phase, in contrast to the constant initial phase in each neuron studied in the above referenced paper. Interestingly, we find that the phase noise can induce resonance even in the single neuronal system with subthreshold signal. Moreover, we reveal that, with the presence of phase noise, the neuron also shows another resonance behavior by varying the period of the external signal. An analysis is conducted to uncover the mechanisms behind these resonance phenomena.

Signal detection is generally related to network structure in both biological and engineering systems, and enormous effort has been putting into understanding the mechanism of amplification of weak signals in the aspects of self-tuning and scale-free topology. Here, we show that a third way of signal amplification exists, which does not require the scale-free topology as a necessary condition. This approach can effectively amplify the signal at an arbitrary node in a complex network by adaptively adjusting the coupling weights between the signal node and its neighbors, and thus can be used in both the local and global areas of a complex network. A theory is provided to explain its mechanism.

The phase-splitting behavior of coupled suprachiasmatic-nucleus neurons has been observed in many mammals, and its mechanism is still not completely understood. Based on our previous work [C. Gu, J. Wang, and Z. Liu, Phys. Rev. E 80, 030904(R) (2009)] on the free-running periods of neurons in the suprachiasmatic nucleus, we present here a modified Goodwin oscillator model to explain the mechanism of phase splitting. In contrast to the previous phase model, the modified Goodwin oscillator model contains the information on both the phase and amplitude and, thus, can show more features than the purely phase model, including all three behaviors of synchronization, phase splitting, and amplitude death and the distributed periodicity in the regions of synchronization and phase splitting, etc. An analytic phase model is extracted from the modified Goodwin oscillator model to explain the dependence of periodicity on the parameters. Moreover, both the modified Goodwin oscillator model and the analytic phase model show that the ensemble frequency can be enhanced or reduced by the time delay.

It is well known that resonance can be induced by external noise or diversity. Here we show that resonance can be induced even by a phase disorder in coupled excitable neurons with subthreshold activity. In contrast to the case of identical phase, we find that phase disorder plays an active role in enhancing neuronal activity. We also uncover that the presence of phase disorder can induce a double resonance phenomenon: phase disorder and coupling strength both can enhance neuronal firing activity. A physical theory is formulated to help understand the mechanism behind this double resonance phenomenon.

It has recently been shown that amplified signal response is possible in scale-free networks of two-state signaling devices [J. A. Acebron, Phys. Rev. Lett. 99, 128701 (2007)]. In the analysis of dynamics in networks, much emphasis is put on the hub, and consequently the applicability thereof is limited to a region of small coupling strength. In this paper, we develop a one-body theory which predicts (1) the behavior of the gain in the whole coupling strength region, and (2) the degree of the unit, which shows maximum response, as a function of the coupling strength. In order to achieve good agreement with numerical experiments effects of finite system size are taken into account when the coupling strength becomes very small and the degree kL of the maximum response unit, predicted by our theory, becomes larger than the maximum degree kmax available to a concrete finite network.

The collective dynamics of coupled oscillators has been well studied in fully occupied networks, but little attention has been paid to the case of partially occupied networks. We study this problem by a dynamic bipartite model and focus on the influence of population mobility. We find that when the density of occupied nodes is smaller than the percolation threshold rho(c), the order parameter will show an effect of mobility with optimal value at a medium moving probability. Its mechanism can be revealed through three factors, i.e., the size of the largest component, the mixing degree in individual components, and the frequency of exchange information among components. When the density of occupied nodes is larger than rho(c), the moving probability will act as a bifurcation parameter to synchronization. The effect of mobility also exists for other dynamics such as the epidemic spreading where the effect is shown through the number of infected agents.

Considering the fact that signal transmission time delays between different pairs of synaptically coupled neurons in the brain are different, we study the effects of distributed time delays on phase synchronization of bursting neurons. We consider the case of inhibitory coupled bursting Hindmarsh-Rose neurons and find that distributed time delays in chemical coupling can induce a variety of phase-coherent dynamic behaviors. The critical mean time delay for the emergence of coherent behaviors is inversely proportional to both the coupling strength and the average degree. This phenomenon is robust to nonidentical external inputs and is independent of network topology. A physical theory is formulated to explain the emergence of coherent neuronal activity.

We present a network model of coupled oscillators to study how a weak signal is transmitted in complex networks. Through both theoretical analysis and numerical simulations, we find that the response of other nodes to the weak signal decays exponentially with their topological distance to the signal source and the coupling strength between two neighboring nodes can be figured out by the responses. This finding can be conveniently used to detect the topology of unknown network, such as the degree distribution, clustering coefficient and community structure, etc., by repeatedly choosing different nodes as the signal source. Through four typical networks, i.e., the regular one dimensional, small world, random, and scale-free networks, we show that the features of network can be approximately given by investigating many fewer nodes than the network size, thus our approach to detect the topology of unknown network may be efficient in practical situations with large network size.

The suprachiasmatic nucleus (SCN) pacemaker shows a free-running period ranging from 20 to 28 h for different species, which was usually explained from the angle of coupling strength. Based on the assumption of nonidentical coupling strengths in SCN, we find an alternative mechanism that the diversity of free-running period can be also caused by the distribution of coupling strengths. The free-running period is proportional to the average coupling strength and inverse proportional to the dispersion of couplings. Moreover, we present an analytic phase model to confirm our finding, which shows a solid foundation of our finding and opens a window to study the collective behaviors of SCN oscillators.

We consider the effects of time-varying packet generation rates in the performance of communication networks. The time variations could be a result of the patterns in human activities. As a model, we study the effects of a degree-dependent packet generation rate that includes a sinusoidal term. Applying a modified traffic awareness protocol (TAP) previously proposed for static packet generation rates to the present situation leads to an altered value of the optimization parameter, when compared to that obtained in the static case. To enhance the performance and to cope with the time-varying effects better, we propose a class of self-adjusting traffic awareness protocols that makes use of instantaneous traffic information beyond that included in the modified TAP. Two special cases that make use of global and local information, respectively, are studied. Comparing results of our proposal schemes with the modified TAP, it is shown that the present self-adjusting schemes perform more effectively.

Considering the accumulation phenomenon in public places, we investigate how the condensation of moving bosonic particles influences the epidemic spreading in scale-free metapopulation networks. Our mean-field theory shows that condensation can significantly enhance the effect of epidemic spreading and reduce the threshold for epidemic to survive, in contrast to the case of without condensation. In the stationary state, the number of infected particles increases with the degree k linearly when kk_{c}, where k_{c} denotes the crossover degree of the nodes with unity particle. The dependence of critical infective rate beta_{c} on the parameters k_{max}, micro, and delta, is figured out, where k_{max}, micro, and delta denote the largest degree, recovery rate, and jumping exponent, respectively. Numerical simulations have confirmed the theoretical predictions.

We study numerically the effect of a scale-free topology on the signal-to-noise ratio of networked two-state systems and find a double resonance phenomenon, i.e., a resonance on coupling strength and a stochastic resonance on noise strength. This finding suggests an alternative approach of self-tuning, i.e., tuning from the scale-free topology, instead of the self-tuning of potential. A heuristic theory through a starlike network is presented to explain the double resonance.

Considering the fact that the distances between coupled oscillators may delay the receiving of signals, we study here the influence of uniformly distributed delays in an array of coupled pendulums instead of studying the influence of the coupling strength. We find that with an increase of the range of distributed delays, the chaotic behaviors of the coupled arrays may be controlled and different synchronized patterns can be induced. An analytic solution is given to confirm the numerical results. This finding may provide further insight into information processing in neurons.

Heat conduction in simple networks consisting of different one dimensional nonlinear chains is studied. We find that the coupling between chains has a different function in heat conduction from that in electric network (circuit). The two coupled particles form an interface and introduce an interface thermal resistance which reduces the heat current. The reduction of heat current depends sensitively on the position and strength of the coupling. This might find application in controlling heat flow in complex networks.

We study the condensation phenomenon in a zero range process on weighted scale-free networks in order to show how the weighted transport influences the particle condensation. Instead of the approach of grand canonical ensemble which is generally used in a zero range process, we introduce an alternate approach of the mean-field equations to study the dynamics of particle transport. We find that the condensation on the scale-free network is easier to occur in the case of weighted transport than in the case of weight-free networks. In the weighted transport, especially, a dynamical condensation is even possible for the case of no interaction among particles, which is impossible in the case of weight-free networks.

We have uncovered a phenomenon in coupled chaotic oscillators where a subset of Lyapunov exponents, which are originally zero in the absence of coupling, can become positive as the coupling is increased. This occurs for chaotic attractors having multiple scrolls, such as the Lorenz attractor. We argue that the phenomenon is due to the disturbance to the relative frequencies with which a trajectory visits different scrolls of the attractor. An algebraic scaling law is obtained which relates the Lyapunov exponents with the coupling strength. The scaling law appears to be universal.

An important quantity characterizing the shadowability of computer-generated trajectories in nonhyperbolic chaotic system is the shadowing time, which measures for how long a numerical trajectory remains valid. This time depends sensitively on an initial condition. Here, we show that for nonhyperbolic systems with unstable-dimension variability, the probability distribution of the shadowing time contains two distinct scaling behaviors: an algebraic scaling for short times and an exponential scaling for long times. The exponential behavior depends on the system details but the small-time algebraic behavior appears to be universal.

We consider the entire spectrum of architectures of general networks, ranging from being heterogeneous (scale-free) to homogeneous (random), and investigate the infection dynamics by using a three-state epidemiological model that does not involve the mechanism of self-recovery. This model is relevant to realistic situations such as the propagation of a flu virus or information over a social network. Our heuristic analysis and computations indicate that (1) regardless of the network architecture, there exists a substantial fraction of nodes that can never be infected and (2) heterogeneous networks are relatively more robust against spreads of infection as compared with homogeneous networks. We have also considered the problem of immunization for preventing wide spread of infection, with the result that targeted immunization is effective for heterogeneous networks.

Results are reported concerning the transition to chaos in random dynamical systems. In particular, situations are considered where a periodic attractor coexists with a nonattracting chaotic saddle, which can be expected in any periodic window of a nonlinear dynamical system. Under noise, the asymptotic attractor of the system can become chaotic, as characterized by the appearance of a positive Lyapunov exponent. Generic features of the transition include the following: (1) the noisy chaotic attractor is necessarily nonhyperbolic as there are periodic orbits embedded in it with distinct numbers of unstable directions (unstable dimension variability), and this nonhyperbolicity develops as soon as the attractor becomes chaotic; (2) for systems described by differential equations, the unstable dimension variability destroys the neutral direction of the flow in the sense that there is no longer a zero Lyapunov exponent after the noisy attractor becomes chaotic; and (3) the largest Lyapunov exponent becomes positive from zero in a continuous manner, and its scaling with the variation of the noise amplitude is algebraic. Formulas for the scaling exponent are derived in all dimensions. Numerical support using both low- and high-dimensional systems is provided.

We uncover a coherence resonance near the Hopf bifurcation from chaos in coupled chaotic oscillators. At the bifurcation, a nearly periodic rotating wave becomes stable as the state of synchronous chaos is destabilized. We find that noise can induce the bifurcation and, more strikingly, it can enhance the temporal regularity of the wave pattern in the coupled system. This resonant phenomenon is expected to be robust and physically observable.

We consider growing networks with algebraic preferential attachment and address two questions: (1) what is the effect of temporal fluctuations in the number of new links acquired by the network? and (2) what is the network tolerance against random failures and intentional attacks? We find that the fluctuations generally have little effect on the network properties, although they lead to a plateau behavior for small degrees in the connectivity distribution. Formulas are derived for the evolution and distribution of the network connectivity, which are tested by numerical simulations. Numerical study of the effect of failures and attacks suggests that networks constructed under algebraic preferential attachment are more robust than scale-free networks.

It has been known that noise can enhance the temporal regularity of dynamical systems that exhibit a bursting behavior--the phenomenon of coherence resonance. But can the phenomenon be expected for nonbursting chaotic systems? We present a theoretical argument based on the idea of time-scale matching and provide experimental evidence with a chaotic electronic circuit for coherence resonance in nonbursting chaotic systems.

We consider a common class of dielectric optical microlasing cavities with quadrupolar deformations and address the question of the maximally allowed amount of deformation for both high-Q operation and a high degree of directionality of light emission. Our approach is to compute the probability for light rays to be trapped in the cavity by examining chaotic scattering dynamics in the classical phase space. We develop a dynamical criterion for high-Q operation and introduce a measure to quantify the directionality of the light emission. Our results suggest that high-Q and directionality can be achieved simultaneously in a wide range of the deformation parameter.