Generalized Projective Synchronization between Two Complex Networks with Time-Varying Coupling Delay

Generalized projective synchronization （GPS） between two complex networks with time-varying coupling delay is investigated. Based on the Lyapunov stability theory, a nonlinear controller and adaptive updated laws are designed. Feasibility of the proposed scheme is proven in theory. Moreover, two numerical examples are presented, using the energy resource system and Lü＇s system [Physica A 382 （2007） 672] as the nodes of the networks. GPS between two energy resource complex networks with time-varying coupling delay is achieved. This study can widen the application range of the generalized synchronization methods and will be instructive for the demand supply of energy resource in some regions of China.     

Small-World Connections to Induce Firing Activity and Phase Synchronization in Neural Networks

We investigate how the firing activity and the subsequent phase synchronization of neural networks with smallworld topological connections depend on the probability p of adding-links. Network elements are described by two-dimensional map neurons （2DMNs） in a quiescent original state. Neurons burst for a given coupling strength when the topological randomness p increases, which is absent in a regular-lattice neural network. The bursting activity becomes frequent and synchronization of neurons emerges as topological randomness further increases. The maximal firing frequency and phase synchronization appear at a particular value of p. However, if the randomness p further increases, the firing frequency decreases and synchronization is apparently destroyed.     

Adaptive Synchronization between Two Different Complex Networks with Time-Varying Delay Coupling

A new general network model for two complex networks with time-varying delay coupling is presented. Then we investigate its synchronization phenomena. The two complex networks of the model differ in dynamic nodes, the number of nodes and the coupling connections. By using adaptive controllers, a synchronization criterion is derived. Numerical examples are given to demonstrate the effectiveness of the obtained synchronization criterion. This study may widen the application range of synchronization, such as in chaotic secure communication.     

Synchronization in complex networks

Synchronization processes in populations of locally interacting elements are the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understanding synchronization phenomena in natural systems now take 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 take an overview of the new emergent features coming out from the interplay between the structure and the function of the underlying patterns 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.     

Improvement of Synchronizability of Scale-Free Networks

We investigate the factors that affect synchronizability of coupled oscillators on scale-free networks. Using the memory Tabu search （MTS） algorithm, we improve the eigen-ratio Q of a coupling matrix by edge intercrossing. The numerical results show that the synchronizatlon-improved scale-free networks should have distinctive both small average distance and larger clustering coefficient, which are consistent with some real-world networks. Moreover, the synchronizability-improved networks demonstrate the disassortative coefficient.     

Projective synchronization in drive-response dynamical networks of partially linear systems with time-varying coupling delay

Recently, projective synchronization (PS) has been widely studied in more than one system. In this Letter, we propose a linear controller and an updated law to realize the PS in drive-response dynamical networks of partially linear systems with time-varying coupling delay, based on the Lyapunov stability theory. A sufficient condition is obtained. Moreover, numerical simulations are provided to verify the correctness and effectiveness of the scheme.     

Synchrony-optimized networks of non-identical Kuramoto oscillators

In this Letter we discuss a method for generating synchrony-optimized coupling architectures of Kuramoto oscillators with a heterogeneous distribution of native frequencies. The method allows us to relate the properties of the coupling network to its synchronizability. These relations were previously only established from a linear stability analysis of the identical oscillator case. We further demonstrate that the heterogeneity in the oscillator population produces heterogeneity in the optimal coupling network as well. Two rules for enhancing the synchronizability of a given network by a suitable placement of oscillators are given: (i) native frequencies of adjacent oscillators must be anti-correlated and (ii) frequency magnitudes should positively correlate with the degree of the node they are placed at.     

Synchronization of Kuramoto oscillators in random complex networks

We study the synchronization of coupled phase oscillators in random complex networks. The topology of the networks is assumed to be vary over time. Here we mainly study the onset of global phase synchronization when the topology switches rapidly over time. We find that the results are, to some extent, different from those in deterministic situations. In particular, the synchronizability of coupled oscillators can be enhanced in ER networks and scale-free networks under fast switching, while in stochastic small-world networks such enhancement is not significant.     

Duplication： a Mechanism Producing Disassortative Mixing Networks in Biology

Assortative/disassortative mixing is an important topological property of a network. A network is called assortative mixing if the nodes in the network tend to connect to their connectivity peers, or disassortative mixing if nodes with low degrees are more likely to connect with high-degree nodes. We have known that biological networks such as protein-protein interaction networks （PPI）, gene regulatory networks, and metabolic networks tend to be disassortative. On the other hand, in biological evolution, duplication and divergence are two fundamental processes. In order to make the relationship between the property of disassortative mixing and the two basic biological principles clear and to study the cause of the disassortative mixing property in biological networks, we present a random duplication model and an anti-preference duplication model. Our results show that disassortative mixing networks can be obtained by both kinds of models from uncorrelated initial networks. Moreover, with the growth of the network size, the disassortative mixing property becomes more obvious.     

Order Parameter Hysteresis on the Complex Network

Collective synchronization is investigated on the small-world network （NW model）. The order parameter is introduced to measure the synchronization of phase. It is found that there are differences between the processes of synchronization and desynchronization. The dependence of order parameter on the coupling strength is shown like a hysteresis loop. The size of the loop demonstrates the non-monotonicity with the change of adding probability, and is relevant to the construction of the network. The area may be maximum, as the adding probability is equal to 0.4. This phenomenon indicates that the clusters in the network play an important role in the processes of synchronization and desynchronization.     

Frequency Sensitivity of Signal Detection in Scale-Free Networks

It has been recently reported that scale-free topology favors the detection of a weak signal because of the higher amplification at the hub node than that at other nodes [Phys. Ref. I？, 78（2008）046111]. We investigate the corresponding synchronization behaviors and find that the favorite detection depends not only on the coupling and noise strengths but also on the frequency of the external signal. We reveal theoretically and numerically that the amplification effect of the hub node will decrease monotonously with the external frequency, which is useful to understand the high sensitivity of animal visual and auditory systems to weak external signals.     

Phase synchronization on scale-free networks with community structure

In this Letter, we propose a growing network model that can generate scale-free networks with a tunable community strength. The community strength, C, is directly measured by the ratio of the number of external edges to that of the internal ones; a smaller C   corresponds to a stronger community structure. By using the Kuramoto model, we investigated the phase synchronization on this network and found an abnormal region (C?0.002$C?0.002$), in which the network has even worse synchronizability than the unconnected case (C=0$C=0$). On the other hand, the community effect will vanish when C exceeds 0.1. Between these two extreme regions, a stronger community structure will hinder global synchronization.     

Synchronization of coupled oscillators on small-world networks

We investigate the synchronous dynamics of Kuramoto oscillators and van der Pol oscillators on Watts-Strogatz type small-world networks. The order parameters to characterize macroscopic synchronization are calculated by numerical integration. We focus on the difference between frequency synchronization and phase synchronization. In both oscillator systems, the critical coupling strength of the phase order is larger than that of the frequency order for the small-world networks. The critical coupling strength for the phase and frequency synchronization diverges as the network structure approaches the regular one. For the Kuramoto oscillators, the behavior can be described by a power-law function and the exponents are obtained for the two synchronizations. The separation of the critical point between the phase and frequency synchronizations is found only for small-world networks in the theoretical models studied.     

Adaptive synchronization of weighted complex dynamical networks with coupling time-varying delays

This Letter considers the problem of controlling a weighted complex dynamical network with coupling time-varying delay toward an assigned evolution. Adaptive controllers have been designed for nodes of the controlled network. Analytical results show that the states of the weighted dynamical network can globally asymptotically synchronize onto a desired orbit under the designed controllers. In comparison with the common linear feedback controllers, the adaptive controllers have strong robustness against asymmetric coupling matrix, time-varying weights, delays, and noise. Numerical simulations illustrated by a nearest-neighbor coupling network verify the effectiveness of the proposed controllers.     

Choosing effective controlled nodes for scale-free network synchronization

Previous studies concerning pinning control of complex-network synchronization have very often demonstrated that in an unweighted symmetrical scale-free network, controlling the high-degree nodes is more efficient than controlling randomly chosen ones; due to the heterogeneity of the node-degree or edge-connection distribution of the scale-free network, small-degree nodes have relatively high probabilities of being chosen at random but their control has less influence on the other nodes through the network. This raises the question of whether or not controlling the high-degree nodes is always better than controlling the small ones in scale-free networks. Our answer to this is yes and no. In this study, we carry out extensive numerical simulations to show that in an unweighted symmetrical Barabasi-Albert scale-free network, when the portion of controlled nodes is relatively large, controlling the small nodes becomes better than controlling the big nodes and controlling randomly chosen nodes has approximately the same effect as controlling the big ones. However, we also show that for normalized weighted scale-free networks, controlling the big nodes is in fact always better than controlling the small ones.     

Adaptive projective synchronization in complex networks with time-varying coupling delay

In this Letter, adaptive projective synchronization (PS) between two complex networks with time-varying coupling delay is investigated by the adaptive control method, and this method has been applied to identify the exact topology of a weighted general complex network. To validate the proposed method, the Lü and Qi systems as the nodes of the networks are detailed analysis, and some numerical results show the effectiveness of the present method.     

Construction principles for highly synchronizable sparse directed networks

In this Letter sparse directed interaction networks of heterogeneous Kuramoto oscillators that give rise to enhanced synchronization properties are generated and analyzed. The particular networks, which allow for the transition to full synchronization for the smallest coupling strength, i.e., optimal networks, are found to be very homogeneous in the in-degree distribution, but exhibit very skewed out-degree distributions. Various correlations between in- and out-degree structure, oscillator heterogeneity and component structure, which are linked to an enhanced synchronizability, are discussed.     

Synchronization of Complex Network with Drivingly Coupled Scheme

We present a network model with a new coupled scheme which is the generalization of drive-response systems called a drivingly coupled network. The synchronization of the network is investigated by numerical simulations based on Lorenz systems. By calculating the largest transversal Lyapunov exponents of such network, the stable and unstable regions of synchronous state for eigenvalues in such network can be obtained and many kinds of drivingly coupled arrays based on Lorenz systems such as all-to-all, star-shape, ring-shape and chain-shape networks are considered.