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.     

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.     

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.     

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.     

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.     

The chaotic phase synchronization in adaptively coupled-delayed complex networks

The notion of chaotic phase synchronization (CPS) in the large-scale delayed scale-free network is discussed in this Letter. The amplitude death (AD) phenomenon is observed and analyzed in terms of energy. AD occurs when the time-delay becomes long enough. The adaptive coupling scheme has better performance in CPS and AD compared with the constant scheme, and simulation results confirm conclusions.     

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.     

Adaptation to synchronization in phase-oscillator networks

We introduce an adaptation algorithm by which an ensemble of coupled oscillators with attractive and repulsive interactions is induced to adopt a prescribed synchronized state. While the performance of adaptation is controlled by measuring a macroscopic quantity, which characterizes the achieved degree of synchronization, adaptive changes are introduced at the microscopic level of the interaction network, by modifying the configuration of repulsive interactions. This scheme emulates the distinct levels of selection and mutation in biological evolution and learning.     

Small-world effect induced by weight randomization on regular networks

The concept of edge weight provides additional depth for describing and adjusting the properties of networks. Redistribution of edge weight can effectively change the properties of networks even though the corresponding binary topology remains unchanged. Based on regular networks with initially homogeneous dissimilarity weights, random redistribution of edge weight can be enough to induce small world phenomena. The effects of random weight redistribution on both static properties and dynamical models of networks are investigated. The results reveal that randomization of weight can enhance the ability of synchronization of chaotic systems dramatically.     

Epidemic outbreaks in growing scale-free networks with local structure

The class of generative models has already attracted considerable interest from researchers in recent years and much expanded the original ideas described in BA model. Most of these models assume that only one node per time step joins the network. In this paper, we grow the network by adding n interconnected nodes as a local structure into the network at each time step with each new node emanating m new edges linking the node to the preexisting network by preferential attachment. This successfully generates key features observed in social networks. These include power-law degree distribution p_k∼k^−(3+μ), where μ=(n−1)/m is a tuning parameter defined as the modularity strength of the network, nontrivial clustering, assortative mixing, and modular structure. Moreover, all these features are dependent in a similar way on the parameter μ. We then study the susceptible-infected epidemics on this network with identical infectivity, and find that the initial epidemic behavior is governed by both of the infection scheme and the network structure, especially the modularity strength. The modularity of the network makes the spreading velocity much lower than that of the BA model. On the other hand, increasing the modularity strength will accelerate the propagation velocity.     

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.     

Edge-based-attack induced cascading failures on scale-free networks

Most previous existing works on cascading failures only focused on attacks on nodes rather than on edges. In this paper, we discuss the response of scale-free networks subject to two different attacks on edges during cascading propagation, i.e., edge removal by either the descending or ascending order of the loads. Adopting a cascading model with a breakdown probability p of an overload edge and the initial load (k_ik_j)^α of an edge ij, where k_i and k_j are the degrees of the nodes connected by the edge ij and α is a tunable parameter, we investigate the effects of two attacks for the robustness of Barabási-Albert (BA) scale-free networks against cascading failures. In the case of α<1, our investigation by the numerical simulations leads to a counterintuitive finding that BA scale-free networks are more sensitive to attacks on the edges with the lowest loads than the ones with the highest loads, not relating to the breakdown probability. In addition, the same effect of two attacks in the case of α=1 may be useful in furthering studies on the control and defense of cascading failures in many real-life networks. We then confirm by the theoretical analysis these results observed in simulations.     

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.     

Synchronization of Coupled Oscillators on Newman-Watts Small-World Networks

We investigate the collection behaviour of coupled phase oscillators on Newman-Watts small-world networks in one and two dimensions. Each component of the network is assumed as an oscillator and each interacts with the others following the Kuramoto model We then study the onset of global synchronization of phases and frequencies based on dynamic simulations and finite-size scaling. Both the phase and frequency synchronization are observed to emerge in the presence of a tiny fraction of shortcuts and enhanced with the increases of nearest neighbours and lattice dimensions.     

Optimal symmetric networks in terms of minimizing average shortest path length and their sub-optimal growth model

Homogeneous entangled networks characterized by small world, large girths, and no community structure have attracted much attention due to some of their favorable performances. However, the optimization algorithm proposed by Donetti et al. is very time-consuming and will lose its efficiency when the size of the target network becomes large. In this paper, an alternative optimization algorithm is provided to get optimal symmetric networks by minimizing the average shortest path length. It is shown that the synchronizability of a symmetric network is enhanced when the average shortest path length of the network is shortened as the optimization proceeds, which suggests that the optimal symmetric networks in terms of minimizing average shortest path length will be very close to those entangled networks. In order to overcome the time-consuming obstacle of the optimization algorithms proposed by us and Donetti et al., a growth model is proposed to get large scale sub-optimal symmetric networks. Numerical simulations show that the symmetric networks derived by our growth model will have small-world property, and besides, these networks will have many other similar favorable performances as entangled networks, e.g., robustness against errors and attacks, very good load balancing ability, and strong synchronizability.     

Estimating the clustering coefficient in scale-free networks on lattices with local spatial correlation structure

When we study the architecture of networks of spatially extended systems the nodes in the network are subject to local correlation structures. In this case, we show that for scale-free networks the traditional way to estimate the clustering coefficient may not be meaningful. Here we explain why and propose an approach that corrects this problem.     

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.     

A model for cascading failures in scale-free networks with a breakdown probability

Considering that not all overload nodes will be removed from networks due to some effective measures to protect them, we propose a new cascading model with a breakdown probability. Adopting the initial load of a node j to be L_j=[k_j(∑_m∈Γjk_m)]^α with k_j and Γ_j being the degree of the node j and the set of its neighboring nodes, respectively, where α is a tunable parameter, we investigate the relationship between some parameters and universal robustness characteristics against cascading failures on scale-free networks. According to a new measure originated from a phase transition from the normal state to collapse, the numerical simulations show that Barabási-Albert (BA) networks reach the strongest robustness level against cascading failures when the tunable parameter α=0.5, while not relating to the breakdown probability. We furthermore explore the effect of the average degree 〈k〉 for network robustness, thus obtaining a positive correlation between 〈k〉 and network robustness. We then analyze the effect of the breakdown probability on the network robustness and confirm by theoretical predictions this universal robustness characteristic observed in simulations. Our work may have practical implications for controlling various cascading-failure-induced disasters in the real world.