Enhancing the network synchronizability

The structural and dynamical properties, particularly the small-world effect and scale-free feature, of complex networks have attracted tremendous interest and attention in recent years. This article offers a brief review of one focal issue concerning the structural and dynamical behaviors of complex network synchronization. In the presentation, the notions of synchronization of dynamical systems on networks, stability of dynamical networks, and relationships between network structure and synchronizability, will be first introduced. Then, various technical methods for enhancing the network synchronizability will be discussed, which are roughly divided into two classes: Structural Modification and Coupling-Pattern Regulation, where the former includes three typical methods—dividing hub nodes, shortening average distances, and deleting overload edges, while the latter mainly is a method of strengthening the hub-nodes’ influence on the network.      

Synchronizability of dynamical scale-free networks subject to random errors

In this paper the robustness of network synchronizability against random deletion of nodes, i.e. errors, in dynamical scale-free networks was studied. To this end, two measures of network synchronizability, namely, the eigenratio of the Laplacian and the order parameter quantifying the degree of phase synchrony were adopted, and the synchronizability robustness on preferential attachment scale-free graphs was investigated. The findings revealed that as the network size decreases, the robustness of its synchronizability against random removal of nodes declines, i.e. the more the number of randomly removed nodes from the network, the worse its synchronizability. We also showed that this dependence of the synchronizability on the network size is different with that in the growing scale-free networks. The profile of a number of network properties such as clustering coefficient, efficiency, assortativity, and eccentricity, as a function of the network size was investigated in these two cases, growing scale-free networks and those with randomly removed nodes. The results showed that these processes are also different in terms of these metrics.     

Average Range and Network Synchronizability

The influence of structural properties of a network on the network synchronizability is studied by introducing a new concept of average range of edges. For both small-world and scale-free networks, the effect of average range on the synchronizability of networks with bounded or unbounded synchronization regions is illustrated through numerical simulations. The relations between average range, range distribution, average distance, and maximum betweenness are also explored, revealing the effects of these factors on the network synchronizability of the small-world and scale-free networks, respectively.     

Are networks with more edges easier to synchronize,or not？

In this paper, the relationship between network synchronizability and the edge-addition of its associated graph is investigated. First, it is shown that adding one edge to a cycle definitely decreases the network synchronizability. Then, since sometimes the synchronizability can be enhanced by changing the network structure, the question of whether the networks with more edges are easier to synchronize is addressed. Based on a subgraph and complementary graph method, it is shown by examples that the answer is negative even if the network structure is arbitrarily optimized. This reveals that generally there are redundant edges in a network, which not only make no contributions to synchronization but actually may reduce the synchronizability. Moreover, a simple example shows that the node betweenness centrality is not always a good indicator for the network synchronizability. Finally, some more examples are presented to illustrate how the network synchronizability varies following the addition of edges, where all the examples show that the network synchronizability globally increases but locally fluctuates as the number of added edges increases.     

Synchronizability of small-world networks generated from ring networks with equal-distance edge additions

This paper investigates the impact of edge-adding number m and edge-adding distance d on both synchronizability and average path length of NW small-world networks generated from ring networks via random edge-adding. It is found that the synchronizability of the network as a function of the distance d is fluctuant and there exist some d that have almost no impact on the synchronizability and may only scarcely shorten the average path length of the network. Numerical simulations on a network of Lorenz oscillators confirm the above results. This phenomenon shows that the contributions of randomly added edges to both the synchronizability and the average path length are not uniform nor monotone in building an NW small-world network with equal-distance edge additions, implying that only if appropriately adding edges when building up the NW small-word network can help enhance the synchronizability and/or reduce the average path length of the resultant network. Finally, it is shown that this NW small-world network has worse synchronizability and longer average path length, when compared with the conventional NW small-world network, with random-distance edge additions. This may be due to the fact that with equal-distance edge additions, there is only one shortcut distance for better information exchange among nodes and for shortening the average path length, while with random-distance edge additions, there exist many different distances for doing so.     

簇间连接方式不同的簇网络的同步过程研究

本文对簇间连接方式不同的三类簇网络的同步能力和同步过程进行研究. 构成簇网络的两个子网均为BA无标度网络, 当簇间连接方式是双向耦合时, 称其为TWD网络模型, 当簇间连接是大子网驱动小子网时, 称其为BDS网络模型, 当簇间连接是小子网驱动大子网时, 称其为SDB网络模型. 研究表明, 当小子网和大子网节点数目的比值大于某一临界值时, TWD网络模型的同步能力大于BDS网络模型的同步能力, 当该比值小于某一临界值时, TWD网络模型的同步能力小于BDS网络模型的同步能力, SDB网络模型的同步能力是三种网络结构中最差的. 对于簇间连接具有方向性的单向驱动网络, 簇网络的整体同步能力与被驱动子网的节点数和簇间连接数有关, 与驱动网络自身节点数无关. 增加簇间连接数在开始时会降低各子网的同步速度, 但最终各子网到达完全同步的时间减少, 网络的整体同步能力增强. 文中以Kuramoto相振子作为网络节点, 研究了不同情况下三种簇网络的同步过程, 证明了所得结论的正确性. 关键词： 簇网络 有向连接 同步能力 Kuramoto振子     

Better synchronizability predicted by a new coupling method

In this paper, inspired by the idea that different nodes should play different roles in network synchronization, we bring forward a coupling method where the coupling strength of each node depends on its neighbors' degrees. Compared with the uniform coupled method and the recently proposed Motter-Zhou-Kurths method, the synchronizability of scale-free networks can be remarkably enhanced by using the present coupling method, and the highest network synchronizability is achieved at β=1 which is similar to a method introduced in [AIP Conf. Proc. 776, 201 (2005)].     

Synchronizability on complex networks via pinning control

It is proved that the maximum eigenvalue sequence of the principal submatrices of coupling matrix is decreasing. The method of calculating the number of pinning nodes is given based on this theory. The findings reveal the relationship between the decreasing speed of maximum eigenvalue sequence of the principal submatrices for coupling matrix and the synchronizability on complex networks via pinning control. We discuss the synchronizability on some networks, such as scale-free networks and small-world networks. Numerical simulations show that different pinning strategies have different pinning synchronizability on the same complex network, and the consistence between the synchronizability with pinning control and one without pinning control in various complex networks.     

Effect of node deleting on network structure

The ever-increasing knowledge of the structure of various real-world networks has uncovered their complex multi-mechanism-governed evolution processes. Therefore, a better understanding of the structure and evolution of these networked complex systems requires us to describe such processes in a more detailed and realistic manner. In this paper, we introduce a new type of network growth rule which comprises addition and deletion of nodes, and propose an evolving network model to investigate the effect of node deleting on network structure. It is found that, with the introduction of node deleting, network structure is significantly transformed. In particular, degree distribution of the network undergoes a transition from scale-free to exponential forms as the intensity of node deleting increases. At the same time, nontrivial disassortative degree correlation develops spontaneously as a natural result of network evolution in the model. We also demonstrate that node deleting introduced in the model does not destroy the connectedness of a growing network so long as the increasing rate of edges is not excessively small. In addition, it is found that node deleting will weaken but not eliminate the small-world effect of a growing network, and generally it will decrease the clustering coefficient in a network.     

两层星形网络的特征值谱及同步能力

多层网络是当今网络科学研究的一个前沿方向.本文深入研究了两层星形网络的特征值谱及其同步能力的问题.通过严格导出的两层星形网络特征值的解析表达式,分析了网络的同步能力与节点数、层间耦合强度和层内耦合强度的关系.当同步域无界时,网络的同步能力只与叶子节点之间的层间耦合强度和网络的层内耦合强度有关;当叶子节点之间的层间耦合强度比较弱时,同步能力仅依赖于叶子节点之间的层间耦合强度;而当层内耦合强度比较弱时,同步能力依赖于层内耦合强度;当同步域有界时,节点数、层间耦合强度和层内耦合强度对网络的同步能力都有影响.当叶子节点之间的层间耦合强度比较弱时,增大叶子节点之间的层间耦合强度会增强网络的同步能力,而节点数、中心节点之间的层间耦合强度和层内耦合强度的增大反而会减弱网络的同步能力;而当层内耦合强度比较弱时,增大层内耦合强度会增强网络的同步能力,而节点数、层间耦合强度的增大会减弱网络的同步能力.进一步,在层间和层内耦合强度都相同的基础上,讨论了如何改变耦合强度更有利于同步.最后,对两层BA无标度网络进行数值仿真,得到了与两层星形网络非常类似的结论.     

Effect of Eliminating Edges on Robustness of Scale-Free Networks under Intentional Attack

We study the robustness of complex networks under edge elimination. We propose three different edge elimination strategies and investigate their effects on the robustness of scale-free networks under intentional attack. We show that deleting a proper fraction of edges connecting hub nodes and hub nodes can enhance the robustness of scale-free networks under intentional attack.     

Synchronization in evolving snowdrift game model

The interaction between the evolution of the game and the underlying network structure with evolving snowdrift game model is investigated. The constructed network follows a power-law degree distribution typically showing scale-free feature. The topological features of average path length, clustering coefficient, degree-degree correlations and the dynamical feature of synchronizability are studied. The synchronizability of the constructed networks changes by the interaction. It will converge to a certain value when sufficient new nodes are added. It is found that initial payoffs of nodes greatly affect the synchronizability. When initial payoffs for players are equal, low common initial payoffs may lead to more heterogeneity of the network and good synchronizability. When initial payoffs follow certain distributions, better synchronizability is obtained compared to equal initial payoff. The result is also true for phase synchronization of nonidentical oscillators.     

The emergence of scale-free networks with a seceding mechanism

In order to explore further the underlying mechanism of scale-free networks, we study stochastic secession as a mechanism for the creation of complex networks. In this evolution the network growth incorporates the addition of new nodes, the addition of new links between existing nodes, the deleting and rewiring of some existing links, and the stochastic secession of nodes. To random growing networks with preferential attachment, the model yields scale-free behavior for the degree distribution. Furthermore, we obtain an analytical expression of the power-law degree distribution with scaling exponent γ ranging from 1.1 to 9. The analytical expressions are in good agreement with the numerical simulation results.     

Two Novel Methods to Enhance Network Synchronizability

In this paper, we propose two methods to enhance the synchronizability of a class of complex networks which do not hold the positive correlation between betweenness centrality （BC） and degree of a node, and observe other topology characteristics of the network affected by the methods. Numerical simulations show that both methods can effectively enhance the synchronizability of this kind of networks. Furthermore, we show that the maximal BC of all edges is an important factor to affect the network synchronizability, although it is not the unique factor.     

Synchronizability of Highly Clustered Scale-Free Networks

We consider the effect of clustering coefficient on the synchronizability of coupled oscillators located on scale-free networks. The analytic result for the value of clustering coefficient aiming at a highly clustered scale-free network model, the Holme-Kim model is obtained, and the relationship between network synchronizability and clustering coefficient is reported. The simulation results strongly suggest that the more clustered the network, the poorer the synchronizability.     

基于相对距离的复杂网络谱粗粒化方法

复杂网络的同步作为一种重要的网络动态特性,在通信、控制、生物等领域起着重要的作用.谱粗粒化方法是一种在保持原始网络的同步能力尽量不变情况下将大规模网络约简为小规模网络的算法.此方法在对约简节点分类时是以每个节点对应特征向量分量间的绝对距离作为判断标准,在实际运算中计算量大,可执行性较差.本文提出了一种以特征向量分量间相对距离作为分类标准的谱粗粒化改进算法,能够使节点的合并更加合理,从而更好地保持原始网络的同步能力.通过经典的三种网络模型(BA无标度网络、ER随机网络、NW小世界网络)和27种不同类型实际网络的数值仿真分析表明,本文提出的算法对比原来的算法能够明显改善网络的粗粒化效果,并发现互联网、生物、社交、合作等具有明显聚类结构的网络在采用谱粗粒化算法约简后保持同步的能力要优于电力、化学等模糊聚类结构的网络.     

Heterogeneity in oscillator networks: are smaller worlds easier to synchronize?

Small-world and scale-free networks are known to be more easily synchronized than regular lattices, which is usually attributed to the smaller network distance between oscillators. Surprisingly, we find that networks with a homogeneous distribution of connectivity are more synchronizable than heterogeneous ones, even though the average network distance is larger. We present numerical computations and analytical estimates on synchronizability of the network in terms of its heterogeneity parameters. Our results suggest that some degree of homogeneity is expected in naturally evolved structures, such as neural networks, where synchronizability is desirable.     

一类高聚类系数的加权无标度网络及其同步能力分析

针对真实世界中大规模网络都具有明显聚类效应的特点, 提出一类具有高聚类系数的加权无标度网络演化模型, 该模型同时考虑了优先连接、三角结构、随机连接和社团结构等四种演化机制. 在模型演化规则中, 以概率p增加单个节点, 以概率1–p增加一个社团. 与以往研究的不同在于新边的建立, 以概率φ在旧节点之间进行三角连接, 以概率1–φ进行随机连接. 仿真分析表明, 所提出的网络度、强度和权值分布都是服从幂律分布的形式, 且具有高聚类系数的特性, 聚类系数的提高与社团结构和随机连接机制有直接的关系. 最后通过数值仿真分析了网络演化机制对同步动态特性的影响, 数值仿真结果表明, 网络的平均聚类系数越小, 网络的同步能力越强. 关键词： 无标度网络 加权网络 聚类系数 同步能力     

Optimizing Synchronizability of Scale-Free Networks in Geographical Space

We investigate the relationship between the structure and the synchronizability of scale-free networks in geographical space. With an optimization approach, the numerical results indicate that when the network synchronizability is improved, the geographical distance becomes larger while the maximal load decreases. Thus the maximal betweenness can be a candidate factor that affects the network synchronizability both in topological space and in geographical space.     

多层单向耦合星形网络的特征值谱及同步能力分析

随着复杂网络同步的进一步发展,对复杂网络的研究重点由单层网络转向更加接近实际网络的多层有向网络.本文分别严格推导出三层、多层的单向耦合星形网络的特征值谱,并分析了耦合强度、节点数、层数对网络同步能力的影响,重点分析了层数和层间中心节点之间的耦合强度对多层单向耦合星形网络同步能力的影响,得出了层数对多层网络同步能力的影响至关重要.当同步域无界时,网络的同步能力与耦合强度、层数有关,同步能力随其增大而增强;当同步域有界时,对于叶子节点向中心节点耦合的多层星形网络,当层内耦合强度较弱时,层内耦合强度的增大会使同步能力增强,而层间叶子节点之间的耦合强度、层数的增大反而会使同步能力减弱;当层间中心节点之间的耦合强度较弱时,层间中心节点之间的耦合强度、层数的增大会使同步能力增强,层内耦合强度、层间叶子节点之间的耦合强度的增大反而会使同步能力减弱.对于中心节点向叶子节点耦合的多层星形网络,层间叶子节点之间的耦合强度、层数的增大会使同步能力增强,层内耦合强度、节点数、层间中心节点之间的耦合强度的增大反而会使同步能力减弱.