On synchronizability and heterogeneity in unweighted networks

Synchronization in classes of continuous-time dynamical unweighted networks with different topologies is investigated. A synchronization-optimal network model based on rewiring of links is proposed. Compared with other networks, it exhibits a stronger synchronizability. We presented link density and investigated the correlation of synchronizability, link density and heterogeneity in degree distribution. In this work, it will be shown that synchronizability of Type I networks is independent of heterogeneity in the degree distribution when the link density is smaller than 0.02. Synchronizability and link density are proportional. When the link density is larger than 0.025, heterogeneity decides the curve slope. The synchronizability of Type II networks is drastically enhanced by enhancing the link density when the link density is smaller than 0.025. However synchronizability increases weakly by enhancing the link density when the link density is larger than 0.025.     

加权方式对网络同步能力的影响

针对真实网络中权值与端点度的相关特性,提出了一种与始点和终点的度 都相关的非对称加权方式.在不同的网络结构下研究加权方式对同步能力的影响. 研究发现网络异质性越强时,通过调节网络权值改变网络同步能力的效果越显著, 而网络越匀质时,调节权值的方式改变网络同步能力的效果越不明显. 仿真实验显示无论在小世界网络还是无标度网络中,网络都是在节点的输入强度为1处获得最优的同步能力.     

Improving consensus and synchronizability of networks of coupled systems via adding links

The problem of improving the consensus and synchronizability for weighted networks of dynamic systems via adding links among them is investigated in this paper. A weighted network entropy is introduced to measure the homogeneity of the weighted networks. The relationship between the homogeneity and the consensus and synchronizability is analyzed. A solution scheme is proposed to improve the consensus and synchronizability of the weighted networks through maximizing their homogeneity via adding as few links as possible. Weighted networks created from the Barrat model are tested by using the proposed scheme, and two commonly used link-adding approaches are selected to compare with this scheme. Computer simulation results show that the proposed link-adding scheme can enhance the consensus and the synchronizability of the weighted networks effectively and efficiently.     

Enhancing synchronizability by rewiring networks

<正>According to different forms of synchronized region,complex networks are divided into typeⅠ(unbounded synchronization region) and typeⅡ(bounded synchronization region) networks.This paper presents a rewiring algorithm to enhance the synchronizability of typeⅠand typeⅡnetworks.By utilizing the algorithm for an unweighted and undirected network,a better synchronizability of network with the same number of nodes and edges can be obtained. Numerical simulations on several different network models are used to support the proposed procedure.The relationship between different topological properties of the networks and the number of rewirings are shown.It finds that the final optimized network is independent of the initial network,and becomes homogeneous.In addition the optimized networks have similar structural properties in the sense of degree,and node and edge betweenness centralities.However,they do not have similar cluster coefficients for typeⅡnetworks.The research may be useful for designing more synchronizable networks and understanding the synchronization behaviour of networks.     

Enhancing synchronizability by weight randomization on regular networks

In weighted networks, redistribution of link weights can effectively change the properties of networks, even though the corresponding binary topology remains unchanged. In this paper, the effects of weight randomization on synchronization of coupled chaotic maps is investigated on regular weighted networks. The results reveal that synchronizability is enhanced by redistributing of link weights, i.e. coupled maps reach complete synchronization with lower cost. Furthermore, we show numerically that the heterogeneity of link weights could improve the complete synchronization on regular weighted networks.     

Synchronizability of network ensembles with prescribed statistical properties

It has been shown that synchronizability of a network is determined by the local structure rather than the global properties. With the same global properties, networks may have very different synchronizability. In this paper, we numerically studied, through the spectral properties, the synchronizability of ensembles of networks with prescribed statistical properties. Given a degree sequence, it is found that the eigenvalues and eigenratios characterizing network synchronizability have well-defined distributions, and statistically, the networks with extremely poor synchronizability are rare. Moreover, we compared the synchronizability of three network ensembles that have the same nodes and average degree. Our work reveals that the synchronizability of a network can be significantly affected by the local pattern of connections, and the homogeneity of degree can greatly enhance network synchronizability for networks of a random nature.     

权重分布对加权局域世界网络动力学同步的影响

加权网络可以更细致地刻画复杂系统中两节点之间的相互作用, 所以加权网络也比无权网络更接近真实的复杂系统. 改变权重分布来调节和改善复杂网络的性质也成为一种新的研究方法. 基于现有无权网络同步的概念, 应用特征值比R来衡量加权局域世界网络的同步能力, 发现权重分布对加权局域世界网络的同步能力有很重要的影响, 权重分布越均匀, 网络的同步能力就越大. 关键词： 权重分布 局域世界 加权网络 同步     

Synchronization speed of identical oscillators on community networks

Synchronizability of complex oscillators networks has attracted much research interest in recent years. In contrast, in this paper we investigate numerically the synchronization speed, rather than the synchronizability or synchronization stability, of identical oscillators on complex networks with communities. A new weighted community network model is employed here, in which the community strength could be tunable by one parameter δ. The results showed that the synchronization speed of identical oscillators on community networks could reach a maximal value when δ is around 0.1. We argue that this is induced by the competition between the community partition and the scale-free property of the networks. Moreover, we have given the corresponding analysis through the second least eigenvalue λ₂ of the Laplacian matrix of the network which supports the previous result that the synchronization speed is determined by the value of λ₂.     

Concentric circular arrays of Josephson junctions as dressed models of granular superconductors

In this paper networks that optimize a combined measure of local and global synchronizability are evolved. It is shown that for low coupling improvements in the local synchronizability dominate network evolution. This leads to an expressed grouping of elements with similar native frequency into cliques, allowing for an early onset of synchronization, but rendering full synchronization hard to achieve. In contrast, for large coupling the network evolution is governed by improvements towards full synchronization, preventing any expressed community structure. Such networks exhibit strong coupling between dissimilar oscillators. Albeit a rapid transition to full synchronization is achieved, the onset of synchronization is delayed in comparison to the first type of networks. The paper illustrates that an early onset of synchronization (which relates to clustering) and global synchronization are conflicting demands on network topology.     

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)].     

Error and attack tolerance of synchronization in Hindmarsh–Rose neural networks with community structure

Synchronization is one of the most important features observed in large-scale complex networks of interacting dynamical systems. As is well known, there is a close relation between the network topology and the network synchronizability. Using the coupled Hindmarsh–Rose neurons with community structure as a model network, in this paper we explore how failures of the nodes due to random errors or intentional attacks affect the synchronizability of community networks. The intentional attacks are realized by removing a fraction of the nodes with high values in some centrality measure such as the centralities of degree, eigenvector, betweenness and closeness. According to the master stability function method, we employ the algebraic connectivity of the considered community network as an indicator to examine the network synchronizability. Numerical evidences show that the node failure strategy based on the betweenness centrality has the most influence on the synchronizability of community networks. With this node failure strategy for a given network with a fixed number of communities, we find that the larger the degree of communities, the worse the network synchronizability; however, for a given network with a fixed degree of communities, we observe that the more the number of communities, the better the network synchronizability.     

Optimal synchronizability of networks

We numerically investigate how to enhance synchronizability of coupled identical oscillators in complex networks with research focus on the roles of the high level of clustering for a given heterogeneity in the degree distribution. By using the edge-exchange method with the fixed degree sequence, we first directly maximize synchronizability measured by the eigenratio of the coupling matrix, through the use of the so-called memory tabu search algorithm developed in applied mathematics. The resulting optimal network, which turns out to be weakly disassortative, is observed to exhibit a small modularity. More importantly, it is clearly revealed that the optimally synchronizable network for a given degree sequence shows a very low level of clustering, containing much fewer small-size loops than the original network. We then use the clustering coefficient as an object function to be reduced during the edge exchanges, and find it a very efficient way to enhance synchronizability. We thus conclude that under the condition of a given degree heterogeneity, the clustering plays a very important role in the network synchronization.     

Undetermination of the relation between network synchronizability and betweenness centrality

Betweenness centrality is taken as a sensible indicator of the synchronizability of complex networks. To test whether betweenness centrality is a proper measure of the synchronizability in specific realizations of random networks,this paper adds edges to the networks and then evaluates the changes of betweenness centrality and network synchronizability. It finds that the two quantities vary independently.     

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     

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.     

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.     

最优的Newman-Watts网络与遍历网络的同步

在Hindmarsh-Rose神经元动力系统中研究了Newman-Watts （NW）网络的同步,给出了一些最优同步网络的拓扑结构. 数值结果表明：NW网络的同步能力主要由耦合点在耦合空间的分布决定,耦合点分布均匀的NW网络一般具有较强的同步能力;在给定连边数的情况下,可能存在多个结构不同的最优同步网络,最优同步网络具有最强的同步能力、均匀的度分布和较好的对称性,但是其对称性不一定是最好的. 最优同步网络一般是非规则网络,但在少数情况下,规则网络也有可能是最优同步网络. 提出了一种新的网络——遍历网络,该网络具有最优同步网络的特点和很强的同步能力. 关键词： Newman-Watts网络 对称度 耦合空间 同步     

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.