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.     

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.     

Rewiring dynamical networks with prescribed degree distribution for enhancing synchronizability

In this paper, we present an algorithm for enhancing synchronizability of dynamical networks with prescribed degree distribution. The algorithm takes an unweighted and undirected network as input and outputs a network with the same node-degree distribution and enhanced synchronization properties. The rewirings are based on the properties of the Laplacian of the connection graph, i.e., the eigenvectors corresponding to the second smallest and the largest eigenvalues of the Laplacian. A term proportional to the eigenvectors is adopted to choose potential edges for rewiring, provided that the node-degree distribution is preserved. The algorithm can be implemented on networks of any sizes as long as their eigenvalues and eigenvectors can be calculated with standard algorithms. The effectiveness of the proposed algorithm in enhancing the network synchronizability is revealed by numerical simulation on a number of sample networks including scale-free, Watts-Strogatz, and Erdo?s-Re?nyi graphs. Furthermore, a number of network's structural parameters such as node betweenness centrality, edge betweenness centrality, average path length, clustering coefficient, and degree assortativity are tracked as a function of optimization steps.     

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.     

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.     

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.     

网络集聚性对节点中心性指标的准确性影响

节点中心性指标是从特定角度对网络某一方面的结构特点进行刻画的度量指标, 因此网络拓扑结构的改变会对节点中心性指标的准确性产生重要影响. 本文利用Holme-Kim模型构建可变集聚系数的无标度网络, 然后采用Susceptible-Infective-Removal模型进行传播影响力的仿真实验, 接着分析了节点中心性指标在不同集聚系数的无标度网络中的准确性. 结果表明, 度中心性和介数中心性的准确性在低集聚系数的网络中表现更好, 特征向量中心性则在高集聚类网络中更准确, 而紧密度中心性的准确性受网络集聚系数的变化影响较小. 因此当网络的集聚系数较低时, 可选择度或者介数作为中心性指标进行网络节点影响力评价; 反之则选择紧密度指标或特征向量指标较好, 尤其当网络的集聚系数接近0.6时特征向量的准确性可以高达到0.85, 是度量小规模网络的较优选择. 另一方面, 传播过程的感染率越高, 度指标和介数指标越可靠, 紧密度和特征向量则相反. 最后Autonomous System实证网络的断边重连实验, 进一步验证了网络集聚性的改变会对节点中心性指标的准确性产生重要影响.     

Routing in scale-free networks based on expanding betweenness centrality

In this paper, an improved routing strategy is proposed for enhancing the traffic capacity of scale-free networks. Instead of using the information of degree and betweenness centrality, the new algorithm is derived on the basis of the expanding betweenness centrality of nodes, which gives an estimate of the traffic handled by the vertex for a certain route set. Since the nodes with large betweenness centrality are more susceptible to traffic congestion, the traffic can be improved by redistributing traffic loads from nodes with large betweenness centrality to nodes with small betweenness centrality in the process of computing the collective routing table. Comparing with results of previous routing strategies, it is shown that the present improved routing performs more effectively.     

Optimization of transport protocols in complex networks

In this paper, an optimal routing strategy is proposed to enhance the traffic capacity of complex networks. In order to avoid nodes overloading, the new algorithm is derived on the basis of generalized betweenness centrality which gives an estimate of traffic handled by the node for a route set. Since the nodes with large betweenness centrality are more susceptible to traffic congestion, the traffic can be improved, as our strategy, by redistributing traffic load from nodes with large betweenness centrality to nodes with small betweenness centrality in the proceeding of computing collective routing table. Particularly, depending on a parameter that controls the optimization scale, the new routing can not only enlarge traffic capacity of networks more, but also enhance traffic efficiency with smaller average path length. Comparing results of previous routing strategies, it is shown that the present improved routing performs more effectively.     

一种复杂网络路由策略的普适优化算法

现有的复杂网络路由策略很多,改进算法也不断涌现,但是目前还没有一个统一的标准来衡量算法是否达到网络最佳传输效果.针对这一问题,本文提出一种适用于现有路由策略的普适优化算法.首先通过理论分析指出制约网络传输能力的关键因素是最大介数中心度,因而"最大介数中心度是否已经最低"成为评判路由策略是否最优的标准.在此基础上,采用"惩罚选择法"避开网络中介数中心度值比较大的节点,使网络介数中心度值分布更均匀,均衡网络中各个节点的传输负载.仿真结果显示,该优化算法针对现有路由策略均能降低最大介数中心度值,大幅度提高网络的传输能力.     

Methods to find community based on edge centrality

Divisive algorithms are of great importance for community detection in complex networks. One algorithm proposed by Girvan and Newman (GN) based on an edge centrality named betweenness, is a typical representative of this field. Here we studied three edge centralities based on network topology, walks and paths respectively to quantify the relevance of each edge in a network, and proposed a divisive algorithm based on the rationale of GN algorithm for finding communities that removes edges iteratively according to the edge centrality values in a certain order. In addition, we gave a comparison analysis of these measures with the edge betweenness and information centrality. We found the principal difference among these measures in the partition procedure is that the edge centrality based on walks first removes the edge connected with a leaf vertex, but the others first delete the edge as a bridge between communities. It indicates that the edge centrality based on walks is harder to uncover communities than other edge centralities. We also tested these measures for community detection. The results showed that the edge information centrality outperforms other measures, the edge centrality based on walks obtains the worst results, and the edge betweenness gains better performance than the edge centrality based on network topology. We also discussed our method’s efficiency and found that the edge centrality based on walks has a high time complexity and is not suitable for large networks.     

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.     

Centrality measure of complex networks using biased random walks

We propose a novel centrality measure based on the dynamical properties of a biased random walk to provide a general framework for the centrality of vertex and edge in scale-free networks (SFNs). The suggested centrality unifies various centralities such as betweenness centrality (BC), load centrality (LC) and random walk centrality (RWC) when the degree, k, is relatively large. The relation between our centrality and other centralities in SFNs is clearly shown by both analytic and numerical methods. Regarding to the edge centrality, there have been few established studies in complex networks. Thus, we also provide a systematic analysis for the edge BC (LC) in SFNs and show that the distribution of edge BC satisfies a power-law. Furthermore we also show that the suggested centrality measures on real networks work very well as on the SFNs.     

Optimized network structure and routing metric in wireless multihop ad hoc communication

Inspired by the Statistical Physics of complex networks, wireless multihop ad hoc communication networks are considered in abstracted form. Since such engineered networks are able to modify their structure via topology control, we search for optimized network structures, which maximize the end-to-end throughput performance. A modified version of betweenness centrality is introduced and shown to be very relevant for the respective modeling. The calculated optimized network structures lead to a significant increase of the end-to-end throughput. The discussion of the resulting structural properties reveals that it will be almost impossible to construct these optimized topologies in a technologically efficient distributive manner. However, the modified betweenness centrality also allows to propose a new routing metric for the end-to-end communication traffic. This approach leads to an even larger increase of throughput capacity and is easily implementable in a technologically relevant manner.     

Identifying important nodes in weighted functional brain networks: a comparison of different centrality approaches

We compare different centrality metrics which aim at an identification of important nodes in complex networks. We investigate weighted functional brain networks derived from multichannel electroencephalograms recorded from 23 healthy subject under resting-state eyes-open or eyes-closed conditions. Although we observe the metrics strength, closeness, and betweenness centrality to be related to each other, they capture different spatial and temporal aspects of important nodes in these networks associated with behavioral changes. Identifying and characterizing of these nodes thus benefits from the application of several centrality metrics.     

Centrality in networks of urban streets

Centrality has revealed crucial for understanding the structural properties of complex relational networks. Centrality is also relevant for various spatial factors affecting human life and behaviors in cities. Here, we present a comprehensive study of centrality distributions over geographic networks of urban streets. Five different measures of centrality, namely degree, closeness, betweenness, straightness and information, are compared over 18 1-square-mile samples of different world cities. Samples are represented by primal geographic graphs, i.e., valued graphs defined by metric rather than topologic distance where intersections are turned into nodes and streets into edges. The spatial behavior of centrality indices over the networks is investigated graphically by means of color-coded maps. The results indicate that a spatial analysis, that we term multiple centrality assessment, grounded not on a single but on a set of different centrality indices, allows an extended comprehension of the city structure, nicely capturing the skeleton of most central routes and subareas that so much impacts on spatial cognition and on collective dynamical behaviors. Statistically, closeness, straightness and betweenness turn out to follow similar functional distribution in all cases, despite the extreme diversity of the considered cities. Conversely, information is found to be exponential in planned cities and to follow a power-law scaling in self-organized cities. Hierarchical clustering analysis, based either on the Gini coefficients of the centrality distributions, or on the correlation between different centrality measures, is able to characterize classes of cities.     

交织型层级复杂网

为研究两个异质关联网络复合后的结构特征与节点中心性特征,本文提出了交织型层级复杂网络的概念,可描述由两个具有部分相同节点,连接边属性近似的子网所构成的层级复杂网络,并定义了节点交织系数、路径交织系数和网络交织系数3种测度用于衡量两个子网之间的密切程度.针对该类网络,研究并改进了节点度中心性和介数中心性的计算方法,同时提出一种新的中心性指标—助联性,用于衡量子网的某一节点对另一子网联通性和流通性的助益.通过实验分析,验证了本文各类指标的有效性.     

基于节点负荷失效的网络可控性研究

Liu和Barabasi将现代控制理论应用到线性系统的网络可控性问题上, 提出了最小驱动节点集的计算方法, 解决了复杂网络控制的可计算问题. 针对现实网络中存在的节点因负荷过载而失效的问题, 本文提出了基于节点负荷失效的网络可控性模型. 通过对网络采用介数和Weibull失效模型, 在随机和目标失效机制下进行仿真, 研究结果表明: 维持无标度网络可控性的难度要明显大于随机网络; 在目标节点失效机制下, 即使对网络输入极少的失效信号, 也能极大地破坏网络的可控性; 使高介数节点失效要比使度高节点失效更能破坏网络的可控性, 说明高介数节点在维持网络可控性上发挥着重要作用; 对不同的负荷失效模型, 要合理采取措施, 防止网络发生阶跃性全不可控现象. 关键词： 网络可控性 结构可控性 节点失效     

Betweenness centrality in finite components of complex networks

We use generating function formalism to obtain an exact formula of the betweenness centrality in finite components of random networks with arbitrary degree distributions. The formula is obtained as a function of the degree and the component size, and is confirmed by simulations for Poisson, exponential, and power-law degree distributions. We find that the betweenness centralities for the three distributions are asymptotically power laws with an exponent 1.5 and are invariant to the particular distribution parameters.