Exploring the robustness of urban bus network: A case from Southern China

The robustness of urban bus network is essential to a city that heavily relies on buses as its main transportation solution. In this paper, the urban bus network has been modeled as a directed and space L network, and Changsha, a transportation hub of nearly 8 million people and hundreds of bus lines in southern China, is taken as a case. Based on the quantitative analyses of the topological properties, it is found that Changsha urban bus network is a scale-free network, not a small-world network. To evaluate the robustness of the network, five scenarios of network failure are simulated, including a random failure and four types of intentional attacks that differed in key node identification methods (i.e., unweighted degree or betweenness centrality) and attack strategies (i.e., normal or cascading attack). It is revealed that intentional attacks are more destructive than a random failure, and cascading attacks are more disruptive than normal attacks in the urban bus network. In addition, the key nodes identification methods are found to play a critical role in the robustness of the urban bus network. Specifically, cascading attack could be more disruptive when the betweenness centrality is used to identify key nodes; in contrast, normal attack could be more disruptive when the unweighted degree is used to identify key nodes. Our results could provide reference for risk management of urban bus network.     

基于超图的超网络相继故障分析

分析了快递超网络和电子元件超网络的相继故障扩散方式, 结合超图理论提出了2-section 图分析法和线图分析法, 并仿真分析了无标度超网络耦合映像格子的相继故障进程. 结果表明: 无标度超网络对外部攻击表现出了既鲁棒又脆弱的特性. 针对相继故障的不同扩散方式, 无标度超网络的相继故障行为表现出不同的特点. 超网络的相继故障行为和超网络的超度以及超边度分布有密切的联系, 也和超网络中超边的个数有关. 通过和同规模的Barabasi-Albert (BA)无标度网络对比, 在同一种攻击方式下同规模的无标度超网络都比BA 无标度网络表现出了更强的鲁棒性. 另外, 基于超边扩散的相继故障进程比基于节点扩散的相继故障进程更加缓慢.     

An effective method to improve the robustness of small-world networks under attack

In this study, the robustness of small-world networks to three types of attack is investigated. Global efficiency is introduced as the network coefficient to measure the robustness of a small-world network. The simulation results prove that an increase in rewiring probability or average degree can enhance the robustness of the small-world network under all three types of attack. The effectiveness of simultaneously increasing both rewiring probability and average degree is also studied, and the combined increase is found to significantly improve the robustness of the small-world network.Furthermore, the combined effect of rewiring probability and average degree on network robustness is shown to be several times greater than that of rewiring probability or average degree individually. This means that small-world networks with a relatively high rewiring probability and average degree have advantages both in network communications and in good robustness to attacks. Therefore, simultaneously increasing rewiring probability and average degree is an effective method of constructing realistic networks. Consequently, the proposed method is useful to construct efficient and robust networks in a realistic scenario.     

负荷作用下相依网络中的级联故障

研究负荷作用下相依网络中的级联故障具有重要的现实意义, 可为提高相依网络的鲁棒性提供参考. 构建了双层相依网络级联故障模型, 主要研究了外部度和内部度对负荷贡献比、耦合因素、层内度-度相关性对相依网络级联故障的影响. 研究表明, 当外部度和内部度对负荷贡献比达到一定值时, 相依网络抵抗级联故障的鲁棒性最强. 而耦合因素的影响是多方面的, 为了达到较高鲁棒性, 建议采用异配耦合方式和尽可能大的平均外部度, 并尽量使外部度保持均匀分布. 另外, 与不考虑负荷作用时相反, 当表征层内度-度相关性的相关系数越大时, 其抵抗级联故障的能力越强.     

Impact of core-periphery structure on cascading failures in interdependent scale-free networks

Core-periphery structure is a typical meso-scale structure in networks. Previous studies on core-periphery structure mainly focus on the improvement of detection methods, while the research on the impact of core-periphery structure on cascading failures in interdependent networks is still missing. Therefore, we investigate the cascading failures of interdependent scale-free networks with different core-periphery structures and coupling preferences in the paper. First, we introduce an evaluation index to calculate the goodness of core-periphery structure. Second, we propose a new scale-free network evolution model, which can generate tunable core-periphery structures, and its degree distribution is analyzed mathematically. Finally, based on a degree-load-based cascading failure model, we mainly investigate the impact of goodness of core-periphery structure on cascading failures in both symmetrical and asymmetrical interdependent networks. Through numerical simulations, we find that with the same average degree, the networks with weak core-periphery structure will be more robust, while the initial load on node will influence the improvement of robustness. In addition, we also find that the inter-similarity coupling performs better than random coupling. These findings may be helpful for building resilient interdependent networks.     

Robustness analysis of static routing on networks

Robustness is one of the crucial properties that needs to be considered in the design of routing strategies on networks. We study the robustness of three typical routing strategies, which are the SP (shortest path), EP (efficient path), and OP (optimal path) strategies, by simulating several different kinds of attacks including random attacks, target attacks and cascading failures on scale-free networks. Results of the average path length, betweenness centrality, network capacity, etc., demonstrate that the EP strategy is more robust than the other two, and the OP strategy is more reliable than the SP strategy in general. However, on the power-grid network, the OP strategy is more resistant against cascading failures than the EP and SP strategies.     

复杂网络中带有应急恢复机理的级联动力学分析

提出带有应急恢复机理的网络级联故障模型,研究模型在最近邻耦合网络,Erdos-Renyi随机网络,Watts-Strogatz小世界网络和Barabasi-Albert无标度网络四种网络拓扑下的网络级联动力学行为.给出了应急恢复机理和网络效率的定义,并研究了模型中各参数对网络效率和网络节点故障率在级联故障过程中变化情况的影响.结果表明,模型中应急恢复概率的增大减缓了网络效率的降低速度和节点故障率的增长速度,并且提高了网络的恢复能力.而且网络中节点负载容量越大,网络效率降低速度和节点故障率的增长速度越慢.同时,随着节点过载故障概率的减小,网络效率的降低速度和节点故障率的增长速度也逐渐减缓.此外,对不同网络拓扑中网络效率和网络节点故障率在级联故障过程中的变化情况进行分析,结果发现网络拓扑节点度分布的异质化程度的增大,提高了级联故障所导致的网络效率的降低速度和网络节点故障率的增长速度.以上结果分析了复杂网络中带有应急恢复机理的网络级联动力学行为,为实际网络中级联故障现象的控制和防范提供了参考.     

Deterministic scale-free small-world networks of arbitrary order

In many real-life networks, both the scale-free distribution of degree and small-world behavior are important features. There are many random or deterministic models of networks to simulate these features separately. However, there are few models that combine the scale-free effect and small-world behavior, especially in terms of deterministic versions. What is more, all the existing deterministic algorithms running in the iterative mode generate networks with only several discrete numbers of nodes. This contradicts the purpose of creating a deterministic network model on which we can simulate some dynamical processes as widely as possible. According to these facts, this paper proposes a deterministic network generation algorithm, which can not only generate deterministic networks following a scale-free distribution of degree and small-world behavior, but also produce networks with arbitrary number of nodes. Our scheme is based on a complete binary tree, and each newly generated leaf node is further linked to its full brother and one of its direct ancestors. Analytical computation and simulation results show that the average degree of such a proposed network is less than 5, the average clustering coefficient is high (larger than 0.5, even for a network of size 2 million) and the average shortest path length increases much more slowly than logarithmic growth for the majority of small-world network models.     

Cascading failures on weighted urban traffic equilibrium networks

In this paper, we study the cascading failure on weighted urban traffic equilibrium networks by introducing a more practical flow assignment mechanism. The whole process including edges overloading to node malfunctioning, dynamic spanning clustering and the phase transitions trigged with O-D flow evolving is simulated. It is found that there are three districts: slow, fast and stationary (collapse for scale-free networks) cascading failure districts. And different topologies have large effects on the ranges of these districts. Simulations also show that, although the latter can support larger traffic flow, homogeneous networks appear to be more robust against cascading failures than heterogeneous ones.     

Cascading failures in congested complex networks with feedback

In this article, we investigate cascading failures in complex networks by introducing a feedback. To characterize the effect of the feedback, we define a procedure that involves a self-organization of trip distribution during the process of cascading failures. For this purpose, user equilibrium with variable demand is used as an alternative way to determine the traffic flow pattern throughout the network. Under the attack, cost function dynamics are introduced to discuss edge overload in complex networks, where each edge is assigned a finite capacity (controlled by parameter α). We find that scale-free networks without considering the effect of the feedback are expected to be very sensitive to α as compared with random networks, while this situation is largely improved after introducing the feedback.     

Two-peak and three-peak optimal complex networks

A central issue in complex networks is tolerance of random failures and intentional attacks. Current literature emphasizes the dichotomy between networks with a power-law node connectivity distribution, which are robust to random failures but fragile to targeted attacks, versus networks with an exponentially decaying connectivity distribution, which are less tolerant to failures but more resilient to attacks. We prove analytically that the optimal network configuration under a classic measure of robustness is altogether different from both of the above: in all cases, failure and/or attack, there are no more than three distinct node connectivities in the optimal network.     

Networked characteristics of the urban rail transit networks

Urban rail transit networks (URTNs) have experienced rapid development and have been receiving much attention recently. In this paper, we comprehensively analyze the topological characteristics of urban rail transit networks, and we find that the average degrees of nodes of urban rail transit networks lie in the interval [2, 2.45], most of the average shortest path lengths between pairs of nodes belong to the interval [10, 16] and the average betweenness of nodes and edges linearly increase with the increase of the number of stations. Moreover, the cumulative probability distributions of the degree and shortest path length can be fitted by exponential distribution and Gauss distribution, respectively. The network failures of the urban rail transit networks are discussed and we also discover that the highest betweenness node-based attack is the most effective method to destroy the network.     

Dynamical robustness analysis of weighted complex networks

Robustness of weighted complex networks is analyzed from nonlinear dynamical point of view and with focus on different roles of high-degree and low-degree nodes. We find that the phenomenon for the low-degree nodes being the key nodes in the heterogeneous networks only appears in weakly weighted networks and for weak coupling. For all other parameters, the heterogeneous networks are always highly vulnerable to the failure of high-degree nodes; this point is the same as in the structural robustness analysis. We also find that with random inactivation, heterogeneous networks are always more robust than the corresponding homogeneous networks with the same average degree except for one special parameter. Thus our findings give an integrated picture for the dynamical robustness analysis on complex networks.     

一种全局同质化相依网络耦合模式

相依网络的相依模式(耦合模式)是影响其鲁棒性的重要因素之一.本文针对具有无标度特性的两个子网络提出一种全局同质化相依网络耦合模式.该模式以子网络的总度分布均匀化为原则建立相依网络的相依边,一方面压缩度分布宽度,提高其对随机失效的抗毁性,另一方面避开对度大节点(关键节点)的相依,提高其对蓄意攻击的抗毁性.论文将其与常见的节点一对一的同配、异配及随机相依模式以及一对多随机相依模式作了对比分析,仿真研究其在随机失效和蓄意攻击下的鲁棒性能.研究结果表明,本文所提全局同质化相依网络耦合模式能大大提高无标度子网络所构成的相依网络抗级联失效能力.本文研究成果能够为相依网络的安全设计等提供指导意义.     

Dynamical robustness of networks based on betweenness against multi-node attack

We explore the robustness of a network against failures of vertices or edges where a fraction f of vertices is removed and an overload model based on betweenness is constructed. It is assumed that the load and capacity of vertex i are correlated with its betweenness centrality B_i as B_i~θ and(1 + α)Bθi(θ is the strength parameter, α is the tolerance parameter).We model the cascading failures following a local load preferential sharing rule. It is found that there exists a minimal αc when θ is between 0 and 1, and its theoretical analysis is given. The minimal αc characterizes the strongest robustness of a network against cascading failures triggered by removing a random fraction f of vertices. It is realized that the minimalαc increases with the increase of the removal fraction f or the decrease of average degree. In addition, we compare the robustness of networks whose overload models are characterized by degree and betweenness, and find that the networks based on betweenness have stronger robustness against the random removal of a fraction f of vertices.     

基于感知流量算法的复杂网络拥塞问题研究

研究了在具有感知流量的路由策略下,复杂网络的拓扑结构对网络中传输流量的影响.为了描述数据包传输过程的有效性,通过引入一个状态参数,利用由稳态到拥塞的指标流量相变值来刻画网络的吞吐量.基于每个节点的数据包处理能力与该节点的度或介数成比例提出两种模型并进行仿真.仿真结果表明,平均度相同的情况下,模型Ⅰ中,WS小世界网络比ER随机网络和BA无标度网络更容易产生拥塞;模型Ⅱ中,所有网络容量都得到较大的提高,尤其是WS小世界网络.但当网络的基本连接参数改变时,哪种模型更利于网络的流量传输,还要依据网络本身的结构特性 关键词： 复杂网络 无标度网络 感知流量 拥塞     

复杂交通运输网络上的拥挤与效率问题研究

本文研究复杂交通运输网络上的拥挤与效率问题. 在无标度网络、随机网络以及小世界网络等不同拓扑结构中, 探讨了不同的能力分配方式和不同的OD (Origin-Destination) 交通需求分布对网络拥挤度和效率的影响. 随着平均交通需求的增加, 分析无标度网络、随机网络以及小世界网络从自由流状态到交通拥堵状态的变化规律. 为便于比较, 本文侧重研究网络拥挤度的倒数, 并将其定义为通畅度. 研究发现网络中的通畅度与效率之间存在线性相关关系, 并且不同网络中的线性比例系数 (或斜率)是不同的, 从而体现了不同网络具有不同的运输性能. 关键词： 复杂网络 拥挤 效率     

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.     

Complex networks: Dynamics and security

This paper presents a perspective in the study of complex networks by focusing on how dynamics may affect network security under attacks. In particular, we review two related problems: attack-induced cascading breakdown and range-based attacks on links. A cascade in a network means the failure of a substantial fraction of the entire network in a cascading manner, which can be induced by the failure of or attacks on only a few nodes. These have been reported for the internet and for the power grid (e.g., the August 10, 1996 failure of the western United States power grid). We study a mechanism for cascades in complex networks by constructing a model incorporating the flows of information and physical quantities in the network. Using this model we can also show that the cascading phenomenon can be understood as a phase transition in terms of the key parameter characterizing the node capacity. For a parameter value below the phase-transition point, cascading failures can cause the network to disintegrate almost entirely. We will show how to obtain a theoretical estimate for the phase-transition point. The second problem is motivated by the fact that most existing works on the security of complex networks consider attacks on nodes rather than on links. We address attacks on links. Our investigation leads to the finding that many scale-free networks are more sensitive to attacks on short-range than on long-range links. Considering that the small-world phenomenon in complex networks has been identified as being due to the presence of long-range links, i.e., links connecting nodes that would otherwise be separated by a long node-to-node distance, our result, besides its importance concerning network efficiency and security, has the striking implication that the small-world property of scale-free networks is mainly due to short-range links.     

Characteristics of real futures trading networks

Futures trading is the core of futures business, and it is considered as one of the typical complex systems. To investigate the complexity of futures trading, we employ the analytical method of complex networks. First, we use real trading records from the Shanghai Futures Exchange to construct futures trading networks, in which nodes are trading participants, and two nodes have a common edge if the two corresponding investors appear simultaneously in at least one trading record as a purchaser and a seller, respectively. Then, we conduct a comprehensive statistical analysis on the constructed futures trading networks. Empirical results show that the futures trading networks exhibit features such as scale-free behavior with interesting odd-even-degree divergence in low-degree regions, small-world effect, hierarchical organization, power-law betweenness distribution, disassortative mixing, and shrinkage of both the average path length and the diameter as network size increases. To the best of our knowledge, this is the first work that uses real data to study futures trading networks, and we argue that the research results can shed light on the nature of real futures business.