Attack vulnerability of scale-free networks due to cascading failures

In this paper, adopting the initial load of a node i to be with k_i being the degree of the node i, we propose a cascading model based on a load local redistribution rule and examine cascading failures on the typical network, i.e., the BA network with the scale-free property. We find that the BA scale-free network reaches the strongest robustness level in the case of α=1 and the robustness of the network has a positive correlation with the average degree 〈k〉, where the robustness is quantified by a transition from normal state to collapse. In addition, we further discuss the effects of two different attacks for the robustness against cascading failures on our cascading model and find an interesting result, i.e., the effects of two different attacks, strongly depending to the value α. These results may be very helpful for real-life networks to avoid cascading-failure-induced disasters.     

A robust matching model of capacity to defense cascading failure on complex networks

How to control the cascading failure has become a hot topic in recent years. In this paper, we propose a new matching model of capacity by developing a profit function to defense cascading failures on artificially created scale-free networks and the real network structure of the North American power grid. Results show that our matching model can enhance the network robustness efficiently, which is particularly important for the design of networks to deduce the damage triggered by the cascading failures.     

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.     

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.     

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.     

How non-uniform tolerance parameter strategy changes the response of scale-free networks to failures

In this paper, we introduce a non-uniform tolerance parameter (TP) strategy (the tolerance parameter is characterized by the proportion between the unused capacity and the capacity of a vertex) and study how the non-uniform TP strategy influences the response of scale-free (SF) networks to cascading failures. Different from constant TP in previous work of Motter and Lai (ML), the TP in the proposed strategy scales as a power-law function of vertex degree with an exponent b. The simulations show that under low construction costs D, when b>0 the tolerance of SF networks can be greatly improved, especially at moderate values of b; When b<0 the tolerance gets worse, compared with the case of constant TP in the ML model. While for high D the tolerance declines slightly with the b, namely b<0 is helpful to the tolerance, and b>0 is harmful. Because for smaller b the cascade of the network is mainly induced by failures of most high-degree vertices; while for larger b, the cascade attributes to damage of most low-degree vertices. Furthermore, we find that the non-uniform TP strategy can cause changes of the structure and the load-degree correlation in the network after the cascade. These results might give insights for the design of both network capacity to improve network robustness under limitation of small cost, and for the design of strategies to defend cascading failures of networks.     

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.     

基于可调负载重分配的无标度网络连锁效应分析

为了深入研究复杂网络抵制连锁故障的全局鲁棒性,针对现实网络上的负载重分配规则常常是介于全局分配与最近邻分配、均匀分配与非均匀分配的特点,围绕负荷这一影响连锁故障发生和传播最重要的物理量以及节点崩溃后的动力学过程,提出了一种可调负载重分配范围与负载重分配异质性的复杂网络连锁故障模型,并分析了该模型在无标度网络上的连锁故障条件.数值模拟获得了复杂网络抵制连锁故障的鲁棒性与模型中参数的关系.此外,基于网络负载分配规则的分析以及理论解析的推导,验证了数值模拟结论,也证明在最近邻与全局分配两种规则下都存在负载分配均匀性参数等于初始负荷强度参数即β=τ使得网络抵御连锁故障的能力最强.     

Dynamics of load entropy during cascading failure propagation in scale-free networks

In this Letter, we introduce the concept of load entropy, which can be an average measure of a network's heterogeneity in the load distribution. Then we investigate the dynamics of load entropy during failure propagation using a new cascading failures load model, which can represent the node removal mechanism in many real-life complex systems. Simulation results show that in the early stage of failure propagation the load entropy for a larger cascading failure increases more sharply than that for a smaller one, and consequently the cascading failure with a larger damage can be identified at the early stage of failure propagation according to the load entropy. Particularly, load entropy can be used as an index to be optimized in cascading failures control and defense in many real-life complex networks.     

Robustness of Complex Networks under Attack and Repair

To study the robustness of complex networks under attack and repair, we introduce a repair model of complex networks. Based on the model, we introduce two new quantities, i.e. attack fraction fa and the maximum degree of the nodes that have never been attacked ~Ka, to study analytically the critical attack fraction and the relative size of the giant component of complex networks under attack and repair, using the method of generating function. We show analytically and numerically that the repair strategy significantly enhances the robustness of the scale-free network and the effect of robustness improvement is better for the scale-free networks with a smaller degree exponent. We discuss the application of our theory in relation to the
understanding of robustness of complex networks with reparability.     

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.     

Mitigation strategies on scale-free networks against cascading failures

According to the dynamic characteristics of the cascading propagation, we introduce a mitigation mechanism and propose four mitigation methods on four types of nodes. By the normalized average avalanche size and a new measure, we demonstrate the efficiencies of the mitigation strategies on enhancing the robustness of scale-free networks against cascading failures and give the order of the effectiveness of the mitigation strategies. Surprisingly, we find that only adopting once mitigation mechanism on a small part of the overload nodes can dramatically improve the robustness of scale-free networks. In addition, we also show by numerical simulations that the optimal mitigation method strongly depends on the total capacities of all nodes in a network and the distribution of the load in the cascading model. Therefore, according to the protection strength for scale-free networks, by the distribution of the load and the protection price of networks, we can reasonably select how many nodes and which mitigation method to efficiently protect scale-free networks at the lower price. These findings may be very useful for avoiding various cascading-failure-induced disasters in the real world and for leading to insights into the mitigation of cascading failures.     

Cascading dynamics in congested complex networks

Cascading failures often occur in congested complex networks. Cascading failures can be expressed as a three phase process: generation, diffusion and dissipation of congestion. Different from betweenness centrality, we propose a congestion function to represent the extent of congestion on a given node. By introducing the concept of “delay time”, we construct an intergradation between permanent removal and nonremoval. We also build a new evaluation function of network efficiency, based on congestion, which measures the damage caused by cascading failures. Finally, based on Statnet and Webgraph topologies we investigate the effects of network structure and size, delay time, processing ability and traffic generation speed on congestion propagation. Also we uncover cascading process composed of three phases and some factors affecting cascade propagation.     

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.     

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.     

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.     

Modeling cascading failures in congested complex networks

Cascading failures occur commonly in congested complex networks, where it may be expressed as the process of generation, diffusion and dissipation of congestion. Different from betweeness centrality, we introduce congestion effects to determine the load on the node. In terms of user equilibrium condition, congestion effects can be described by cost functions or link performance functions, which map link flows to travel times. By introducing conceptual “practical capacity” dynamics to cost functions, cascading failures are well discussed in terms of the degree of congestion in complex networks. Moreover, the efficiency dynamics of the network due to cascading failures is also investigated, and a transition phenomenon is uncovered independent of clustering effect.     

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.     

Unified index to quantifying heterogeneity of complex networks

Although recent studies have revealed that degree heterogeneity of a complex network has significant impact on the network performance and function, a unified definition of the heterogeneity of a network with any degree distribution is absent. In this paper, we define a heterogeneity index 0≤H<1 to quantify the degree heterogeneity of any given network. We analytically show the existence of an upper bound of H=0.5 for exponential networks, thus explain why exponential networks are homogeneous. On the other hand, we also analytically show that the heterogeneity index of an infinite power law network is between 1 and 0.5 if and only if its degree exponent is between 2 and 2.5. We further show that for any power law network with a degree exponent greater than 2.5, there always exists an exponential network such that both networks have the same heterogeneity index. This may help to explain why 2.5 is a critical degree exponent for some dynamic behaviors on power law networks.     

A Robustness Model of Complex Networks with Tunable Attack Information Parameter

We introduce a novel model for robustness of complex with a tunable attack information parameter. The random failure and intentional attack known are the two extreme cases of our model. Based on the model, we study the robustness of complex networks under random information and preferential information, respectively. Using the generating function method, we derive the exact value of the critical removal fraction of nodes for the disintegration of networks and the size of the giant component. We show that hiding just a small fraction of nodes randomly can prevent a scale-free network from collapsing and detecting just a small fraction of nodes preferentially can destroy a scale-free network.