Narrow Results

Understanding the robustness of networks is a key step to design stable networked systems and protect them from being collapsed under small initial failures. In a two-layer multiplex network with flow, where physical quantities such as traffic flow or electricity move, a global cascading failure may be triggered out when edges in one layer fail and their loads are redistributed within the layer and to the other layer. Here we propose an edge pressure index to identify the critical edges which are the most likely to initiate a cascading failure in the network with flow. By reinforcing the critical edges, the robustness of the multiplex networks can be significantly enhanced. Based on large simulations on both synthetic networks and real-world networks, we find that protecting critical edges identified by the edge pressure index outperforms protecting edges with the highest betweenness centrality and the randomly selected edges in enhancing the robustness of the network. Furthermore, preventing the initial cascading failures between layers is critical to reduce the cascading failure size, which is due to the accumulative effects of redistributed loads. When the layer under initial attack is a heterogeneous BA network, which is more robust under flow redistribution, the entire system is more robust because the edge failures in the BA network create less intra- and inter-layer load redistribution.

In this paper, adopting the initial load of a node j to be , where k_j is the degree of the node j and α is a tunable parameter that controls the strength of the initial load of a node, we propose a cascading model with a breakdown probability and explore cascading failures on a typical network, i.e., the Barabási–Albert (BA) network with scale-free property. Assume that a failed node leads only to a redistribution of the load passing through it to its neighboring nodes. According to the simulation results, we find that BA networks reach the strongest robustness level against cascading failures when α = 1 and the robustness of networks has a positive correlation with the average degree 〈k〉, not relating to the different breakdown probabilities. In addition, it is found that the robustness against cascading failures has an inversely proportional relationship with the breakdown probability of an overload node. Finally, the numerical simulations are verified by the theoretical analysis.

Assume the initial load of an edge ij in a network to be L_ij =[(k_i ∑_{a ∈ Γi} k_a)(k_j ∑_{b ∈ Γj} k_b)]^α with k_i and k_j being the degrees of the nodes connected by the edge, where α is a tunable parameter which controls the strength of the edge initial load, and Γ_i and Γ_j are the sets of neighboring nodes of i and j, respectively. We investigate the cascading phenomenon of uncorrelated scale-free networks subject to two different attacking strategies on edges, i.e. attacking on the edges with the highest loads or the lowest loads (LL). By the critical threshold of edge capacity quantifying the network robustness, we numerically discuss the effects of two attacks for the network vulnerability. Interestingly, it is found that the attack on the edge with the LL is highly effective in disrupting the structure of the attacked network when α < 0.5. In the case of α = 0.5, the effects of two attacks for the network robustness against cascading failures are almost identical. We furthermore provide the theoretical prediction support for the numerical simulations. These results may be very helpful for real-life networks to protect the key edges selected effectively to avoid cascading-failure-induced disasters.

According to the dynamical characteristics of the local redistribution of the load on a removal node, by the reconnection of the neighboring edge of the most vulnerable node, we propose an effective method to improve the network robustness against cascading failures. Under two constraints, i.e. keeping the degree of each node unchanged and fixing the total protective cost of a network, we investigate the efficiency of the swap method on scale-free networks and analyze the correlation between the optimized network and the Pearson correlation coefficient. We numerically show that effective swapping of the small part of connections can dramatically improve the network robust level against cascading failures and find that the optimized networks obtained by the swap method exhibit an extremely disassortative degree–degree correlation, that is, the disassortativity decreases the robustness of the optimized network against cascading failures. While the extent of the disassortative mixing is decided by the parameters in the cascading model. In addition, we also compare the average path length and the diameter of the optimized and the original networks.

Many real-world networks interact with other networks by only several links, for example, transportation networks and aviation networks among cities, internet router network among the different regions, power supply network among cities and so on. Understanding how to protect these coupled networks and improve their robustness against cascading failures is very important. By protecting the edges between coupled networks, we investigate its efficiency on improving the robustness of coupled networks against cascading failures. Fixing the total protective cost of coupled networks, we find that adjusting the capacities of the edges among coupled networks can better improve the robustness of coupled networks against cascading failures and observe that the more uniform the distribution of the edge load, the more effective the protection strategy. In addition, by immunizing the edges among coupled networks, we compare two protecting methods and find that the immunization strategy can better protect coupled networks. Our results are useful not only for how to protect coupled networks from the local perspective, but also for significantly improving the robustness of a single network by protecting some key edges.

Considering the weight of a node and the coupled strength of two interdependent nodes in the different networks, we propose a method to assign the initial load of a node and construct a new cascading load model in the interdependent networks. Assuming that a node in one network will fail if its degree is 0 or its dependent node in the other network is removed from the network or the load on it exceeds its capacity, we study the influences of the assortative link (AL) and the disassortative link (DL) patterns between two networks on the robustness of the interdependent networks against cascading failures. For better evaluating the network robustness, from the local perspective of a node we present a new measure to qualify the network resiliency after targeted attacks. We show that the AL patterns between two networks can improve the robust level of the entire interdependent networks. Moreover, we obtain how to efficiently allocate the initial load and select some nodes to be protected so as to maximize the network robustness against cascading failures. In addition, we find that some nodes with the lower load are more likely to trigger the cascading propagation when the distribution of the load is more even, and also give the reasonable explanation. Our findings can help to design the robust interdependent networks and give the reasonable suggestion to optimize the allocation of the protection resources.

Cascading failure is ubiquitous in many networked infrastructure systems, such as power grids, Internet and air transportation systems. In this paper, we extend the cascading failure model to a scale-free network with tunable clustering and focus on the effect of clustering coefficient on system robustness. It is found that the network robustness undergoes a nonmonotonic transition with the increment of clustering coefficient: both highly and lowly clustered networks are fragile under the intentional attack, and the network with moderate clustering coefficient can better resist the spread of cascading. We then provide an extensive explanation for this constructive phenomenon via the microscopic point of view and quantitative analysis. Our work can be useful to the design and optimization of infrastructure systems.

In railway traffic, safety analysis is a key issue for controlling train operation. Here, the identification and order of key factors are very important. In this paper, a new network model is constructed for analyzing the railway safety, in which nodes are regarded as causation factors and links represent possible relationships among those factors. Our aim is to give all these nodes an importance order, and to find the in-depth relationship among these nodes including how failures spread among them. Based on the constructed network model, we propose a control method to ensure the safe state by setting each node a threshold. As the results, by protecting the Hub node of the constructed network, the spreading of railway accident can be controlled well. The efficiency of such a method is further tested with the help of numerical example.

Recently the robustness of coupled network under cascading failure has attracted a lot of attention. In this paper, we investigate the cascading failure of the interconnected weighted networks based on the state of link. The load on one link is defined by a function of the strength of the two nodes at the ends of that link, using four intentional attack strategies, we study the invulnerability of the interconnected weighted networks when cascading failure occurs. Our studies show that when the link with highest load is attacked, the damage to the network will be more serious by attacking the inner-link with highest load than that caused by attacking the coupling link with highest load, and no matter how the coupling links distribute, there are two thresholds. In addition, we find that the larger the weight increment in the model or the smaller the network’s mean clustering coefficient, the stronger the ability of the network to resist cascading failure when the inner-link with highest load is attacked, while the weaker the ability of the network to suppress the cascading failure when the inner-link with lowest load is attacked.

We study the problem of universal resilience patterns in complex networks against cascading failures. We revise the classical betweenness method and overcome its limitation of quantifying the load in cascading model. Considering that the generated load by all nodes should be equal to the transported one by all edges in the whole network, we propose a new method to quantify the load on an edge and construct a simple cascading model. By attacking the edge with the highest load, we show that, if the flow between two nodes is transported along the shortest paths between them, then the resilience of some networks against cascading failures inversely decreases with the enhancement of the capacity of every edge, i.e. the more capacity is not always better. We also observe the abnormal fluctuation of the additional load that exceeds the capacity of each edge. By a simple graph, we analyze the propagation of cascading failures step by step, and give a reasonable explanation of the abnormal fluctuation of cascading dynamics.

We study the structural robustness of the scale free network against the cascading failure induced by overload. In this paper, a failure mechanism based on betweenness-degree ratio distribution is proposed. In the cascading failure model we built the initial load of an edge which is proportional to the node betweenness of its ends. During the edge random deletion, we find a phase transition. Then based on the phase transition, we divide the process of the cascading failure into two parts: the robust area and the vulnerable area, and define the corresponding indicator to measure the performance of the networks in both areas. From derivation, we find that the vulnerability of the network is determined by the distribution of betweenness-degree ratio. After that we use the connection between the node ability coefficient and distribution of betweenness-degree ratio to explain the cascading failure mechanism. In simulations, we verify the correctness of our derivations. By changing connecting preferences, we find scale free networks with a slight assortativity, which performs better both in robust area and vulnerable area.

To control the spread of cascading failure on scale-free networks, we propose a new model with the betweenness centrality and the degrees of the nodes which are combined. The effects of the parameters of the edge weight on cascading dynamics are investigated. Five metrics to evaluate the robustness of the network are given: the threshold parameter ($T_C$), the proportion of collapsed edges ($C{F}_E$), the proportion of collapsed nodes ($C{F}_N$), the number of nodes in the largest connected component ($S_G$) and the number of the connected component ($S_C$). Compared with the degrees of nodes’ model and the betweenness of the nodes’ model, the new model could control the spread of cascading failure more significantly. This work might be helpful for preventing and mitigating cascading failure in real life, especially for small load networks.

Revealing how the interdependency between different networks affects the load-induced cascading failures is a new interesting topic recently. A new cascading failure model for interconnected network system is introduced in this paper, in which the loads of the failed nodes can be transmitted between different layer networks or among each layer network, the ratio is adjusted with a defined parameter $T_f$ which can be viewed as a measure of coupling strength. Based on the proposed model, we find that under different attacking strategies and different value of $T_f$, different coupling types have different effects on the robustness of interconnected network systems. We mainly focus on the robustness of interconnected network with random coupling under intentional attack, we find there being a threshold $T_f^{\ast }$, when $T_f<T_f^{\ast }$, the interconnected network systems will be more vulnerable with larger value of $T_f$; however, when $T_f>T_f^{\ast }$, if the capacity redundancy of interconnected network is low, increasing $T_f$ tends to enhance the robustness of interconnected network system. In addition, we also get the conclusion that increasing the total amount of loads in one layer network or the unbalance of load distribution between different layer networks will decline the robustness of the interconnected network system. We hope our work will shed some light on designing high-robust interconnected network systems or improving the robustness of interconnected network systems.

Considering congestion effects in realistic network environments, we give a new method to adjust dynamically adjust the weight of the congested edge. We calculate the load on an edge based on the revised betweenness method and propose a novel model with three states of edges to investigate the dynamics of cascading failures in the ring network, the BA scale-free network, and the real traffic networks in London. By two robust metrics, we surprisingly observe the abnormal dynamics of cascading propagation, especially compared with that in the unadjustable weight, the curves of cascading dynamics in the adjustable weight are more irregular, which means that enhancing the capacity of each edge is not always better to avoid the cascading propagation. In addition, our simulation results show that the dynamical change of the edge’s weight makes the heterogeneous BA networks more vulnerable.

Cascading failure is widespread in the networks. In order to solve practical problems, it is necessary to study the cascading failure in the complex network. In this paper, we mainly focus on the cascading failure in multilayer networks with asymmetric dependence group and discuss the influence factors of networks’ robustness. We build a model with recover mechanism within the layer and define an effect between the layers. Under the effect, if the fraction of failure nodes between layers is greater than or equal to the failure–recover threshold, other connected nodes in other layers will recover and if the fraction of failure nodes between layers is less than the failure–recover threshold, the connected nodes in other layers will become invalid. It is worth mentioning that in our model, the impact of the proportion $v$ of the dependency groups and the group size $g$ on the robustness of the networks is related to the probability $p$ of initial removal. Because the failure and recovery mechanism exist, the networks robustness will be reversed by changing the value of $p$ for different values of $v$ and $g$. In addition, the number of layers $T$ has different effects on networks robustness due to the effect between the layers.

In the context of urban rail evacuation analysis, little work addresses the mechanism of passenger propagation between urban rail and bus systems. This paper attempts to quantify the propagation mechanism to make evacuation scheduling reasonable and effective. First, it provides a set of methods for characterizing the law of passenger propagation in the region affected by a rail station failure, based on the theory of interdependent networks. An interdependent public transport network between the rail transit and buses in the affected region is constructed using the methods of Space L and Space P. In the network, the initial capacities are modeled, the redistribution rules of passengers and the status identification rules of stations are established, and the propagation results are characterized. Second, using the passenger propagation impact of a faulty rail station, an emergency bus-scheduling model is set up, aimed at minimizing the operating costs of emergency buses and maximizing the benefits for passengers. A genetic algorithm is devised to solve the scheduling model. Finally, an example is put forward to verify the feasibility of the designed methods. The example shows that the methods provide theoretical support and a decision aid for passenger evacuation strategies at faulty urban rail stations.

Reality networks such as power grids and social networks can be spatially embedded. In this paper, we focus on the spatial cascading effect in such networks. The spatial cascading effect is that the failure of one node may cause other nodes that are close to it in space to fail. The phenomenon is very common, such that a person is more likely to have an impact on his neighbors even if he is not connected with his neighbors via social networks. Based on this, we construct a spatial cascading model to simulate the spatial cascading effect. In addition, we apply the exponential distribution $P(l)\sim {exp}^{-\frac{l}{\zeta }}$ to fit the real link distances. The networks are generated by two-dimensional lattices. We define two kinds of connections, namely actual spatial connections. The actual connections are links generated by the exponential distribution. The spatial connections are links in the lattice. Simulations show that the spatial embeddedness makes networks more robust in our model, which is different from previous research results. We put forward an algorithm to alter the link distances in the networks without changing node degree values. Using the algorithm verifies our conclusion that if nodes tend to connect with local nodes, networks will be robust to the spatial cascading effect. We further extend our model to a more general form. The nodes embedded in lattice can be sparse, which means that the existing probability of nodes in the lattice is not always 1. The networks in the extension model are more vulnerable compared to those in the original model.

Recent work on the cascading failure of networks with dependence groups assumes that the number of nodes in each dependence group is equal. In this paper, we construct a model on interdependent networks with dependence groups against cascading failure. The size of dependence group is supposed to obey the Poisson Distribution and the Truncated Normal Distribution, respectively. By applying the tools of mean-field approximation and the generating function techniques, the cascading model is theoretically analyzed and the theoretical solutions are nearly consistent with the simulation values. Besides, we define three kinds of coupling preferences based on node degree, i.e. assortative coupling, disassortative coupling and random coupling. The connection between layers is no longer one-to-one correspondence of nodes, but fully connection of some groups. In addition, some factors affecting the network robustness are discussed and extensive simulations are realized on two-layer BA networks. The simulation results show that the coupling preference has influence on the network robustness and the network with dependence groups obeying the Truncated Normal Distribution performs better than the Poisson Distribution.

The robustness of complex networks responding to attacks has long been the focus of network science researching. Nonetheless, the precious studies mostly focus on network performance when facing malicious attacks and random failures while rarely pay attention to the influences of scales of attacking. It is wondering if it is an actual fact that the network is more fragile when attacking scale is exacerbated. In this paper, we are committed to exploring the influences related to the very factor of attacking scale from the perspective of cascading failure problem of dynamic network theory. We construct the model with a regular ranking edge deletion method by simulating attacking scale with $N_a$ and $a$ is denoted as attacked edge number. To be specific, we rank the edges according to initial distributed loads and delete edges in the ranked list, and subsequently observe the changes of robustness in the networks, including BA scale-free network, WS small-world network and several real traffic networks. During the process, an unusual counterintuitive phenomenon captures our attention that the network damages caused by attacks do not always grow with the increase of attacked edges number. We specifically demonstrate and analyze this abnormal cascading propagation phenomenon, ascribing this paradox to the dynamics of the load and the connections of the network structure. Our work may offer a new angle on better controlling the spread of cascading failure and remind the importance of effectively protecting networks from underlying dangers.

Robustness studies on integrated urban public transport networks have attracted growing attention in recent years due to the significant influence on the overall performance of urban transport system. In this paper, topological properties and robustness of a bus–subway coupled network in Beijing, composed of both bus and subway networks as well as their interactions, are analyzed. Three new models depicting cascading failure processes on the coupled network are proposed based on an existing binary influence modeling approach. Simulation results show that the proposed models are more accurate than the existing method in reflecting actual passenger flow redistribution in the cascading failure process. Moreover, the traffic load influence between nodes also plays a vital role in the robustness of the network. The proposed models and derived results can be utilized to improve the robustness of integrated urban public transport systems in traffic planning.