Narrow Results

In this work, we study the effect of congestion on the behavior of cascading failures in scale-free networks, where a capacity is assigned on each node (controlled by a tolerance parameter α), and traffic flows are governed by user equilibrium instead of going along the shortest paths. The effect of congestion can be described by link cost function, which denotes the time needed to travel along the link. Here we focus on studying the effect of link's practical capacity, which is a parameter in link cost function. Two different kinds of link's practical capacity are investigated, i.e. uniform case and nonuniform case. In the uniform case, each link has the same value of practical capacity. While in the nonuniform case, we assume that link's practical capacity and degrees of the link's endpoints are correlated (controlled by parameter θ, which governs the heterogeneity of link's practical capacity). Simulation results show that, in the uniform case, scale-free networks are more prone to cascading failures when increasing the value of link's practical capacity. In the nonuniform case, cascading failures in scale-free networks are very sensitive to α when θ > 0; while θ < 0, scale-free networks may suffer from serious cascading failures, regardless of α.

Much empirical evidence shows that when attacked with cascading failures, scale-free or even random networks tend to collapse more extensively when the initially deleted node has higher betweenness. Meanwhile, in networks with strong community structure, high-betweenness nodes tend to be bridge nodes that link different communities, and the removal of such nodes will reduce only the connections among communities, leaving the networks fairly stable. Understanding what will affect cascading failures and how to protect or attack networks with strong community structure is therefore of interest. In this paper, we have constructed scale-free Community Networks (SFCN) and Random Community Networks (RCN). We applied these networks, along with the Lancichinett–Fortunato–Radicchi (LFR) benchmark, to the cascading-failure scenario to explore their vulnerability to attack and the relationship between cascading failures and the degree distribution and community structure of a network. The numerical results show that when the networks are of a power-law distribution, a stronger community structure will result in the failure of fewer nodes. In addition, the initial removal of the node with the highest betweenness will not lead to the worst cascading, i.e. the largest avalanche size. The Betweenness Overflow (BOF), an index that we developed, is an effective indicator of this tendency. The RCN, however, display a different result. In addition, the avalanche size of each node can be adopted as an index to evaluate the importance of the node.

To control counterparty risk, financial regulations such as the Dodd Frank Act are increasingly requiring standardized derivatives trades to be cleared by central counterparties (CCPs). It is anticipated that in the near-term future, CCPs across the world will be linked through interoperability agreements that facilitate risk-sharing but also serve as a conduit for transmitting shocks. This paper theoretically studies a network with CCPs that are linked through interoperability arrangements, and studies the properties of the network that contribute to cascading failures. The magnitude of the cascading is theoretically related to the strength of network linkages, the size of the network, the logistic mapping coefficient, a stochastic effect and CCP's defense lines. Simulations indicate that larger network effects increase systemic risk from cascading failures. The size of the network N raises the threshold value of shock sizes that are required to generate cascades. Hence, the larger the network, the more robust it will be.

As the cascading failures in networked traffic systems are becoming more and more serious, research on cascade defense in complex networks has become a hotspot in recent years. In this paper, we propose a traffic-based cascading failure model, in which each packet in the network has its own source and destination. When cascade is triggered, packets will be redistributed according to a given routing strategy. Here, a global hybrid (GH) routing strategy, which uses the dynamic information of the queue length and the static information of nodes' degree, is proposed to defense the network cascade. Comparing GH strategy with the shortest path (SP) routing, efficient routing (ER) and global dynamic (GD) routing strategies, we found that GH strategy is more effective than other routing strategies in improving the network robustness against cascading failures. Our work provides insight into the robustness of networked traffic systems.

To control counterparty risk, financial regulations such as the Dodd–Frank Act are increasingly requiring standardized derivatives trades to be cleared by central counterparties (CCPs). It is anticipated that in the near term future, CCPs across the world will be linked through interoperability agreements that facilitate risk sharing but also serve as a conduit for transmitting shocks. This paper theoretically studies a networked network with CCPs that are linked through interoperability arrangements. The major finding is that the different configurations of networked network CCPs contribute to the different properties of the cascading failures.

Cascading failures of loads in isolated networks have been studied extensively over the last decade. Since 2010, such research has extended to interdependent networks. In this paper, we study cascading failures with local load redistribution in interdependent Watts–Strogatz (WS) networks. The effects of rewiring probability and coupling strength on the resilience of interdependent WS networks have been extensively investigated. It has been found that, for small values of the tolerance parameter, interdependent networks are more vulnerable as rewiring probability increases. For larger values of the tolerance parameter, the robustness of interdependent networks firstly decreases and then increases as rewiring probability increases. Coupling strength has a different impact on robustness. For low values of coupling strength, the resilience of interdependent networks decreases with the increment of the coupling strength until it reaches a certain threshold value. For values of coupling strength above this threshold, the opposite effect is observed. Our results are helpful to understand and design resilient interdependent networks.

The hierarchical structure, $k$-core, is common in various complex networks, and the actual network always has successive layers from 1-core layer (the peripheral layer) to $k_m$-core layer (the core layer). The nodes within the core layer have been proved to be the most influential spreaders, but there is few work about how the depth of $k$-core layers (the value of $k_m$) can affect the robustness against cascading failures, rather than the interdependent networks. First, following the preferential attachment, a novel method is proposed to generate the scale-free network with successive $k$-core layers (KCBA network), and the KCBA network is validated more realistic than the traditional BA network. Then, with KCBA interdependent networks, the effect of the depth of $k$-core layers is investigated. Considering the load-based model, the loss of capacity on nodes is adopted to quantify the robustness instead of the number of functional nodes in the end. We conduct two attacking strategies, i.e. the RO-attack (Randomly remove only one node) and the RF-attack (Randomly remove a fraction of nodes). Results show that the robustness of KCBA networks not only depends on the depth of $k$-core layers, but also is slightly influenced by the initial load. With RO-attack, the networks with less $k$-core layers are more robust when the initial load is small. With RF-attack, the robustness improves with small $k_m$, but the improvement is getting weaker with the increment of the initial load. In a word, the lower the depth is, the more robust the networks will be.

Critical infrastructures are tightly connected and extremely fragile multilayer coupled networks. This paper discusses the cross-networks impact of subnetworks and global network of networks on robustness by taking a critical infrastructures with three-layer interdependent networks as an example. The percolation theory is applied to capture the flow characteristics of cascading failures and evaluate the robustness of multilayer networks. And further discuss and compare the situation of each subnetwork affecting or being affected. The quantitative evaluation model of the interaction of multilayer networks is proposed based on cascading failures, where the influence expansion matrix and the dependency matrix are obtained. The results show that the power network has a high influence on other networks, and it is difficult to be affected. Meanwhile the influence ability of water network and gas network is limited.

Many realistic networked systems may face sustained attack over a sequence of time. In this paper, we propose a cascading failure model in coupled map lattices (CMLs) with sustained attack, where a number of crucial nodes will be attacked one by one at each time step. The effects of four attacking strategies: high-degree strategy (HDS), high-clustering coefficient strategy, high-closeness strategy and high-betweenness strategy are compared. This shows that the performance of HDS is better than that of other attacking strategies when the value of the outside attack is small. The effectiveness of HDS on Watts–Strogatz (WS) small-world networks and two real-world networks is extensively investigated. The results indicate that increasing the value of the rewiring probability of WS networks can make the network more robust to resist sustained attack. The sparser the network structure is, the faster the diffusion velocity of cascading failures is. A smaller coupling strength in CMLs can restrain the diffusion velocity of cascades. We also extend our CML-based model with sustained attack from one node strategy (ONS) to two nodes strategy (TNS) and compare the effect of ONS and TNS with different values of the outside attack. It is found that ONS outperforms TNS for small values of the outside attack, but TNS would be a better choice when the outside attack and the time step are all large. Our work will provide new insights into the problem of protecting complex networks against cascading failures with respect to malicious attack.

Traditional research studies on interdependent networks with groups ignore the relationship between nodes in dependency groups. In real-world networks, nodes in the same group may support each other through cooperation and tend to fail or survive together. In this paper, based on the framework of group percolation, a cascading failure model on interdependent networks with cooperative dependency groups under targeted attacks is proposed, and the effect of group size distributions on the robustness of interdependent networks is investigated. The mutually giant component and phase transition point of networks with different group size distributions are analyzed. The effectiveness of the theory is verified through simulations. Results show that the robustness of interdependent networks with cooperative dependency groups can be enhanced by increasing the heterogeneity between groups under targeted attacks. The theory can well predict the numerical simulation results. This model provides some theoretical guidance for designing robust interdependent systems in real world.

The load is a very important physical quantity for cascading failures in critical infrastructure networks. However, in previous studies, regardless of the node cascading failure model or the edge cascading failure model, all load in the network is born on the nodes ignoring the possibility of load pouring into the network from edges. For example, almost all vehicles from communities, units, parking lots, etc. are flooding into the transportation network from roads. To this end, considering that the starting point and the destination of the transferred load are located on the “edge”, we give a new method to calculate the initial load on the node and construct a node cascading model with source-sink edges. According to the information about the degree of two nodes connecting the edge, we define the edge weight and give the preferential mechanism of load destination selection. In addition, different from previous models, the load transmitted between two edges will be evenly distributed on all nodes of the shortest paths. By removing the node with the highest load in some artificial networks and two real networks, we study the cascading dynamics. We find the interesting phenomenon of the capacity paradox that the higher capacities of all nodes do not mean that the network has the higher robustness. By two metrics to quantify the network robustness, we further evaluate the robust mode of the network under different parameters, and study the optimal cost to suppress cascading failures in some networks.

In interdependent networks, link addition strategies can enhance the connectivity of networks, thus improving robustness in the face of cascading failures. In this paper, first, interdependent networks models under various coupling methods and cascading failures model are proposed. Moreover, link addition strategies and node importance metrics are obtained. Finally, this paper analyzes the influence of link addition strategies and node importance metrics on robustness under three coupling methods. Besides, the effects of coupling ratio and link addition ratio on robustness in the interdependent network under different coupling methods are also analyzed. The simulation results show in partial coupled networks, intra-layer link addition strategy (Intra LAS) is more robust. In one-to-one coupled networks, inter-layer link addition strategy (Inter LAS) yields better performance in improving robustness when the number of initial attacked nodes is fewer, and intra-layer link addition strategy (Intra LAS) is more robust when more nodes are attacked initially. The effect of network size and the average degree of network on this result are also discussed. The variation trend of networks with multiple dependencies is similar to that of partial coupled network with high coupling rate, but it has less robust than partial coupled network. The conclusions can provide guidance on how to select link addition strategy and node importance metric under different coupling methods.

A coupled network model consisting of bus and subway systems is proposed, and the statistic properties of the three networks: bus, subway and coupled networks of Beijing are studied with the theory of complex network. The result shows that the three networks have typical properties of small-world. We propose three parameters to depict the coupled network, they are: the coupled parameter β the influence parameter S and the node tolerance parameter γ. We use the binary influence model to simulate a feedback process and cascading failure in the coupled network and we obtain the conclusions: (1) The cascading size grows with β; (2) The cascading size grows with S, but it has a critical point; (3) The cascading size grows with the decrease of γ, when γ≤0.3, the cascading failure will extent to the whole network.

In this paper, we investigate the trade-off problem between the high robustness against cascading failures and the low costs of network constructions in complex networks. Since the important nodes with highly connected components usually play a key role in the network structure and network dynamics, we propose an optimal capacity allocation model based on node importance. The novel model will increase the capacities of those important nodes but reduce the network construction cost with the appropriate capacity allocation parameter. Moreover, we also discover that our matching model can enhance the robustness against cascading failures on the IEEE 300 network.

With load-based model, considering the loss of capacity on nodes, we investigate how the coupling strength (many-to-many coupled pattern) and link patterns (one-to-one coupled pattern) can affect the robustness of interdependent networks. In one-to-one coupled pattern, we take into account the properties of degree and betweenness, and adopt four kinds of inter-similarity link patterns and random link pattern. In many-to-many coupled pattern, we propose a novel method to build new networks via adding inter-links (coupled links) on the existing one-to-one coupled networks. For a full investigation on the effects, we conduct two types of attack strategies, i.e. RO-attack (randomly remove only one node) and RF-attack (randomly remove a fraction of nodes). We numerically find that inter-similarity link patterns and bigger coupling strength can effectively improve the robustness under RO-attacks and RF-attacks in some cases. Therefore, the inter-similarity link patterns can be applied during the initial period of network construction. Once the networks are completed, the robustness level can be improved via adding inter-links appropriately without changing the existing inter-links and topologies of networks. We also find that BA–BA topology is a better choice and that it is not useful to infinitely increase the capacity which is defined as the cost of networks.

Much empirical evidence shows that many real-world networks fall into the broad class of small-world networks and have a modular structure. The modularity has been revealed to have an important effect on cascading failure in isolated networks. However, the corresponding results for interdependent modular small-world networks remain missing. In this paper, we investigate the relationship between cascading failures and the intra-modular rewiring probabilities and inter-modular connections under different coupling preferences, i.e. random coupling with modules (RCWM), assortative coupling in modules (ACIM) and assortative coupling with modules (ACWM). The size of the largest connected component is used to evaluate the robustness from global and local perspectives. Numerical results indicate that increasing intra-modular rewiring probabilities and inter-modular connections can improve the robustness of interdependent modular small-world networks under intra-attacks and inter-attacks. Meanwhile, experiments on three coupling strategies demonstrate that ACIM has a better effect on preventing the cascading failures compared with RCWM and ACWM. These results can be helpful to allocate and optimize the topological structure of interdependent modular small-world networks to improve the robustness of such networks.

Infrastructure networks are usually spatially embedded, which makes them even more susceptible to spatially localized failures, i.e. a set of nodes located within a spatially localized area is subject to damage while other nodes do not directly fail. This paper mainly presents a methodological approach for analyzing vulnerability of interdependent infrastructure networks under localized attack. Two types of interdependencies are considered in the cascading failures propagation: functional and geographic interdependencies. The roles of different nodes and geographical proximity are employed to establish the functional interdependency. A novel attack failure model, i.e. localized failure model, is proposed to model geographic interdependency and quantify the failure probabilities of nodes under localized attack. Both topology-based and efficiency-based vulnerabilities of infrastructure networks are investigated. An artificial interdependent power and water networks are generated and their pertinent vulnerability is analyzed through using the presented methodological approach. Our method can help the stakeholders increase their knowledge of the vulnerability of the interdependent spatially embedded infrastructure networks under localized attack and protect them more efficiently.

As one of the most common mesoscale structures in real-life networks, k-core hierarchical structure has attracted a lot of attention. Recent research about k-core always focuses on detecting influential nodes determining failure or epidemic propagation. However, few studies have attempted to understand how k-core structural properties can affect dynamic characteristics of network. In this paper, the influences of depth and coupling preferences of k-core on the cascading failures of interdependent scale-free networks are investigated. First, k-core structures of some real-life networks are analyzed, and a scale-free network evolution model with rich and successive k-core layers is proposed. Then, based on a load-based cascading model, the influence of the depth of k-core is investigated with a new evaluation index. In the end, two coupling preferences are analyzed, i.e. random coupling (RC) and assortative coupling (AC). Results show that the lower the depth is, the more robust the interdependent networks will be, and we find AC and RC perform dissimilarly when the capacity varies. Furthermore, all the effects will be affected by the initial load.

The robustness of complex networks against cascading failures has been of great interest, while most of the researchers have considered undirected networks. However, to be more realistic, a part of links of many real systems should be described as unidirectional. In this paper, by applying three link direction-determining (DD) strategies, the tolerance of cascading failures is investigated in various networks with both unidirectional and bidirectional links. By extending the utilization of a classical global betweenness method, we propose a new cascading model, taking into account the weights of nodes and the directions of links. Then, the effects of unidirectional links on the network robustness against cascaded attacks are examined under the global load-based distribution mechanism. The simulation results show that the link-directed methods could not always lead to the decrease of the network robustness as indicated in the previous studies. For small-world networks, these methods certainly make the network weaker. However, for scale-free networks, the network robustness can be significantly improved by the link-directed method, especially for the method with non-random DD strategies. These results are independent of the weight parameter of the nodes. Due to the strongly improved robustness and easy realization with low cost on networks, the method for enforcing proper links to the unidirectional ones may be useful for leading to insights into the control of cascading failures in real-world networks, like communication and transportation networks.

Cascading failures have been widely analyzed in interdependent networks with different coupling preferences from microscopic and macroscopic perspectives in recent years. Plenty of real-world interdependent infrastructures, representing as interdependent networks, exhibit community structure, one of the most important mesoscopic structures, and partial coupling preferences, which can affect cascading failures in interdependent networks. In this paper, we propose the partial random coupling in communities, investigating cascading failures in interdependent modular scale-free networks under inner attacks and hub attacks. We mainly analyze the effects of the discoupling probability and the intermodular connection probability on cascading failures in interdependent networks. We find that increasing either the dicoupling probability or the intermodular connection probability can enhance the network robustness under both hub attacks and inner attacks. We also note that the community structure can prevent cascading failures spreading globally in entire interdependent networks. Finally, we obtain the result that if we want to efficiently improve the robustness of interdependent networks and reduce the protection cost, the intermodular connection probability should be protected preferentially, implying that improving the robustness of a single network is the fundamental method to enhance the robustness of the entire interdependent networks.