The influence of the depth of k-core layers on the robustness of interdependent networks against cascading failures
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The hierarchical structure, [Formula: see text]-core, is common in various complex networks, and the actual network always has successive layers from 1-core layer (the peripheral layer) to [Formula: see text]-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 [Formula: see text]-core layers (the value of [Formula: see text]) 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 [Formula: see text]-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 [Formula: see text]-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 [Formula: see text]-core layers, but also is slightly influenced by the initial load. With RO-attack, the networks with less [Formula: see text]-core layers are more robust when the initial load is small. With RF-attack, the robustness improves with small [Formula: see text], 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.Keywords:
Robustness
Interdependent networks
Cascading failure
Core network
In real complex systems, the overall function is maintained through the connections among nodes. Failures of some nodes may destroy the connectivity of the system and thus damage the function of the system. In some complex systems, some nodes can form “interdependency groups” through hidden interdependency. The failure of one node may damage the rest of the nodes in the interdependency group. In this paper, we investigate the effects of the interdependency strength of the nodes, the size distribution, and the size of the interdependency groups on the cascading dynamics and the robustness of complex networks. Through numerical simulation and theoretical analysis, it is found that the cascading failures of the networks can be divided into two processes at a scale level: “intra-group cascading” and “inter-group cascading”. In the intra-group cascading process, the failure of one node will result in damage to the other nodes in the group through the interdependence among nodes, thus inducing more nodes to be unworkable and resulting in greater destructive force. In the inter-group cascading process, the failed nodes will cause the networks to be fragmented, which leads some nodes outside the interdependency group to isolate from the giant component and go to failure. Under the synergistic effects of these two processes, it is found that there are continuous and discontinuous phase transition phenomena in the cascade dynamics of the network. The occurrence of these two kinds of phase transition phenomena is related to the interdependency strength of nodes, the network degree distribution and the size distribution of the interdependency group. This means that by controlling the characteristics of interdependency groups, such as the interdependence strength of the nodes in the interdependency group or the size distribution of interdependency groups, the system can avoid collapsing suddenly and thus the robustness of the network can be improved.
Interdependent networks
Cascading failure
Robustness
Complex system
Degree distribution
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On complex networks,large-scale cascading failures that are triggered by some small disturbances can lead to disastrous consequences.In order to satisfy demand of the people on the security and reliability of complex networks related to the national economy and people's livelihood,the study on cascading failures on complex networks becomes a hot branch of complex networks research in recent years.The theoretical modeling is the basic and key problem for analysis,prevention and control of cascading failures.Main developments of cascading failures on complex networks were surveyed,mainly including several types of cascading failures models and the relevant research results.Both the existing problems at present and the development trend were pointed out.
Cascading failure
Interdependent networks
Complex system
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Interdependent networks are emerging in the critical infrastructures such as telecommunication and Internet networks, electrical smart grids, transportation networks, and many more. There are advantages of introducing interdependency into critical infrastructures to enhance the functional and operational capability. On the other hand, there is a critical tradeoff which is the cascading failures that propagate from one network to another network. The occurrence of this phenomena will cause major catastrophe to our lives that is hugely dependent on such critical infrastructure. Thus, there is a need to understand how cascading failures are spreading in interdependent networks and the sensitive factors that have a severe impact on these phenomena. In this paper, we have developed a cascade model to describe and quantify the effects of cascading failures in interdependent networks using Sandpile dynamic models as load distribution. The extensive analysis and numerical results have shown that the increase of interdependency links can enlarge the global cascade probability from one network to another network. In addition, the failures that triggered from a dense network can cause more severe damage to another network than the failures triggered from a sparse network.
Cascading failure
Interdependent networks
Network model
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Cascading failures often occur in real systems, and the modeling method of weighted interdependent networks can be closer to the physical reality. The cascading failure in complex network is rarely studied in weighted interdependent networks. On the one hand, the influence of network weight on node capacity and load-sharing strategy is seldom considered in the design of cascading failure model. On the other hand, the cascading failure process in weighted interdependent networks is seldom considered. In order to resist the cascade effect in the network, a new cascading failure model is proposed based on the construction of weighted interdependent networks, in which the initial load is constructed by incorporating the node strength, and the load-sharing strategy is designed by combining the residual capacity of nodes and the node importance. Secondly, considering the time-varying characteristics of load and the inter-layer dependence, the cascading failure algorithm of weighted interdependent networks is designed. Finally, the effect of parameters in the model on the network robustness and the performance of the model in different weighted interdependent networks are obtained through the analysis of network cascading failures, which verifies the effectiveness of the proposed method.
Cascading failure
Interdependent networks
Robustness
Network model
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Our society nowadays is governed by complex networks, examples being the power grids, telecommunication networks, biological networks, and social networks. It has become of paramount importance to understand and characterize the dynamic events (e.g. failures) that might happen in these complex networks. For this reason, in this paper, we propose two measures to evaluate the vulnerability of complex networks in two different dynamic multiple failure scenarios: epidemic-like and cascading failures. Firstly, we present epidemic survivability (ES), a new network measure that describes the vulnerability of each node of a network under a specific epidemic intensity. Secondly, we propose cascading survivability (CS), which characterizes how potentially injurious a node is according to a cascading failure scenario. Then, we show that by using the distribution of values obtained from ES and CS it is possible to describe the vulnerability of a given network. We consider a set of 17 different complex networks to illustrate the suitability of our proposals. Lastly, results reveal that distinct types of complex networks might react differently under the same multiple failure scenario.
Survivability
Cascading failure
Vulnerability
Interdependent networks
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Cascading failure
Interdependent networks
Power grid
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Due to interdependencies among networks,cascading failure can spread from one network to others,which is a kind of relative failure based on function dependency by nature.A cascading failure model for interdependent networks is proposed after modeling two interdependent networks.Cascading failure in interdependent networks can be divided into internal cascading and cascading among networks.Degree-based cascading scale evaluation method is adopted to study the cascading failure phenomenon of two interdependent scale-free networks.It can be found that the bigger the average degree of the correlation rate and correlation nodes is,the more easily the cascading failure among networks occurs.The model can be extended to serve the turn for multiple cascade networks.
Cascading failure
Interdependent networks
Network model
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Cascading failures in isolated networks have been widely studied in the past decade, cascading failures in interdependent networks are attracting more and more attention recently, but previous studies focus on the interdependent networks topological cascading effects, neglecting the loads which present in most real networks. Considering the effect of loads, the two-layered interdependent BA network model and the cascading failure model were reestablished in this paper, based on these, the robustness of interdependent networks under random failures and intentional attacks and the effects of average degree on the suppressing of cascading failures were researched. The simulation results show that compared to random failures, interdependent network is more vulnerable under intentional attacks; the network in coupled state is more vulnerable than in isolated state under random failures, while therere no obvious differences under intentional attacks; the network performs more robust to resist cascading failures if each layer network of the interdependent network hold larger average degree.
Cascading failure
Interdependent networks
Robustness
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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.
Cascading failure
Interdependent networks
Vulnerability
Preferential attachment
Degree distribution
Giant component
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Cascading failure
Interdependent networks
Robustness
Complex system
Power-system protection
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