Figures
Abstract
Modern infrastructure networks are often coupled together and thus could be modeled as interdependent networks. Overload and interdependent effect make interdependent networks more fragile when suffering from attacks. Existing research has primarily concentrated on the cascading failure process of interdependent networks without load, or the robustness of isolated network with load. Only limited research has been done on the cascading failure process caused by overload in interdependent networks. Redundant design is a primary approach to enhance the reliability and robustness of the system. In this paper, we propose two redundant methods, node back-up and dependency redundancy, and the experiment results indicate that two measures are effective and costless. Two detailed models about redundant design are introduced based on the non-linear load-capacity model. Based on the attributes and historical failure distribution of nodes, we introduce three static selecting strategies-Random-based, Degree-based, Initial load-based and a dynamic strategy-HFD (historical failure distribution) to identify which nodes could have a back-up with priority. In addition, we consider the cost and efficiency of different redundant proportions to determine the best proportion with maximal enhancement and minimal cost. Experiments on interdependent networks demonstrate that the combination of HFD and dependency redundancy is an effective and preferred measure to implement redundant design on interdependent networks. The results suggest that the redundant design proposed in this paper can permit construction of highly robust interactive networked systems.
Citation: Liu L, Yin Y, Zhang Z, Malaiya YK (2016) Redundant Design in Interdependent Networks. PLoS ONE 11(10): e0164777. https://doi.org/10.1371/journal.pone.0164777
Editor: Peter Csermely, Semmelweis University, HUNGARY
Received: June 12, 2016; Accepted: September 30, 2016; Published: October 20, 2016
Copyright: © 2016 Liu et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper and its Supporting Information files.
Funding: This work was supported by National Aerospace Science Foundation of China (No.20140751008); Beihang University Innovation and Practice Fund for Graduate (YCSJ-02-2016-09).
Competing interests: The authors have declared that no competing interests exist.
Introduction
With the rapid development in modern information and energy technologies, infrastructure networks have become larger and more complex, and show an interdependent relationship between them because of functional dependence. For example, electricity system and communication system are highly coupled with together, where communication system relies on the electricity system to provide electricity, and electricity system needs communication system to provide control[1].
Compared with an isolated network, interdependent networks are prone to be more susceptible to attacks (random or intentional attack) because of coupling between networks [1–3]. Interdependent network has two different types of links, connectivity link and dependency link. Connectivity link is the connection of paths in isolated network whereas the dependency link reflects the coupling between two isolated networks. If a node in network NA failed and be disconnected from network NA, which will lead to the failure of another node in another network NB and may result in a further failure of network NA. The process described above occurs recursively, which may lead to a complete collapse of the interdependent systems.
Robust analysis of interdependent network and designing anti-vulnerable topology have received considerable attention in recent years. Buldyrev[1] firstly constructed an interdependent network model and defined the failure mode of network. He used the generation function and percolation theory to derive a percolation value and performed experiments on three interdependent networks (ER-ER, SF-SF, and ER-SF) to verify the theoretical value. After this pioneering study, researchers have investigated methods to enhance the robustness of interdependent networks. A typical model, introduced by Parshani[2], through reducing coupled strength by removing a part of dependency links. Furthermore, the results demonstrated that robustness of interdependent network had considerably increased by decoupling 40% of nodes. Another method is generating autonomy nodes that would not be non-functional under attack. The experiment found that interdependent networks will maintain their function by generating 10% autonomy nodes. Gong[4] introduced a technique to protect certain key nodes, in which 5% of key nodes can be determined using six strategies (random, degree, betweenness centrality, leader-rank, local and page-rank) and protected them from failure when its coupled nodes collapsed. Hongshen[5] presented a defensive measure in which removing certain nodes with lower degree in interdependent networks to suppress cascading failure process.
Based on a comprehensive analysis of the above studies, we determined that initial studies on the robustness of interdependent networks mainly focus on isolated network with a free load[6–8]. However, in reality, most systems have a load, e.g., electricity network, transportation network, etc. The cascading failure of these networks are primarily caused by overload[9], such as a traffic jam or congestion in communication system. Researches on overload failure model primarily focused on isolated network and rarely addressed interdependent networks[10, 11]. Fei Tan et al[12] introduced three coupling preference (assortative, disassortative and random coupling) in interconnected networks, and determined assortative coupling had better advantages in enhancing the robustness of interconnected networks. After a node fails, its load will redistribute to other nodes. In conclusion, there are two types of redistribution strategies: local distribution and global distribution. In local distribution model, the load of failure nodes would redistribute to its adjacent nodes, such as sandpile model introduced by Brummitt et al[13]; however, the load of remaining nodes are dependent on the betweenness centrality of nodes in updated network topology in global distribution model[12, 14].
Studies on cascading failure process of interdependent networks caused by overload are limted and incomprehensive[14]. The hypothesis in the model introduced by Fei Tan et al[12] is that two isolated networks are homogeneous, and load of failure nodes would be redistributed between two networks, which is not suitable for interdependent networks. Brummitt et al[15] assumed that load of failure nodes would only redistribute to its adjacent nodes. The protecting strategy presented by Hongsheng[5] is plausible but in reality difficult to implement because cascading process is extremely fast, and the interval time is too short to use this method. The main reason of nodes failed is interdependent failure or overload. Previous studies, such as the autonomous node[16] generation, focused on reducing the scope of interdependent failures. Thus, this study aims at minimizing the scope of overload failure.
Redundant design, which is an important measure used to improve the reliability of the system, is used more commonly in reliability design[17, 18]. Thus, in this paper, to suppress overload failure of nodes, we introduce a novel method-redundant design in construction of interdependent networks and analyze different effects of two kinds of redundant strategies. The remainder of this paper can be described as follows: section 2 presents two redundant design models; section 3 discusses the results of simulation; and section 4 gives the conclusion.
Methods Study
Cascading process
Without loss of generality, two isolated networks (labeled NA and NB) with the same nodes (NA = NB = N = 300) are investigated. The typical topology of interdependent networks is illustrated in Fig 1. Connectivity links (blue or green solid line) in two isolated networks could transfer traffic but dependent links (black dash line) could not. Dependent links between two isolated networks are bidirectional and only provide functional and logical connectivity between two networks. If a node in network NA failed, which would lead to another node in network NB collapsed because of functional dependency, which means interdependent failure.
A isolated network is a pair of (V, E), with V the set of all nodes {v1, v2, …, vn} and E the set of all edges {e1, e2, …, en}. A path P(vi, vk) between node vi and vk is a subset of consecutive edges, i.e. P(vi, vk) = {e1, e2, …, ej}∈E. The length |P(vi, vk)| of a path P(vi, vk) is given by the number of edges in it. There are several paths from node vi to vk, and the distance d(vi, vk) between vi and vk is defined as the shortest path(i.e. minimal length of |P(vi, vk)|). From the definition of d(vi, vk), perhaps several path P(vi, vk) exist and whose length |P(vi, vk)| are all equal to d(vi, vk).
Each node vj in interdependent network is allocated load Lt(vj) which fluctuates timely with the topology of network. For initial load L0(vj) (t = 0) of each node, we can take betweenness centrality to calculate it and its formula is expressed as follows:
(1)
Where, d(vi, vk: vj) represents the shortest path P(vi, vk) which pass through node vj and N(d(vi, vk: vj)) is the number of d(vi, vk: vj)
The load of node vj at time t, Lt(vj), varies with the topology of whole network. After a node failed, its load would delivery to other functional nodes. Here we assume that the load of failed nodes transfer evenly to remaining functional nodes in network which is expressed as follows:
(2)
Where, Vfunctional and Vfailed are set of nodes that remain functional at time t and failed at time t-1, respectively. |Vfunctional| denotes number of node in Vfunctional.
Due to restriction of cost, the capacity of a node is limited. A node’s capacity means the ability to bear load on it mostly. For studying cascading process on complex network with traffic, Motter and Lai[11] proposed that the capacity of node C(vi) was proportional to its initial load L0(vi), as showed as follows
(3)
Where, α ≥ 0 is the tolerance coefficient of node vi, and greater of it means node vi could handle more load on it. However, Kim and Motter[19] found that there was no linear relationship between node’s load and capacity by analyzing four real networks. And Dou B.[20] proposed a non-linear load-capacity model, which seemed more suitable for real system. In this paper, we adopt this model to determine the capacity of nodes, and the model can be described as follows
(4)
Here, α ≥ 0 and β ≥ 0. When α = 1, this model degenerates to ML model.
In addition, the cost of network is depend on nodes capacity, which means that with the increase of α and β, we should allocate more resource on network. In this paper, we define cost of network as follows
(5)
And according to the findings of Kim and Motter[19], due to network traffic fluctuations, real systems tend to have larger unoccupied portions of the capacities—smaller load-to-capacity ratios—on network elements with smaller capacities. To evaluate the efficiency of node’s capacity, we propose a modified model based on Kim and Motter and it is described as follows
(6)
Where, E_Value and Ln(vi) are the efficiency coefficient and load of node vi when interdependent networks in a stable state, respectively. And if node vi failed after cascading process, the value of Ln(vi)/C(vi) is 1. Different from KM model, our modified model make a weighted average of all nodes’ efficiency.
Once the load of a node surpass its capacity, it will lead to overload failure, which is another failure mode of node in interdependent network. Compared to isolated network, interdependent network exhibits more obvious vulnerability[12]. A small amount of nodes in the network are overload or suffering attack, which may cause cascading failure and lead to further damage. Fig 1 depicts an intuitive example for explaining cascading failure process of interdependent networks when experiencing a random attack. In this cascading process, node labeled 3 in network NA was attacked, which lead to node labeled 4 in NB failed. Then, we threat node labeled 1 in NB failed because of out of giant component, which lead to node labeled 2 in NA collapsed. After interdependent failure ending, load of all failed nodes in two networks would evenly be distributed to remaining two nodes (as shown in Fig 1(B)), which lead to node labeled 3 in NB overload. Failure of node labeled 3 in NB causes a brandnew interdependent failure, resulting a complete breakdown of whole networks (as shown in Fig 1(D)).
Previous studies proposed that the relative value of average network size after the cascading failure process can be selected to evaluate the robustness of interdependent networks under random/intentional attack[1, 4, 14]. The relative value of average network size after cascading failure is defined as follows
(7)
Where, S′(NA) and S′(NB) stand for the number of node in NA and NB after cascading process, respectively. To analyze the failure distribution of cascading failure process, we also focus on number of failed nodes caused by interdependency and overload, respectively.
We firstly construct two Erdős-Rényi networks according to the literature[21], whose degree distribution obeys the Poisson distribution and parameter NA = NB = N = 300 and average degree <kA> = <kB> = <k> = 6. Then constructing an ER-ER interdependent network by coupling two Erdős-Rényi networks. We randomly attack a proportion, 5%, of nodes in each network and follow the iterative process of cascading failure, and do simulation over 50000 times on different value of α and β. The simulation results of cascading failure process with diverse α and β are demonstrated in Fig 2. Obviously, the behavior of G is characteristic of a first-order phase transition in ER-ER interdependent network.
(Underlying data are in Text 8-Text 13 in S2 File).
In ER-ER interdependent network, for a certain β, a lower limit αL and upper limit αU are determined. When α ≤ αL, whole networks collapsed and all nodes failed caused by interdependency or overload and overload failed nodes account for most. While α > αU, despite those nodes be attacked at first, no more nodes in networks failed. [αL,αU] is a phase transition interval and during this interval, the value of G increases rapidly with the growing of α and overload failed nodes decreases quickly at the same time and the cascading failure process come to its end in a short time.
According to the results of Fig 2(A) and Fig 2(B), we can conclude that overload failed nodes have been accounted for 70% when β = 6 and α = 0.4, the number of interdependent failed nodes is large simultaneously. Thus, we fix β = 6 and α = 0.4 and do simulation over 100000 times, the results are shown in Fig 3. We have counted the failed time for each node over 100000 times, including interdependent failure and overload failure. Interesting, when β = 6 and α = 0.4, the frequency of most nodes in NA or NB failed because of overload is about 70%, which is the proportion of overload failed nodes, and several nodes failed with a slightly lower frequency compared to most nodes. In contrast, some nodes failed due to interdependency with a higher frequency compared to other nodes.
(Underlying data are in Text 14-Text 17 in S3 File).
Redundant design
As described in Fig 2(C), we can increase the value of α and β to reduce the size of cascade avalanche, which also improving the cost of networks and reducing the value of E_Value at the same time. And perhaps it is a troublesome problem to enhance the capacity of node in the current technology level. Thus, it is a worth considering question that how we can utilize existing resources to heighten the robustness of whole networks. The redundancy allocation is one of the most advantageous methods to optimize system reliability [22–25]. For some key components in system, designer will allocate more resources to those to enhance their ability to handle catastrophic events. Because of aging or suffering attack, a component failed and it will still remain be functional if it has a backup and it would collapse only its back-up failed. With this measure, the failure rate of component in systems would decrease drastically and reliability of whole system enhances simultaneously.
Nodes back-up
To suppress overload failure of nodes and inspired by redundant design in reliability domain, this paper proposes a redundant design to some nodes in interdependent network. As shown in Fig 4(A), compared to other four nodes (labeled 1, 2, 4 and 5), node labeled 3 is determined with a key node according to the value of betweenness centrality, its failure would break down four nodes to two disconnected parts and lead to whole network topology collapsed accordingly. If we make an addition of node labeled 3 and two joint nodes (filled green) whose reliability value is 1, and two joint nodes could switch on successfully to connect back-up if necessary, which would reduce the failure rate of node labeled 3 and maintain whole network connected like before.
In this model, we introduce the concept of node unit U(vi) which represents set of nodes with same function (red dashed ellipse in Fig 4) and define U(vi) as follows:
(8)
where n represents redundant level and it means that node vi has no back-up if n = 1;
Then, we can determine the capacity of node unit U(vi), C(U(vi)), according to our redundant model. Clearly C(U(vi)) is the sum of capacity of all node
(9)
And the cost of network after redundant design is increment to
(10)
We define initial node as main node and it should be activate firstly. If main node could bear load that passes through on it, it is unnecessary to activate its back-up, and back-up would be in work immediately to balance load once main node couldn’t handle all load on it. With this mechanism, the value of E_Value of main node and back-up which has been activated will maintain a larger level and node unit U(vi) would not collapse at the same time. However, the reliability and successful rate of activating back-up are two important question. This paper assumes that joint nodes are reliable and the successful rate of activating back-up is always equal to 1.
Determining which nodes would have a back-up is really important, which directly effects the result of implementation. In this paper, we introduce four kinds of select strategies, Random-based, Degree-based, Initial Load-based, and HFD-based (historical failure distribution).
The first three strategies, mainly based on static attributes of node, can be classified as static method. Random-based, which means we randomly choose some nodes in network. Degree and Initial Load-based, which according to the ranking of degree and initial load value of nodes and then pick those nodes listed in the forefront with a proportion. HFD (as shown in Fig 4 and Fig 5), which based on the simulation results of cascading failure over large-scale simulation times, is classified as a dynamic method and we can implement redundant design more accurately with this method. In HFD, we select those nodes with higher frequency of overload or interdependent failure compared to other nodes.
Dependency redundancy
To reduce the number of nodes caused by interdependent failure, we introduce a model named dependency redundancy. According to the failure mode of interdependent network, a node failed in NA will lead to another node in NB collapsed, which may cause an iterative cascading process. Because each node in NA only can find a node in NB to be coupled with, in other word, only a node in NB could provide functional or logical dependency to node in NA.
As shown in Fig 5, network NA (whose nodes filled blue) and network NB (whose nodes filled green) are coupled together, and each node is coupled with a certain node in another network through dependency link (black dotted line). Before adopting dependency redundancy, if node labeled 1 in NA suffering from attack or overload, which would lead to failure of node labeled 2 in NB. Here we randomly add two dependency links (red dotted line) in interdependent networks. If the same thing happened (node labeled 1 in NA failed), node labeled 2 in NB would not collapse because node labeled 3 in NA could provide functional support. Similarly, failure of node labeled 4 in NA would not result in node labeled 3 disconnected from NB.
Adding some dependent links in networks would increase cost of design, which is a question that should be studied. In this paper, we assume that cost of dependent link is depend on capacity of two nodes, and we defined it as follows
(11)
Where, C(l(vi,vj)) means the cost of adding a dependent links between node vi and node vj.
Simulation algorithm
To study the impact of redundant design in interdependent networks, we have applied the following simple simulated algorithm. Firstly we give some necessary definitions about some symbols and operations.
AM (NA), AM (NB)←adjacency matrix of NA and NB, respectively
CN(NA), CN(NB) ← 1×N vector (out of order), node pair (CN(NA)(1,i),CN(NB)(1,i)) (i = 1:N) represents that they are interdependent in networks
CN (CN(NA), CN(NB))←interdependence relation between NA and NB
AP←attack proportion
VA←attack nodes
,
set of failed nodes caused by dependency at time t in NA and NB, respectively
,
set of failed nodes caused by overload at time t in NA and NB, respectively
,
set of failed nodes because of out of giant component at time t in NA and NB, respectively
V(NA), V(NB)←set of failed nodes during cascading process in NA and NB, respectively
V(NA)⨂CN (CN(NA), CN(NB))←failed nodes in NB because of coupled with V(NA)
V(NB)⨂CN (CN(NA), CN(NB))←failed nodes in NA because of coupled with V(NB)
The pseudo-code of simulated algorithm is as follows
Program: Initial Configuration Module
NA = (V, E, AM (NA))←generating NA.
NB = (V, E, AM (NB))←generating NB.
CN(NA), CN(NB) ←generating coupling nodes pair
L0(vi) ←calculating initial load of nodes in NA and NB
C(vi) ←calculating capacity of nodes in NA and NB
End Initial Configuration Module
Program: Cascading Process Module
VA = AP × 2N,
V(NA) = VA(1:length(VA)/2)
V(NB) = VA(length(VA)/2 + 1:length(VA))
= V(NB)⨂CN (CN(NA), CN(NB))
= V(NA)⨂CN (CN(NB), CN(NA))
V(NA) = V(NA)+
V(NB) = V(NB)+
Load_A = 0
Load_B = 0
t = 0
Do While
Do While
Remove {} and links that connected with those
Nodes, and update
CN (CN(NA), CN(NB))
CN (CN(NB), CN(NA))
V(NA) = V(NA)+
V(NB) = V(NB)+
Load_A = Load_A+
Load_B = Load_B+
EndWhile
t = t+1
Vt = V(NA)∩ V in NA
For i = 1 to length(Vt)
Lt(vi) = Lt−1(vi) + Load_A/ length(Vt)
If Lt(vi) > C(vi)
EndIf
EndFor
Vt = V(NB)∩ V in NB
For i = 1 to length(Vt)
Lt(vi) = Lt−1(vi) + Load_B/ length(Vt)
If Lt(vi) > C(vi)
EndIf
EndFor
EndWhile
Results
To analyze the role of the redundant design, we have applied two redundant design methods on ER-ER interdependent networks. For a better simulation result, each redundant proportion is simulated over 5000 times.
The simulation results of cascading process with different proportion and four kinds of node back-up strategies are demonstrated in Fig 6. The simulations parameters are NA = NB = 300, <kA> = <kB> = 6, β = 6, α = 0.4. If we don’t consider the cost of redundant design, obviously, HFD back-up strategy is a preferred method on the whole when the redundant proportion is a constant. As pictured in Fig 6(A), when 5% of nodes in whole network have back-up, it would get almost 60% of enhancement on G compared to no redundant proportion, which means we could maintain 70% of nodes when 5% of nodes have a back-up. In comparison, degree-based and random-based back-up strategy have similar result and perform worst. But once we take consideration of the cost of redundant design, the results will be a big difference. From Fig 6(B), we note that HFD is still the best way and dependency redundancy has gradually show its advantage, while initial load-based back-up strategy becomes the worst because it seemly costly. And we noticed that the cost of redundant design has a shape rise when we want to maintain 90% of initial network survival.
(Underlying data are in Text 18-Text 32 in S4 File).
HFD, a new and distinct method that based on the distribution of failed nodes, performs better with a low cost and high efficiency. In Fig 3, it is indicated that most nodes have a higher frequency of overload failure, which means it is unwise to select nodes based on this results. While the distribution of interdependent failure is different from overload failure-several nodes have a higher frequency of interdependent failure compared to the rest. If we control the failure of nodes which coupled with those nodes with higher frequency of interdependent failure, then interdependent failure would eliminate a lot. For example, when 8% of nodes have a back-up based on historical failure distribution, which will keep 78% of nodes in initial network survival with the least cost. And from Fig 6, it illustrates that HFD is the most effective and costless method compared to others. In HFD-based method, the number of nodes that have a back-up are determined by the results of interdependent failure. Thus only a part of nodes can be determined by HFD, which could not protect whole network from failure.
The simulation results of dependency redundancy are also demonstrated in Fig 6, which illustrates that dependency redundancy is not an ideal method to improve the value of G compared with other four node back-up strategies when robustness is the first consideration. However, if we consider cost of implementation, dependency redundancy has more advantages to others. And we notice that it performs similar with HFD, and perhaps combination these two measures will get unexpected results.
To illustrate this hypothesis, we combine two redundant measures and the results are demonstrated in Fig 7 and Fig 8. In the experiment, 5% and 10% of nodes are selected based four measures, respectively, meanwhile 5% and 10% of dependency links are randomly constructed. In Fig 7 and Fig 8, G0 and C0 stand for value of G and Cost after cascading process when only 5% of nodes have a back-up, respectively, while δG and δC are the increment of G and Cost from 5% and 10% dependency redundancy. From Fig 7 and Fig 8, it is clearly that adding a small proportion of dependent links will obtain a significant enhancement on G while with a tiny cost incrementation. Thus through a combination of node back-up and dependency redundancy, we can construct more robust interdependent network with lower cost and resource.
Thus, in ER-ER interdependent network, HFD, a dynamic method, is the first choice to determine which nodes that would have back-up. And dependency redundancy is also suggested with minimal cost when no back-up could deploy. If possible, combination of two measures is the most effective and costless scheme.
Conclusion
In this paper, we have studied potential cascading failures of interdependent networks with traffic under random attacks when system incorporates redundant design.
Based on the non-linear load-capacity model, we can enhance the invulnerability of interdependent networks under random attack through following two methods:
- If technology allows, we can increase value of α and β, and a phase transition interval [αL,αU] for each β could help us to determine the most suitable value of α;
- If conditions are limited, the redundant design proposed in this paper could be a nice suggestion to be considered.
In ER-ER interdependent networks, the experimental results indicated that HFD is a preferred measure to decide which nodes could have a back-up with lower cost. While we also could select nodes based on its initial load if robust is more important than cost. And if cost of design is limited and no more back-up could be deployed, dependency redundancy may be an ideal way with a minimal cost to get higher G value.
If we want to construct a more robust interdependent system, node back-up and dependency redundancy should be adopted in the beginning. Combination of two kinds of measures is attractive by its higher efficiency and lower cost, which could suppress overload failure more accurately and reduce the scope of interdependent failure.
Therefore, the proposed preventive measure for enhancing the survivability of interdependent networks can be used effectively when constructing robust networked systems. This approach provides a practicable technical method to design and optimize these networks with lower cost. And combination of node back-up and dependency redundancy could be studied deeply in the future.
Supporting Information
S1 File.
Text 1: Adjacency matrix of network A. Text 2: Adjacency matrix of network B. Text 3: Initial load of each nodes in network A. Text 4: Initial load of each nodes in network B. Text 5: Coupling between two networks. Text 6: Initial degree of each nodes in network A. Text 7: Initial degree of each nodes in network B.
https://doi.org/10.1371/journal.pone.0164777.s001
(ZIP)
S2 File.
Text 8: Interdependent failed nodes at different α and β duiring 100 simulation times of Network A. Text 9: Interdependent failed nodes at different α and β duiring 100 simulation times of Network B. Text 10: Nodes that are not belong to gaint component in network A at different α and β duiring 100 simulation times of Network A. Text 11: Nodes that are not belong to gaint component in network B at different α and β duiring 100 simulation times of Network B. Text 12: Overload failed nodes at different α and β duiring 100 simulation times of Network A. Text 13: Overload failed nodes at different α and β duiring 100 simulation times of Network B.
https://doi.org/10.1371/journal.pone.0164777.s002
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S3 File.
Text 14: Interdependent failed nodes accounts in network A over 100000 simulations times. Text 15: Interdependent failed nodes accounts in network B over 100000 simulations times. Text 16: Overload failed nodes accounts in network A over 100000 simulations times. Text 17: Overload failed nodes accounts in network B over 100000 simulations times.
https://doi.org/10.1371/journal.pone.0164777.s003
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S4 File.
Text 18: Redundant proportion and nodes in network A with Random-based node back-up strategy. Text 19: Redundant proportion and nodes in network B with Random-based node back-up strategy. Text 20: Failed nodes in different redundant proportion in network A with Random-based node back-up strategy. Text 21: Redundant proportion and nodes in network A with Degree-based node back-up strategy. Text 22: Redundant proportion and nodes in network B with Degree-based node back-up strategy. Text 23: Failed nodes in different redundant proportion in network A with Degree -based node back-up strategy. Text 24: Redundant proportion and nodes in network A with InitialLoad-based node back-up strategy. Text 25: Redundant proportion and nodes in network B with InitialLoad-based node back-up strategy. Text 26: Failed nodes in different redundant proportion in network A with InitialLoad-based node back-up strategy. Text 27: Redundant proportion and nodes in network A with HFD-based node back-up strategy. Text 28: Redundant proportion and nodes in network B with HFD-based node back-up strategy. Text 29: Failed nodes in different redundant proportion in network A with HFD-based node back-up strategy. Text 30: Redundant proportion and nodes in network A with Dependency redundancy. Text 31: Redundant proportion and nodes in network B with Dependency redundancy. Text 32: Failed nodes in different redundant proportion in network A with Dependency redundancy.
https://doi.org/10.1371/journal.pone.0164777.s004
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Acknowledgments
We owe our thanks to the National Aerospace Science Foundation of China and Beihang University Innovation and Practice Fund for Graduate. The authors would also like to thank the anonymous reviewers for their feedback, which has helped to improve the clarity of our manuscript.
Author Contributions
- Conceptualization: LL YY.
- Data curation: LL YY ZZ.
- Formal analysis: LL ZZ.
- Funding acquisition: YY ZZ.
- Investigation: LL YY.
- Methodology: LL YY.
- Project administration: YY.
- Resources: YY YM.
- Software: LL YY ZZ.
- Supervision: YY.
- Validation: YY ZZ.
- Visualization: LL.
- Writing – original draft: LL.
- Writing – review & editing: LL YY YM.
References
- 1. Buldyrev SV, Parshani R, Paul G, Stanley HE, Havlin S. Catastrophic cascade of failures in interdependent networks. Nature. 2010;464(7291):1025–8. pmid:WOS:000276635000035.
- 2. Parshani R, Buldyrev SV, Havlin S. Interdependent Networks: Reducing the Coupling Strength Leads to a Change from a First to Second Order Percolation Transition. Physical review letters. 2010;105(4). pmid:WOS:000280237400014.
- 3. Shao J, Buldyrev SV, Havlin S, Stanley HE. Cascade of failures in coupled network systems with multiple support-dependence relations. Phys Rev E. 2011;83(3). pmid:WOS:000288898300001.
- 4. Gong M, Ma L, Cai Q, Jiao L. Enhancing robustness of coupled networks under targeted recoveries. Scientific reports. 2015;5:8439. pmid:25675980; PubMed Central PMCID: PMC4327128.
- 5.
Hong S, Wang B, Wang J, editors. Cascading failure propagation in interconnected networks with tunable load redistribution strategy. Prognostics and System Health Management Conference (PHM), 2015; 2015: IEEE.
- 6. Hong S, Yang H, Li G, Huang N, Ma X, Trivedi KS. Analysis of propagation dynamics in complex dynamical network based on disturbance propagation model. International Journal of Modern Physics B. 2014;28(22):1450149.
- 7. Hong S, Yang H, Zhao T, Ma X. Epidemic spreading model of complex dynamical network with the heterogeneity of nodes. International Journal of Systems Science. 2015;47(11):2745–52.
- 8. Hong S, Yang H, Zio E, Huang N. A novel dynamics model of fault propagation and equilibrium analysis in complex dynamical communication network. Applied Mathematics and Computation. 2014;247:1021–9.
- 9. Hong S, Lv C, Zhao T, Wang B, Wang J, Zhu J. Cascading failure analysis and restoration strategy in an interdependent network. Journal of Physics A: Mathematical and Theoretical. 2016;49(19):195101.
- 10. Hong S, Wang B, Ma X, Wang J, Zhao T. Failure cascade in interdependent network with traffic loads. Journal of Physics A: Mathematical and Theoretical. 2015;48(48):485101.
- 11. Motter AE, Lai YC. Cascade-based attacks on complex networks. Physical review E, Statistical, nonlinear, and soft matter physics. 2002;66(6 Pt 2):065102. pmid:12513335.
- 12. Tan F, Xia Y, Zhang W, Jin X. Cascading failures of loads in interconnected networks under intentional attack. EPL (Europhysics Letters). 2013;102(2):28009.
- 13. Liu C, Li D, Fu B, Yang S, Wang Y, Lu G. Modeling of self-healing against cascading overload failures in complex networks. EPL (Europhysics Letters). 2014;107(6):68003.
- 14. Hong S, Zhang XJ, Zhu JX, Zhao TD, Wang BQ. Suppressing failure cascades in interconnected networks: Considering capacity allocation pattern and load redistribution. Mod Phys Lett B. 2016;30(5). pmid:WOS:000371065200017.
- 15. Brummitt CD, D'Souza RM, Leicht EA. Suppressing cascades of load in interdependent networks. P Natl Acad Sci USA. 2012;109(12):E680–E9. pmid:WOS:000301712600003.
- 16. Schneider CM, Moreira AA, Andrade JS, Havlin S, Herrmann HJ. Mitigation of malicious attacks on networks. P Natl Acad Sci USA. 2011;108(10):3838–41. pmid:WOS:000288120400009.
- 17. Qun L, De-huan Z, Chen C. Study of Reliability Redundancy Design of Underwater Sensor Networks. Fire Control & Command Control. 2012;8:019.
- 18. Clark L, Shirinzadeh B, Zhong Y, Tian Y, Zhang D. Design and analysis of a compact flexure-based precision pure rotation stage without actuator redundancy. Mechanism and Machine Theory. 2016;105:129–44.
- 19. Kim DH, Motter AE. Resource allocation pattern in infrastructure networks. J Phys a-Math Theor. 2008;41(22). pmid:WOS:000256387900020.
- 20. Dou B-L, Wang X-G, Zhang S-Y. Robustness of networks against cascading failures. Physica A: Statistical Mechanics and its Applications. 2010;389(11):2310–7.
- 21. ERDdS P WI A. On random graphs I. Publ Math Debrecen. 1959;6:290–7.
- 22. Niyato D, Wang P, Hossain E. Reliability analysis and redundancy design of smart grid wireless communications system for demand side management. IEEE Wireless Communications. 2012;19(3):38–46.
- 23. Su H-P, Shen S-M, Liu W-Y, Ye Y. Real-time Communication System Based on Redundancy CAN-Bus. Automation & Instrumentation. 2013;1:011.
- 24. Jin Z, Tongzhen W, Qunhai H. Reliability model analysis and redundancy design of A2MC VSC-HVDC power transmission system. Transactions of China Electrotechnical Society. 2013;28(2):319–23.
- 25. WANG Q, CUI W. Study of Redundancy Reliability Design Based on the Genetic Algorithm. Journal of Nanchang Hangkong University (Natural Sciences). 2015;2:003.