Cascading Model of Infrastructure Networks based ... - Semantic Scholar
1448
JOURNAL OF NETWORKS, VOL. 8, NO. 6, JUNE 2013
Cascading Model of Infrastructure Networks based on Complex Network Shaolin Wang Institute of Higher Education Research, Jilin Business and Technology College, Changchun, China, E-mail: [email protected]
Abstract—The paper focuses on the cascading failure phenomenon of infrastructure networks. A new cascading model is proposed to analyze the cascading failure characteristics of the systems. Based on the model, we present a good evaluation method of node importance to identify the key nodes which may cause cascading failures of infrastructure networks. Finally, the damage degree of infrastructure networks considering cascading failures is investigated through our cascading model. The simulation results indicate the validity and practicality of the proposed model. The research is helpful to protect the key components of infrastructure networks and decrease the network vulnerability. Meanwhile, the proposed method provides valuable reference to many actual networks in nature. Index Terms—cascading failure, infrastructure system, node importance, network damage degree, complex network theory
I. INTRODUCTION Many complex systems in real life can be described by various networks [1][2][3]. The people have been more and more aware of the importance of safety and robustness of networks [4][5][6][7]. In a network, a node or an edge will collapse due to the overload. And then the load at a neighboring node will change. If the load of the neighboring node increases and exceeds its corresponding capacity, the node is prone to malfunction. As a result, a new redistribution of loads occurs, and thus may lead to the failures of a new round of nodes. This kind of stepby-step process is so-called the cascading failure. The cascading failures often spread across the entire network like a plague, resulting in serious damage. Therefore, the cascading failure phenomenon on all kinds of networks has become the research focus [8]. Infrastructure systems are becoming more interdependent to other systems. The malfunctions within a given infrastructure system are more likely to impact the effective operation of other systems [9][10]. The increased interdependencies and reduced robustness margins induce the performance evaluation of complex systems less tractable. The networked systems are more complex due to the interactions among network members working at various information-to-capacity regimens. Therefore, it is difficult to anticipate emergent incidents from the information of single component nodes since there are so many interacting members. The systems are more and more vulnerable just because of these complex relations. Meanwhile, the cascading failure phenomenon © 2013 ACADEMY PUBLISHER doi:10.4304/jnw.8.6.1448-1454
of complex networks has been the focus of many studies, for the reason that cascading failures are common in reallife systems and especially can cause huge damages to our life and society development [11][12]. The infrastructure networks are also the very typical networks including cascading failure phenomenon. Even though the failures emerge very locally in an infrastructure network, the entire network can be largely affected, even resulting in global collapse. For instance, the earthquake in Taiwan resulted in a significant loss to many electronic enterprises because their suppliers are Taiwan enterprises in 1999. The shortage of 4600 million influenza vaccine happened in the United State because the only supplier of Chiron company was sued for bacterial contamination problems [13][14]. The blackout in North American was initiated by an untrimmed tree too close to high voltage transmission lines in the Midwest in 2003. From these serious results, we can see that the malfunction of only a component in an infrastructure system may trigger the serious results of the whole system due to cascading failure phenomenon. Recently, the research in infrastructure systems has attracted much attention. For a single infrastructure, many papers investigate the safety and vulnerability of a given system, as well as the impact of intentional attack and random failure on the system. The probabilistic risk analysis methods are presented to investigate the risk of infrastructure systems when there is little information of an infrastructure system [12][15]. Then some scholars analyze the impact of infrastructure performance triggered by natural disasters by using statistical learning theory [16]. However, the two methods have not considered the topology structure of infrastructure systems. Then some papers study the measurement of infrastructure vulnerability against disasters [17][18]. The relation between the network topology and the safety of a power system under natural disasters was explored by combining the power network topology with a component model [20][21]. However, all studies for risk and vulnerability analysis focus on only the single noninteracting infrastructure system. Since infrastructure systems have become interconnected with each other, a risk and safety research should not be investigated in isolation. Based on the background of infrastructure systems, this paper studies the cascading failure phenomenon in infrastructure networks. Different from the previous studies, we build a cascading model according to the
JOURNAL OF NETWORKS, VOL. 8, NO. 6, JUNE 2013
1449
topology structure and features of infrastructure systems first. Then an evaluation method of node importance is presented to identify the key nodes of infrastructure networks. The largest connected component is introduced to measure the damage degree of infrastructure networks after cascading failures. The simulation results indicate the validity and practicality of our proposed model. It is known that various nodes have various impacts on infrastructure systems. Especially, the failures of key nodes will be easier to cause the serious damage of infrastructure systems. If we identify the key nodes and provide protection with them in advance, the safety of infrastructure networks will be improved well. Our research therefore provides theoretical basis and valuable reference to actual networks in nature. II. PROPOSED CASCADING MODEL A. Dynamic Network Model Our society depends heavily on many complex systems. The safety and robustness of complex systems have been an important issue [19][20]. Therefore, the model and method to explore the cascading phenomenon of infrastructure networks become necessary. In this article, a graph (network) G is defined by a non-empty set of nodes A and a non-empty set of . A {v1 , v2 , ⋅ ⋅ ⋅, vN } is the set of edges K : G = ( A, K )=
K {l1 , l2 , ⋅ ⋅ ⋅, lM } is the set of edges. The total nodes and= number of nodes in the set A of the network is represented by N, while the total number of edges in the set K of the network is M . Here we consider the cascading failures triggered by removing a node of infrastructure network. If the initial load of the node is small, the removal cannot induce the disastrous results, thus the network can recover from the load redistribution. However, if the load of the removed node is large, the local redistribution will trigger cascading failures across the system and even lead to serious consequences, eventually result in the entire network collapsing. Distribution of load on the networks is determined by several factors. The network topology structure is one of the main factors. In order to different from the actual physical load, we regard that the load of a node is completely determined by the topology structure of the node within networks. In this article, the structure load of a node is defined as its betweenness. Because the actual physical load within networks is difficult to be determined, it is reasonable that this dimensionless structural load is used to study the vulnerability and node importance assessment. Therefore, the shortest path betweenness is adopted in this paper. That is to say, the more the shortest path through a node is, the load on the node is higher. We define that the load of node i is as following. δ vv′ (i ) Li C= = B (i )
© 2013 ACADEMY PUBLISHER
∑
v ≠ v ′∈V
δ vv′
n(n − 1)
(1)
where δ vv′ is the number of all the shortest paths between node v and node v′ , δ vv′ (i ) is the number of the shortest paths through node i between node v and node v′ . In order to avoid for the load of a node being zero, node v or node v′ is allowed to become node i in the Eq. (1). The detailed situation is as follows. δ vv′ (i )
∑
v ≠ v ′∈V
δ vv′
δ vv′ (i ) δ (i ) δ (i ) + ∑ vv′ + ∑ vv′ δ δ v =i ≠ v′ v ≠ i =v′ v ≠ i ≠ v ′ δ vv ′ vv ′ vv ′ δ (i ) = ∑ vv′ + 2(n − 1) v ≠ i ≠ v ′ δ vv ′ =
∑
(2)
From Eq.(2), 2 / n ≤ Li ≤ 1 . In a star network, the shortest paths between any two node pairs must pass through the center node, so the load of the central node is a maximum of 1. The node capacity refers to the maximum load that the node can handle. In infrastructure networks, the capacity of a node is limited by its cost. In our model, we assume the capacity Ci of node i to be proportional to its initial load Li (0) , i.e.,
C= (1 + β ) Li (0) i = (1 + β )
δ vv0 ′ (i ) ∑ 0 v ≠ v ′∈V δ vv ′ n(n − 1)
(3)
where the parameter β is the tolerance parameter which can characterize the resistance to the natural disasters. According to the actual situation, Li (0) is calculated by Eq. (1) and the capacity Ci of node i is calculated by Eq. (3) in the simulations. Because of the limitation of the capacity, when the node i malfunctions, the load of node i will be redistributed to its neighbor nodes. In this way, the additional load will be achieved by the neighbor node j of node i , i.e. (4) L j (t + = 1) L j (t ) + ∆Lij If L j (t + 1) ≤ C j , then the whole network can restore balance. If L j (t + 1) > C j , the node j will also collapse, which will lead to further redistributing the load Li of node j . This may result in the other breakdowns, therefore the cascading failures occur. The visualization of the local load redistribution rule is illustrated in Figure 1. The removal probability is adopted as following: L j (t + 1) ≤ C j , 0 pj = (5) L j (t + 1) ≥ C j , 1 where p j denotes the removal probability of node j .
1450
JOURNAL OF NETWORKS, VOL. 8, NO. 6, JUNE 2013
process is over, according to Eq. (6), we will calculate the value of ek , which denotes the network efficiency induced by removing k. In this case, N k′ is the numbers of the nodes which can maintain the normal operations after cascading failures in the whole complex network and dij is shortest path of any remaining node pair. In
i j
Figure 1. The visualization of load redistribution after the breakdown of node i.
B. Evaluation Method Network efficiency is used to describe the capacity on information transfer in networks. By investigating the network efficiency, we can try to improve the network structure so as to optimize the network efficiency. And then the resilience of infrastructure networks against cascading failures will be improved. Assume that the connected efficiency eij between node i and node j is inversely proportional to the shortest path dij , i.e.
eij =
1 dij
(6)
From Eq. (6), if there is no path that connects node node i and node j, dij = +∞ , thus eij = 0 . If no connection between node i and node j , the value of eij tends to 0. The network efficiency is defined by the connected efficiency eij . The detailed formula is as follows:
e(G ) =
=
∑e
i ≠ j∈G
ij
N ( N − 1) 1 ∑ i ≠ j∈G d ij
(7) N ( N − 1) where e(G ) denotes the network efficiency, and N is the number of nodes in network. From Eq. (7), e(G ) is normalized considering the case in which a network with the same number of nodes is fully connected. Thus, e(G ) is normalized by N ( N − 1) / 2 edges, case in which information communication is most efficient. Therefore, 0 ≤ e(G ) ≤ 1 , and e(G ) = 1 just when the network is fully connected. When an infrastructure network is on normal operation initially, the network efficiency is denoted by e0 which is calculated according to Eq. (7). In this case, N is the number of nodes during normal operation. Let dij be the shortest path between two nodes. After the cascading
© 2013 ACADEMY PUBLISHER
order to obtain the values of N k′ and ek , the above cascading model will be adopted to calculate the values. In the simulations, we introduce two indicators which are the node load and node capability respectively. In the following, we define the node importance by adopting the network efficiency before and after cascading failures. The importance value of node k is denoted by Rk . According to Eq. (7), the detailed expression of Rk is as follows.
Rk = 1 −
ek e0
1 ) N k′ ( N k′ − 1) i ≠ j∈Gk′ d ij = 1− 1 (∑ ) N ( N − 1) i ≠ j∈G d ij (
∑
(8)
where ek and e0 are the network efficiency before and after cascading failures triggered by node k in the whole network respectively, and Gk′ is the network after cascading failures. The weight of each node is obtained by the standardization for Rk in this paper. The algorithm steps are given below: Step 1: The initial load Lk (0) and the load capacity
Ck for all nodes are calculated according to Eq. (1) and Eq. (3). Step 2: The values of network efficiency for all nodes in the infrastructure network are calculated according to Eq. (7). Step 3: For k = 1 to n , the node importance Rk of all nodes are evaluated. Step 4: The node k is removed. The new nodes in a failure state are checked. Step 5: The load of all nodes is calculated again according to Eq. (4) after cascading failures. Step 6: Calculate Rk on the basis of Eq. (8) after cascading failures. C. Damage Degree In order to reflect the phenomenon associated to the cascading damage process, we need to give a quantitative measurement of network damage degree. There is always the largest connected component in networks which contains more nodes than other connected component in networks and there are pathways between any two nodes within this subgraph. The relative size of the largest connected component of the remaining network after cascading failures under our model can be expressed as:
JOURNAL OF NETWORKS, VOL. 8, NO. 6, JUNE 2013
Q=
N′ N
1451
(9)
where N is the initial size of the network before cascading failures and N ′ is the size of the largest connected component after cascading failures. Therefore, the simplest quantitative measurement of damage degree is given as follows: N′ H = 1− (10) N Obviously, if H = 1 , the network has been broken into many small disconnected parts and eventually resulted in the entire network collapsing. If H = 0 , the network works well. Obviously, H is the residual rate of the largest connected component after cascading failures, which can characterize the damage degree of infrastructure networks.
are investigated the between members. When there is the exchange of matter or energy between the two component nodes existing in the road transport infrastructure system, an edge between them is determined. In order to obtain the corresponding adjacency matrices of this road transport infrastructure system, we collect data and investigate the relations among component nodes in the road transport infrastructure system. Then the adjacency matrix is obtained. The network topology of this road transport infrastructure system is shown in Figure 2 under UCINET 6.0, which consists of 16 nodes.
III. NUMERIAL RESULTS A. Illustration of Infrastructure Networks Infrastructures networks can be represented effectively as a matrix. We define the matrix as adjacency matrix which characterizes an infrastructure network. The adjacency matrix is a square matrix. In a square matrix, values are 1 if there exists a information communication between the components to which the interception of row and column refers, and the values are 0 if the two components are not related within an infrastructure network. In the following, we formalize the adjacency matrix in infrastructure systems: the adjacency matrix Bij = {bij } is a N × N matrix which is defined such as
bij = 1 if vi , v j ∈ A and bij = 0 if vi , v j ∉ A . If the adjacency matrix is showing a directed network, then there is the probability that the matrix is asymmetric, i.e. bij ≠ b ji . Here, we regard an infrastructure system as an undirected network because any two components influence each other. If the adjacency matrix is representing an undirected graph, then bij = b ji . In this case, the matrix which reflects the infrastructure is symmetric. Furthermore, the exchange data of information and energy between component nodes is not so easy to obtain and it is difficult to unify the measurement standard, an infrastructure system is extracted to be the unweighted network which consist of nodes and edges. Here, the road transport infrastructure network is taken as an example to test our model and method. We make ensure the components of the road transport infrastructure system. The area components are the foundation nodes. Adopting the administrative boundary as the network boundary, we supply some component nodes outside the road transport infrastructure system who exchange resources with components within the system as the supplement component nodes. Then the exchange relations of information and energy among components
© 2013 ACADEMY PUBLISHER
Figure 2. Virtual infrastructure network representation
B. Results The dijkstra algorithm of shortest path and the Ucinet 6.0 are adopted in the paper. Various nodes have different impact on infrastructure networks. By the simulations, we see that the failure of a single node 4 to node 14 causes cascading failures of the network. To show the effectiveness and practical applicability of our model, the method under no considering cascading failures is also used to evaluate node importance as a comparison. The specific process and evaluation results are shown in Table 1. We use the network efficiency before and after cascading failures triggered by nodes to evaluate node importance of the infrastructure network according to Eq. (8). From Table 1, we see that Rk = 1 for node13. The result implies that node 13 is the most important node. The failure of node 13 will cause the collapse of the entire infrastructure network. Node 10 and node 7, 9 rank 2th, 3th respectively because the values of Rk for the three nodes are 0.786 and 0.733 respectively. Their failures cause the failures of five nodes. The results indicate that they are also key component nodes. We should provide more protection with these components. However, the failures of nodes 1, 2, 3, 11 and 15 cannot lead to cascading failures. Therefore, their importance is relatively weak. From Table 1, we also see that the assessment considering cascade failure has a significant impact on the assessment results of node importance in the infrastructure network. Under the condition of not considering cascading failure, the nodes 8, 12, 10 are key
1452
JOURNAL OF NETWORKS, VOL. 8, NO. 6, JUNE 2013
of Q . However, in the actual situation, the tolerate parameter β is usually less than 0.3 due to cost constraints. Therefore, we can say that the damage degree triggered by removing the biggest load node is more serious than the highest degree node. 1 0.9 0.8 0.7 0.6
Q
nodes. Additionally, the importance of node 4 and node 5 is not obvious. However, by our method under considering cascading failure, we find that the failures triggered by the node 4, 5 lead to the cascading failures of other nodes, so that the overall network performance decreases sharply. It implies that these nodes are the potential key nodes. Especially for node 4, even if its degree and initial load are very low as is shown in Figure 2, the failure of node 4 leads to the failure of node 12 due to the overload of its capacity first. And then other nodes further failure due to the cascading phenomenon. It is shown that our evaluation method on node importance is more effective than other method because of the identification on the key nodes.
0.5 0.4 0.3 0.2
TABLE I. RESULTS OF EVALUATION OF NODE IMPORTANCE ENDER CONSIDERING CASCADING FAILURES
0.1 0 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Cascading process None None None 4-12-6 5-10-6 6-10-16 7-1-9,12,8 8-11-2 9-1-8,3,15 10-8,5,6,16 None 12-3-9 13-127,16,3, 5,9,8 14-2,11 None None
ek
Rank
Weight
0.195 0.195 0.195 0.086 0.086 0.179 0.061 0.080 0.061 0.049 0.196 0.068
0.125 0.125 0.125 0.621 0.621 0.196 0.735 0.649 0.735 0.788 0.117 0.706
12 12 12 7 7 10 3 6 3 2 15 5
0.015 0.015 0.015 0.089 0.089 0.028 0.101 0.092 0.101 0.112 0.016 0.099
0
1
1
0.140
0.134 0.205 0.193
0.401 0.077 0.133
9 16 11
0.056 0.010 0.016
TABLE II. RESULTS OF EVALUATION OF NODE IMPORTANCE ENDER NOT CONSIDERING CASCADING FAILURES
Other method under not considering cascading failures Node Rank Weight Node Rank Weight 1 11 0.027 9 12 0.022 2 8 0.058 10 3 0.103 3 6 0.087 11 6 0.089 4 15 0.003 12 2 0.107 5 14 0.002 13 5 0.093 6 13 0.011 14 4 0.097 7 14 0.002 15 8 0.058 8 1 0.138 16 8 0.058
C. Result of Damage Degree of Infrastructure Network The largest connected component is used to measure the damage degree of networks. In this paper, we adopt two removals: the removal of the node with the highest degree, and the removal of the node with the biggest load. By simulations, we obtain the relation between the relative size of the largest connected component of the remaining network and the tolerate parameter β . As shown in Figure 3 and Figure 4, the removal of the biggest load node triggers more serious damage than the removal of the highest degree node according to the value
© 2013 ACADEMY PUBLISHER
0.1
0.2
0.3
0.4
0.5 β
0.6
0.7
0.8
0.9
1
Figure 3. The relation between the relative size Q and β by removing the highest degree node under our model 1 0.9 0.8 0.7 0.6
Q
Node
Our method Rk
0.5 0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5 β
0.6
0.7
0.8
0.9
1
Figure 4. The relation between the relative size Q and β by removing the highest load node under our model
From the above results, the removal of the biggest load node has a big impact on the network than the removal of the highest degree node. Therefore, we could illustrate the vulnerability of infrastructure network subjected to intentional attacks according to the node loads in the future. The largest load node-based removal is adopted to investigate the cascading phenomenon of the infrastructure network. From Figure 3 and Figure 4, it is also shown that the tolerance parameter β influence the performance of the network significantly. The infrastructure network is more resilient with the increase of the tolerate parameter. We know that the parameter is limited by the network cost. Therefore, the simulation results is consistent with the actual situation well. Considering that few researchers discuss the role of different nodes in infrastructure networks under cascading failures, the following aim of this article is to compare the damage degree results against cascading failures under different attacks. We adopt two attacks according to the node importance in the proposed model. First, we attack on the nodes which have the highest node importance. The attack strategy is to select the nodes in the descending order of the node importance of all nodes in the infrastructure network. And then the nodes are attacked one by one starting from the node with the highest node importance. In the following, the nodes
JOURNAL OF NETWORKS, VOL. 8, NO. 6, JUNE 2013
1453
network increases sharply with the decrease of the tolerance parameter β in two cases, respectively. This simulation results are also consistent with the actual situation. Comparing Figures 5, 6 and Figures 7, 8, we see that the damage degree under considering is bigger than the result under not considering cascading. Actually, the cascading failures spread across the entire network like a plague. In this way, the damages which influence our life severely often happen. The simulation results show the validity and practicality of the proposed model and method. 1 0.9 0.8 0.7 0.6 H
which are with the lowest node importance are attacked comparing the attack strategy with the highest node importance with this attack strategy, rarely used to the infrastructure networks. We select the nodes in the ascending order of node importance in the infrastructure network and then we remove these nodes one by one starting from the nodes with the lowest node importance. Figure 5 illustrates the size of the damage degree after cascading failures of all attacked nodes in the descending order of the node importance, as a function of the tolerance parameter β , for the infrastructure network. From the simulation results, the attack on the nodes with the highest node importance may be easier to induce to a cascade of overload failures, and even lead to the collapse of the network than the attack on the nodes with the lowest node importance. Therefore, in this case the node with the highest node importance plays more important in infrastructure network safety than the one with the lowest node importance. It is original expected that the presence of a few nodes with higher node importance has a disturbing side effect, as shown in Figure 5 and Figure6.
0.4 0.3 0.2 0.1 0 0
1 0.9
0.7 0.6 0.5
1
0.4
0.9
0.2
0.3
0.4
0.5 β
0.6
0.7
0.8
0.9
1
0.8
0.3
0.7
0.2
0.6 H
0.1 0 0
0.1
Figure.7 The relation between the damage degree H and β by removing the highest node importance under not considering cascading failures
0.8
H
0.5
0.1
0.2
0.3
0.4
0.5 β
0.6
0.7
0.8
0.9
1
Figure.5 The relation between the damage degree H and β by removing the highest node importance under our method
0.5 0.4 0.3 0.2 0.1
1
0 0
0.9
0.2
0.3
0.4
0.5 β
0.6
0.7
0.8
0.9
1
Figure.8 The relation between the damage degree H and β by removing the lowest node importance under not considering cascading failures
0.8 0.7 0.6 H
0.1
0.5
IV. CONCLUSION
0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5 β
0.6
0.7
0.8
0.9
1
Figure.6 The relation between the damage degree H and β by removing the lowest node importance under our method
The damage degree of two attacks on nodes according to the node importance is explored with the higher or lower importance for the infrastructure network damage degree through different methods. We furthermore try to explain this cascading phenomenon by adopting comparison our method with the method under not considering cascading failures. Figure 7 and Figure 8 illustrate the results through attacking the nodes with the highest node importance and the nodes with the lowest node importance. We find that the damage degree of the © 2013 ACADEMY PUBLISHER
There are more frequent cascading phenomenon due to the susceptibility of different systems to increased vulnerability and decreased local interconnections. Service malfunction even from simple disruptions could be extremely serious. The networked systems are more complex due to the interactions among network members working at various information-to-capacity regimens. And these large-scale cascading failures are more common. Meanwhile, the reduced safety and increased interdependencies trigger the performance evaluation of complex systems less tractable. Therefore, this dynamic behavior has been focused on many practical test systems and distribution and transmission systems such as power systems, supply chain systems, society systems [22][23]. This paper studies the cascading failure phenomenon of infrastructure networks. A new cascading model is presented to analyze the cascading failure characteristics of infrastructure networks. Under the model, we
1454
JOURNAL OF NETWORKS, VOL. 8, NO. 6, JUNE 2013
introduce an effective evaluation method of node importance to identify the key components within infrastructure networks. As a result, the critical nodes which may lead to the serious damage of the entire network due to cascading failures are identified effectively. Furthermore, this paper investigates the damage degree of infrastructure networks subjected to different removal strategies. In the simulations, we introduce the local load redistribution rule to redistribute the load from the attacked node to its neighbor nodes. The damage degree of infrastructure networks against cascading failures is measured by the scale of largest connected component. Finally, the road transport infrastructure network is taken as an example to test the validity and practicality of our model. The final example analysis also shows that the evaluation results considering cascade failure in infrastructure networks have important impacts to the identification of key nodes. Through our results, the key nodes can be protected in advance, and then the robustness and security of networks will be enhanced. Our findings have an important implication because it can provide guidance and good reference in the protection of some key nodes selected effectively, as well as the improvement of the security in infrastructure systems. Thus, we can avoid cascading-failure-induced disasters and decrease vulnerability according to the different cases of real-life complex systems. Future work is to advance the area of infrastructure system risk and enhance the resilience including the following aspects: the improvement of performance under natural and intentional attacks, recovery processes on the susceptible systems including cascading failure phenomenon, the progress of analytical methods to construct a resilient infrastructure network, the optimization based on system safety so as to control cascading failures beyond the path evaluations, and the measurement on the resilience of infrastructure systems triggered by different cascading failures . REFERENCES [1] M. E. J. Newman, “The structure and function of complex networks,” Siam Review, vol. 45, pp. 167-256, 2003. [2] H. Jeong, B. Tombor and R. Albert, “The large-scale organization of metabolic networks,” Nature, vol. 407, pp. 651-654, 2000. [3] H. Jeong, S. P. Mason, A. L. Barabasi and Z. N. Oltvai, “Lethality and centrality in protein networks,” Nature, vol. 411, pp. 41-42, 2001. [4] X. F. Zhang and N. Zhao, “The Model and Simulation of the Invulnerability of Scale-free Networks Based on Honeypot,” Journal of Networks, vol. 6(6), pp. 928-931, 2011. [5] Q. Wei, J. S. He and X. Zhang, “Privacy Enhanced Key Agreement for Public Safety Communication in Wireless Mesh Networks,” Journal of Networks, vol. 6(9), pp. 13511358, 2011. [6] S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, and S. Havlin, “Catastrophic cascade of failures in interdependent networks,” Nature, vol. 464, pp. 1025, 2010.
© 2013 ACADEMY PUBLISHER
[7] X. X. Liu, J. S. Huang and Z. J. Chen, “The Research of Ranging with Timing over Packet Network for the Mine Safety Application,” Journal of Networks, vol. 7(7), pp. 1054-1062, 2012. [8] J. Ash and D. Newth, “Optimizing complex networks for resilience against cascading failure,” Physica A, vol. 380, pp. 673- 683, 2007. [9] Y. Y. Haimes, B. M. Horowitz, J. H. Lambert, etc., “Inoperability input-output model for interdependent infrastructure sectors,” Journal of Infrastructure Systems, vol. 11(2), pp. 67-79, 2005. [10] T. Adachi and B. Ellingwood, “Serviceability of earthquake-damaged systems: effects of electrical power availability and back-up systems on system vulnerability,” Reliability Engineering and System Safety, vol. 93(1), pp. 78-88, 2008. [11] L. Zhao, K. Park and Y. C. Lai, “Attack vulnerability of scale-free networks due to cascading breakdown,” Phys Rev E, vol. 70, pp. 035101. 1-4, 2004. [12] D.-O. Leonardo, and M.V. Srivishnu, 2009. “Cascading failures in complex infrastructure systems,” Structural Safety, vol. 31(2), pp. 157-167, 2009. [13] Y. Sheffi and J. Adibi, “Discovering important nodesthrough graph entropy: The case of biology database,” Transportation Research. Part E: Logistics and Transportation Review, vol. 43(6), pp. 737-754, 2007. [14] H. Pearson, “Protecting critical assets: The interdiction median problem with fortification,” Physica Review E, vol. 37(19), pp. 352-367, 2004. [15] R. A. Davidson, H. Liu, I. Sarpong, P. Sparks and D. V. Rosowsky, “Electric power distribution system performance in Carolina hurricanes,” Nat. Hazard. Rev., vol. 4 (1), pp. 36-45, 2003. [16] H. Liu, R. A. Davidson, D. V. Rosowsky and J. R. Stedinger, “Negative binomial regression of electric power outages in hurricanes,” J. Infrastruct. Syst, vol. 11 (4), pp. 258-267, 2005. [17] P. S. Koutsourelakis, “Assessing structural vulnerability against earthquakes using multi-dimensional fragility surfaces: a Bayesian framework,” Probab. Eng. Mech, vol. 25, pp. 49-60, 2010. [18] R. Billinton, W. Li, Reliability assessment of electric power systems using Monte Carlo methods. New York: Springer, 2006. [19] S. Lee, “Probabilistic Reliability Assessment for transmission planning and operation including cascading outages,” IEEE PES Power Systems Conference and Exposition, 2009. [20] Dobson, I., et al., 2007. “Complex systems analysis of series of blackouts: cascading failure, critical points, and self-organization,” Chaos, vol. 17(2), pp. 026103.1-13, 2007. [21] A. A. Moreira, J. S. Andrade, H. J. Herrmann, and J. O. Indekeu, “How to make a fragile network robust and vice versa,” Physical Review Letters, vol. 102, pp. 018701, 2009. [22] R. Cohen and S. Havlin, Complex networks: structure, robustness and function, Cambridge University Press, Cambridge, 2010. [23] J. Lorenza, S. Battiston and F. Schweitzer, “Systemic risk in a unifying framework for cascading processes on networks,” European Physical Journal B, vol. 71, pp. 441460, 2009.
JOURNAL OF NETWORKS, VOL. 8, NO. 6, JUNE 2013
Cascading Model of Infrastructure Networks based on Complex Network Shaolin Wang Institute of Higher Education Research, Jilin Business and Technology College, Changchun, China, E-mail: [email protected]
Abstract—The paper focuses on the cascading failure phenomenon of infrastructure networks. A new cascading model is proposed to analyze the cascading failure characteristics of the systems. Based on the model, we present a good evaluation method of node importance to identify the key nodes which may cause cascading failures of infrastructure networks. Finally, the damage degree of infrastructure networks considering cascading failures is investigated through our cascading model. The simulation results indicate the validity and practicality of the proposed model. The research is helpful to protect the key components of infrastructure networks and decrease the network vulnerability. Meanwhile, the proposed method provides valuable reference to many actual networks in nature. Index Terms—cascading failure, infrastructure system, node importance, network damage degree, complex network theory
I. INTRODUCTION Many complex systems in real life can be described by various networks [1][2][3]. The people have been more and more aware of the importance of safety and robustness of networks [4][5][6][7]. In a network, a node or an edge will collapse due to the overload. And then the load at a neighboring node will change. If the load of the neighboring node increases and exceeds its corresponding capacity, the node is prone to malfunction. As a result, a new redistribution of loads occurs, and thus may lead to the failures of a new round of nodes. This kind of stepby-step process is so-called the cascading failure. The cascading failures often spread across the entire network like a plague, resulting in serious damage. Therefore, the cascading failure phenomenon on all kinds of networks has become the research focus [8]. Infrastructure systems are becoming more interdependent to other systems. The malfunctions within a given infrastructure system are more likely to impact the effective operation of other systems [9][10]. The increased interdependencies and reduced robustness margins induce the performance evaluation of complex systems less tractable. The networked systems are more complex due to the interactions among network members working at various information-to-capacity regimens. Therefore, it is difficult to anticipate emergent incidents from the information of single component nodes since there are so many interacting members. The systems are more and more vulnerable just because of these complex relations. Meanwhile, the cascading failure phenomenon © 2013 ACADEMY PUBLISHER doi:10.4304/jnw.8.6.1448-1454
of complex networks has been the focus of many studies, for the reason that cascading failures are common in reallife systems and especially can cause huge damages to our life and society development [11][12]. The infrastructure networks are also the very typical networks including cascading failure phenomenon. Even though the failures emerge very locally in an infrastructure network, the entire network can be largely affected, even resulting in global collapse. For instance, the earthquake in Taiwan resulted in a significant loss to many electronic enterprises because their suppliers are Taiwan enterprises in 1999. The shortage of 4600 million influenza vaccine happened in the United State because the only supplier of Chiron company was sued for bacterial contamination problems [13][14]. The blackout in North American was initiated by an untrimmed tree too close to high voltage transmission lines in the Midwest in 2003. From these serious results, we can see that the malfunction of only a component in an infrastructure system may trigger the serious results of the whole system due to cascading failure phenomenon. Recently, the research in infrastructure systems has attracted much attention. For a single infrastructure, many papers investigate the safety and vulnerability of a given system, as well as the impact of intentional attack and random failure on the system. The probabilistic risk analysis methods are presented to investigate the risk of infrastructure systems when there is little information of an infrastructure system [12][15]. Then some scholars analyze the impact of infrastructure performance triggered by natural disasters by using statistical learning theory [16]. However, the two methods have not considered the topology structure of infrastructure systems. Then some papers study the measurement of infrastructure vulnerability against disasters [17][18]. The relation between the network topology and the safety of a power system under natural disasters was explored by combining the power network topology with a component model [20][21]. However, all studies for risk and vulnerability analysis focus on only the single noninteracting infrastructure system. Since infrastructure systems have become interconnected with each other, a risk and safety research should not be investigated in isolation. Based on the background of infrastructure systems, this paper studies the cascading failure phenomenon in infrastructure networks. Different from the previous studies, we build a cascading model according to the
JOURNAL OF NETWORKS, VOL. 8, NO. 6, JUNE 2013
1449
topology structure and features of infrastructure systems first. Then an evaluation method of node importance is presented to identify the key nodes of infrastructure networks. The largest connected component is introduced to measure the damage degree of infrastructure networks after cascading failures. The simulation results indicate the validity and practicality of our proposed model. It is known that various nodes have various impacts on infrastructure systems. Especially, the failures of key nodes will be easier to cause the serious damage of infrastructure systems. If we identify the key nodes and provide protection with them in advance, the safety of infrastructure networks will be improved well. Our research therefore provides theoretical basis and valuable reference to actual networks in nature. II. PROPOSED CASCADING MODEL A. Dynamic Network Model Our society depends heavily on many complex systems. The safety and robustness of complex systems have been an important issue [19][20]. Therefore, the model and method to explore the cascading phenomenon of infrastructure networks become necessary. In this article, a graph (network) G is defined by a non-empty set of nodes A and a non-empty set of . A {v1 , v2 , ⋅ ⋅ ⋅, vN } is the set of edges K : G = ( A, K )=
K {l1 , l2 , ⋅ ⋅ ⋅, lM } is the set of edges. The total nodes and= number of nodes in the set A of the network is represented by N, while the total number of edges in the set K of the network is M . Here we consider the cascading failures triggered by removing a node of infrastructure network. If the initial load of the node is small, the removal cannot induce the disastrous results, thus the network can recover from the load redistribution. However, if the load of the removed node is large, the local redistribution will trigger cascading failures across the system and even lead to serious consequences, eventually result in the entire network collapsing. Distribution of load on the networks is determined by several factors. The network topology structure is one of the main factors. In order to different from the actual physical load, we regard that the load of a node is completely determined by the topology structure of the node within networks. In this article, the structure load of a node is defined as its betweenness. Because the actual physical load within networks is difficult to be determined, it is reasonable that this dimensionless structural load is used to study the vulnerability and node importance assessment. Therefore, the shortest path betweenness is adopted in this paper. That is to say, the more the shortest path through a node is, the load on the node is higher. We define that the load of node i is as following. δ vv′ (i ) Li C= = B (i )
© 2013 ACADEMY PUBLISHER
∑
v ≠ v ′∈V
δ vv′
n(n − 1)
(1)
where δ vv′ is the number of all the shortest paths between node v and node v′ , δ vv′ (i ) is the number of the shortest paths through node i between node v and node v′ . In order to avoid for the load of a node being zero, node v or node v′ is allowed to become node i in the Eq. (1). The detailed situation is as follows. δ vv′ (i )
∑
v ≠ v ′∈V
δ vv′
δ vv′ (i ) δ (i ) δ (i ) + ∑ vv′ + ∑ vv′ δ δ v =i ≠ v′ v ≠ i =v′ v ≠ i ≠ v ′ δ vv ′ vv ′ vv ′ δ (i ) = ∑ vv′ + 2(n − 1) v ≠ i ≠ v ′ δ vv ′ =
∑
(2)
From Eq.(2), 2 / n ≤ Li ≤ 1 . In a star network, the shortest paths between any two node pairs must pass through the center node, so the load of the central node is a maximum of 1. The node capacity refers to the maximum load that the node can handle. In infrastructure networks, the capacity of a node is limited by its cost. In our model, we assume the capacity Ci of node i to be proportional to its initial load Li (0) , i.e.,
C= (1 + β ) Li (0) i = (1 + β )
δ vv0 ′ (i ) ∑ 0 v ≠ v ′∈V δ vv ′ n(n − 1)
(3)
where the parameter β is the tolerance parameter which can characterize the resistance to the natural disasters. According to the actual situation, Li (0) is calculated by Eq. (1) and the capacity Ci of node i is calculated by Eq. (3) in the simulations. Because of the limitation of the capacity, when the node i malfunctions, the load of node i will be redistributed to its neighbor nodes. In this way, the additional load will be achieved by the neighbor node j of node i , i.e. (4) L j (t + = 1) L j (t ) + ∆Lij If L j (t + 1) ≤ C j , then the whole network can restore balance. If L j (t + 1) > C j , the node j will also collapse, which will lead to further redistributing the load Li of node j . This may result in the other breakdowns, therefore the cascading failures occur. The visualization of the local load redistribution rule is illustrated in Figure 1. The removal probability is adopted as following: L j (t + 1) ≤ C j , 0 pj = (5) L j (t + 1) ≥ C j , 1 where p j denotes the removal probability of node j .
1450
JOURNAL OF NETWORKS, VOL. 8, NO. 6, JUNE 2013
process is over, according to Eq. (6), we will calculate the value of ek , which denotes the network efficiency induced by removing k. In this case, N k′ is the numbers of the nodes which can maintain the normal operations after cascading failures in the whole complex network and dij is shortest path of any remaining node pair. In
i j
Figure 1. The visualization of load redistribution after the breakdown of node i.
B. Evaluation Method Network efficiency is used to describe the capacity on information transfer in networks. By investigating the network efficiency, we can try to improve the network structure so as to optimize the network efficiency. And then the resilience of infrastructure networks against cascading failures will be improved. Assume that the connected efficiency eij between node i and node j is inversely proportional to the shortest path dij , i.e.
eij =
1 dij
(6)
From Eq. (6), if there is no path that connects node node i and node j, dij = +∞ , thus eij = 0 . If no connection between node i and node j , the value of eij tends to 0. The network efficiency is defined by the connected efficiency eij . The detailed formula is as follows:
e(G ) =
=
∑e
i ≠ j∈G
ij
N ( N − 1) 1 ∑ i ≠ j∈G d ij
(7) N ( N − 1) where e(G ) denotes the network efficiency, and N is the number of nodes in network. From Eq. (7), e(G ) is normalized considering the case in which a network with the same number of nodes is fully connected. Thus, e(G ) is normalized by N ( N − 1) / 2 edges, case in which information communication is most efficient. Therefore, 0 ≤ e(G ) ≤ 1 , and e(G ) = 1 just when the network is fully connected. When an infrastructure network is on normal operation initially, the network efficiency is denoted by e0 which is calculated according to Eq. (7). In this case, N is the number of nodes during normal operation. Let dij be the shortest path between two nodes. After the cascading
© 2013 ACADEMY PUBLISHER
order to obtain the values of N k′ and ek , the above cascading model will be adopted to calculate the values. In the simulations, we introduce two indicators which are the node load and node capability respectively. In the following, we define the node importance by adopting the network efficiency before and after cascading failures. The importance value of node k is denoted by Rk . According to Eq. (7), the detailed expression of Rk is as follows.
Rk = 1 −
ek e0
1 ) N k′ ( N k′ − 1) i ≠ j∈Gk′ d ij = 1− 1 (∑ ) N ( N − 1) i ≠ j∈G d ij (
∑
(8)
where ek and e0 are the network efficiency before and after cascading failures triggered by node k in the whole network respectively, and Gk′ is the network after cascading failures. The weight of each node is obtained by the standardization for Rk in this paper. The algorithm steps are given below: Step 1: The initial load Lk (0) and the load capacity
Ck for all nodes are calculated according to Eq. (1) and Eq. (3). Step 2: The values of network efficiency for all nodes in the infrastructure network are calculated according to Eq. (7). Step 3: For k = 1 to n , the node importance Rk of all nodes are evaluated. Step 4: The node k is removed. The new nodes in a failure state are checked. Step 5: The load of all nodes is calculated again according to Eq. (4) after cascading failures. Step 6: Calculate Rk on the basis of Eq. (8) after cascading failures. C. Damage Degree In order to reflect the phenomenon associated to the cascading damage process, we need to give a quantitative measurement of network damage degree. There is always the largest connected component in networks which contains more nodes than other connected component in networks and there are pathways between any two nodes within this subgraph. The relative size of the largest connected component of the remaining network after cascading failures under our model can be expressed as:
JOURNAL OF NETWORKS, VOL. 8, NO. 6, JUNE 2013
Q=
N′ N
1451
(9)
where N is the initial size of the network before cascading failures and N ′ is the size of the largest connected component after cascading failures. Therefore, the simplest quantitative measurement of damage degree is given as follows: N′ H = 1− (10) N Obviously, if H = 1 , the network has been broken into many small disconnected parts and eventually resulted in the entire network collapsing. If H = 0 , the network works well. Obviously, H is the residual rate of the largest connected component after cascading failures, which can characterize the damage degree of infrastructure networks.
are investigated the between members. When there is the exchange of matter or energy between the two component nodes existing in the road transport infrastructure system, an edge between them is determined. In order to obtain the corresponding adjacency matrices of this road transport infrastructure system, we collect data and investigate the relations among component nodes in the road transport infrastructure system. Then the adjacency matrix is obtained. The network topology of this road transport infrastructure system is shown in Figure 2 under UCINET 6.0, which consists of 16 nodes.
III. NUMERIAL RESULTS A. Illustration of Infrastructure Networks Infrastructures networks can be represented effectively as a matrix. We define the matrix as adjacency matrix which characterizes an infrastructure network. The adjacency matrix is a square matrix. In a square matrix, values are 1 if there exists a information communication between the components to which the interception of row and column refers, and the values are 0 if the two components are not related within an infrastructure network. In the following, we formalize the adjacency matrix in infrastructure systems: the adjacency matrix Bij = {bij } is a N × N matrix which is defined such as
bij = 1 if vi , v j ∈ A and bij = 0 if vi , v j ∉ A . If the adjacency matrix is showing a directed network, then there is the probability that the matrix is asymmetric, i.e. bij ≠ b ji . Here, we regard an infrastructure system as an undirected network because any two components influence each other. If the adjacency matrix is representing an undirected graph, then bij = b ji . In this case, the matrix which reflects the infrastructure is symmetric. Furthermore, the exchange data of information and energy between component nodes is not so easy to obtain and it is difficult to unify the measurement standard, an infrastructure system is extracted to be the unweighted network which consist of nodes and edges. Here, the road transport infrastructure network is taken as an example to test our model and method. We make ensure the components of the road transport infrastructure system. The area components are the foundation nodes. Adopting the administrative boundary as the network boundary, we supply some component nodes outside the road transport infrastructure system who exchange resources with components within the system as the supplement component nodes. Then the exchange relations of information and energy among components
© 2013 ACADEMY PUBLISHER
Figure 2. Virtual infrastructure network representation
B. Results The dijkstra algorithm of shortest path and the Ucinet 6.0 are adopted in the paper. Various nodes have different impact on infrastructure networks. By the simulations, we see that the failure of a single node 4 to node 14 causes cascading failures of the network. To show the effectiveness and practical applicability of our model, the method under no considering cascading failures is also used to evaluate node importance as a comparison. The specific process and evaluation results are shown in Table 1. We use the network efficiency before and after cascading failures triggered by nodes to evaluate node importance of the infrastructure network according to Eq. (8). From Table 1, we see that Rk = 1 for node13. The result implies that node 13 is the most important node. The failure of node 13 will cause the collapse of the entire infrastructure network. Node 10 and node 7, 9 rank 2th, 3th respectively because the values of Rk for the three nodes are 0.786 and 0.733 respectively. Their failures cause the failures of five nodes. The results indicate that they are also key component nodes. We should provide more protection with these components. However, the failures of nodes 1, 2, 3, 11 and 15 cannot lead to cascading failures. Therefore, their importance is relatively weak. From Table 1, we also see that the assessment considering cascade failure has a significant impact on the assessment results of node importance in the infrastructure network. Under the condition of not considering cascading failure, the nodes 8, 12, 10 are key
1452
JOURNAL OF NETWORKS, VOL. 8, NO. 6, JUNE 2013
of Q . However, in the actual situation, the tolerate parameter β is usually less than 0.3 due to cost constraints. Therefore, we can say that the damage degree triggered by removing the biggest load node is more serious than the highest degree node. 1 0.9 0.8 0.7 0.6
Q
nodes. Additionally, the importance of node 4 and node 5 is not obvious. However, by our method under considering cascading failure, we find that the failures triggered by the node 4, 5 lead to the cascading failures of other nodes, so that the overall network performance decreases sharply. It implies that these nodes are the potential key nodes. Especially for node 4, even if its degree and initial load are very low as is shown in Figure 2, the failure of node 4 leads to the failure of node 12 due to the overload of its capacity first. And then other nodes further failure due to the cascading phenomenon. It is shown that our evaluation method on node importance is more effective than other method because of the identification on the key nodes.
0.5 0.4 0.3 0.2
TABLE I. RESULTS OF EVALUATION OF NODE IMPORTANCE ENDER CONSIDERING CASCADING FAILURES
0.1 0 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Cascading process None None None 4-12-6 5-10-6 6-10-16 7-1-9,12,8 8-11-2 9-1-8,3,15 10-8,5,6,16 None 12-3-9 13-127,16,3, 5,9,8 14-2,11 None None
ek
Rank
Weight
0.195 0.195 0.195 0.086 0.086 0.179 0.061 0.080 0.061 0.049 0.196 0.068
0.125 0.125 0.125 0.621 0.621 0.196 0.735 0.649 0.735 0.788 0.117 0.706
12 12 12 7 7 10 3 6 3 2 15 5
0.015 0.015 0.015 0.089 0.089 0.028 0.101 0.092 0.101 0.112 0.016 0.099
0
1
1
0.140
0.134 0.205 0.193
0.401 0.077 0.133
9 16 11
0.056 0.010 0.016
TABLE II. RESULTS OF EVALUATION OF NODE IMPORTANCE ENDER NOT CONSIDERING CASCADING FAILURES
Other method under not considering cascading failures Node Rank Weight Node Rank Weight 1 11 0.027 9 12 0.022 2 8 0.058 10 3 0.103 3 6 0.087 11 6 0.089 4 15 0.003 12 2 0.107 5 14 0.002 13 5 0.093 6 13 0.011 14 4 0.097 7 14 0.002 15 8 0.058 8 1 0.138 16 8 0.058
C. Result of Damage Degree of Infrastructure Network The largest connected component is used to measure the damage degree of networks. In this paper, we adopt two removals: the removal of the node with the highest degree, and the removal of the node with the biggest load. By simulations, we obtain the relation between the relative size of the largest connected component of the remaining network and the tolerate parameter β . As shown in Figure 3 and Figure 4, the removal of the biggest load node triggers more serious damage than the removal of the highest degree node according to the value
© 2013 ACADEMY PUBLISHER
0.1
0.2
0.3
0.4
0.5 β
0.6
0.7
0.8
0.9
1
Figure 3. The relation between the relative size Q and β by removing the highest degree node under our model 1 0.9 0.8 0.7 0.6
Q
Node
Our method Rk
0.5 0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5 β
0.6
0.7
0.8
0.9
1
Figure 4. The relation between the relative size Q and β by removing the highest load node under our model
From the above results, the removal of the biggest load node has a big impact on the network than the removal of the highest degree node. Therefore, we could illustrate the vulnerability of infrastructure network subjected to intentional attacks according to the node loads in the future. The largest load node-based removal is adopted to investigate the cascading phenomenon of the infrastructure network. From Figure 3 and Figure 4, it is also shown that the tolerance parameter β influence the performance of the network significantly. The infrastructure network is more resilient with the increase of the tolerate parameter. We know that the parameter is limited by the network cost. Therefore, the simulation results is consistent with the actual situation well. Considering that few researchers discuss the role of different nodes in infrastructure networks under cascading failures, the following aim of this article is to compare the damage degree results against cascading failures under different attacks. We adopt two attacks according to the node importance in the proposed model. First, we attack on the nodes which have the highest node importance. The attack strategy is to select the nodes in the descending order of the node importance of all nodes in the infrastructure network. And then the nodes are attacked one by one starting from the node with the highest node importance. In the following, the nodes
JOURNAL OF NETWORKS, VOL. 8, NO. 6, JUNE 2013
1453
network increases sharply with the decrease of the tolerance parameter β in two cases, respectively. This simulation results are also consistent with the actual situation. Comparing Figures 5, 6 and Figures 7, 8, we see that the damage degree under considering is bigger than the result under not considering cascading. Actually, the cascading failures spread across the entire network like a plague. In this way, the damages which influence our life severely often happen. The simulation results show the validity and practicality of the proposed model and method. 1 0.9 0.8 0.7 0.6 H
which are with the lowest node importance are attacked comparing the attack strategy with the highest node importance with this attack strategy, rarely used to the infrastructure networks. We select the nodes in the ascending order of node importance in the infrastructure network and then we remove these nodes one by one starting from the nodes with the lowest node importance. Figure 5 illustrates the size of the damage degree after cascading failures of all attacked nodes in the descending order of the node importance, as a function of the tolerance parameter β , for the infrastructure network. From the simulation results, the attack on the nodes with the highest node importance may be easier to induce to a cascade of overload failures, and even lead to the collapse of the network than the attack on the nodes with the lowest node importance. Therefore, in this case the node with the highest node importance plays more important in infrastructure network safety than the one with the lowest node importance. It is original expected that the presence of a few nodes with higher node importance has a disturbing side effect, as shown in Figure 5 and Figure6.
0.4 0.3 0.2 0.1 0 0
1 0.9
0.7 0.6 0.5
1
0.4
0.9
0.2
0.3
0.4
0.5 β
0.6
0.7
0.8
0.9
1
0.8
0.3
0.7
0.2
0.6 H
0.1 0 0
0.1
Figure.7 The relation between the damage degree H and β by removing the highest node importance under not considering cascading failures
0.8
H
0.5
0.1
0.2
0.3
0.4
0.5 β
0.6
0.7
0.8
0.9
1
Figure.5 The relation between the damage degree H and β by removing the highest node importance under our method
0.5 0.4 0.3 0.2 0.1
1
0 0
0.9
0.2
0.3
0.4
0.5 β
0.6
0.7
0.8
0.9
1
Figure.8 The relation between the damage degree H and β by removing the lowest node importance under not considering cascading failures
0.8 0.7 0.6 H
0.1
0.5
IV. CONCLUSION
0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5 β
0.6
0.7
0.8
0.9
1
Figure.6 The relation between the damage degree H and β by removing the lowest node importance under our method
The damage degree of two attacks on nodes according to the node importance is explored with the higher or lower importance for the infrastructure network damage degree through different methods. We furthermore try to explain this cascading phenomenon by adopting comparison our method with the method under not considering cascading failures. Figure 7 and Figure 8 illustrate the results through attacking the nodes with the highest node importance and the nodes with the lowest node importance. We find that the damage degree of the © 2013 ACADEMY PUBLISHER
There are more frequent cascading phenomenon due to the susceptibility of different systems to increased vulnerability and decreased local interconnections. Service malfunction even from simple disruptions could be extremely serious. The networked systems are more complex due to the interactions among network members working at various information-to-capacity regimens. And these large-scale cascading failures are more common. Meanwhile, the reduced safety and increased interdependencies trigger the performance evaluation of complex systems less tractable. Therefore, this dynamic behavior has been focused on many practical test systems and distribution and transmission systems such as power systems, supply chain systems, society systems [22][23]. This paper studies the cascading failure phenomenon of infrastructure networks. A new cascading model is presented to analyze the cascading failure characteristics of infrastructure networks. Under the model, we
1454
JOURNAL OF NETWORKS, VOL. 8, NO. 6, JUNE 2013
introduce an effective evaluation method of node importance to identify the key components within infrastructure networks. As a result, the critical nodes which may lead to the serious damage of the entire network due to cascading failures are identified effectively. Furthermore, this paper investigates the damage degree of infrastructure networks subjected to different removal strategies. In the simulations, we introduce the local load redistribution rule to redistribute the load from the attacked node to its neighbor nodes. The damage degree of infrastructure networks against cascading failures is measured by the scale of largest connected component. Finally, the road transport infrastructure network is taken as an example to test the validity and practicality of our model. The final example analysis also shows that the evaluation results considering cascade failure in infrastructure networks have important impacts to the identification of key nodes. Through our results, the key nodes can be protected in advance, and then the robustness and security of networks will be enhanced. Our findings have an important implication because it can provide guidance and good reference in the protection of some key nodes selected effectively, as well as the improvement of the security in infrastructure systems. Thus, we can avoid cascading-failure-induced disasters and decrease vulnerability according to the different cases of real-life complex systems. Future work is to advance the area of infrastructure system risk and enhance the resilience including the following aspects: the improvement of performance under natural and intentional attacks, recovery processes on the susceptible systems including cascading failure phenomenon, the progress of analytical methods to construct a resilient infrastructure network, the optimization based on system safety so as to control cascading failures beyond the path evaluations, and the measurement on the resilience of infrastructure systems triggered by different cascading failures . REFERENCES [1] M. E. J. Newman, “The structure and function of complex networks,” Siam Review, vol. 45, pp. 167-256, 2003. [2] H. Jeong, B. Tombor and R. Albert, “The large-scale organization of metabolic networks,” Nature, vol. 407, pp. 651-654, 2000. [3] H. Jeong, S. P. Mason, A. L. Barabasi and Z. N. Oltvai, “Lethality and centrality in protein networks,” Nature, vol. 411, pp. 41-42, 2001. [4] X. F. Zhang and N. Zhao, “The Model and Simulation of the Invulnerability of Scale-free Networks Based on Honeypot,” Journal of Networks, vol. 6(6), pp. 928-931, 2011. [5] Q. Wei, J. S. He and X. Zhang, “Privacy Enhanced Key Agreement for Public Safety Communication in Wireless Mesh Networks,” Journal of Networks, vol. 6(9), pp. 13511358, 2011. [6] S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, and S. Havlin, “Catastrophic cascade of failures in interdependent networks,” Nature, vol. 464, pp. 1025, 2010.
© 2013 ACADEMY PUBLISHER
[7] X. X. Liu, J. S. Huang and Z. J. Chen, “The Research of Ranging with Timing over Packet Network for the Mine Safety Application,” Journal of Networks, vol. 7(7), pp. 1054-1062, 2012. [8] J. Ash and D. Newth, “Optimizing complex networks for resilience against cascading failure,” Physica A, vol. 380, pp. 673- 683, 2007. [9] Y. Y. Haimes, B. M. Horowitz, J. H. Lambert, etc., “Inoperability input-output model for interdependent infrastructure sectors,” Journal of Infrastructure Systems, vol. 11(2), pp. 67-79, 2005. [10] T. Adachi and B. Ellingwood, “Serviceability of earthquake-damaged systems: effects of electrical power availability and back-up systems on system vulnerability,” Reliability Engineering and System Safety, vol. 93(1), pp. 78-88, 2008. [11] L. Zhao, K. Park and Y. C. Lai, “Attack vulnerability of scale-free networks due to cascading breakdown,” Phys Rev E, vol. 70, pp. 035101. 1-4, 2004. [12] D.-O. Leonardo, and M.V. Srivishnu, 2009. “Cascading failures in complex infrastructure systems,” Structural Safety, vol. 31(2), pp. 157-167, 2009. [13] Y. Sheffi and J. Adibi, “Discovering important nodesthrough graph entropy: The case of biology database,” Transportation Research. Part E: Logistics and Transportation Review, vol. 43(6), pp. 737-754, 2007. [14] H. Pearson, “Protecting critical assets: The interdiction median problem with fortification,” Physica Review E, vol. 37(19), pp. 352-367, 2004. [15] R. A. Davidson, H. Liu, I. Sarpong, P. Sparks and D. V. Rosowsky, “Electric power distribution system performance in Carolina hurricanes,” Nat. Hazard. Rev., vol. 4 (1), pp. 36-45, 2003. [16] H. Liu, R. A. Davidson, D. V. Rosowsky and J. R. Stedinger, “Negative binomial regression of electric power outages in hurricanes,” J. Infrastruct. Syst, vol. 11 (4), pp. 258-267, 2005. [17] P. S. Koutsourelakis, “Assessing structural vulnerability against earthquakes using multi-dimensional fragility surfaces: a Bayesian framework,” Probab. Eng. Mech, vol. 25, pp. 49-60, 2010. [18] R. Billinton, W. Li, Reliability assessment of electric power systems using Monte Carlo methods. New York: Springer, 2006. [19] S. Lee, “Probabilistic Reliability Assessment for transmission planning and operation including cascading outages,” IEEE PES Power Systems Conference and Exposition, 2009. [20] Dobson, I., et al., 2007. “Complex systems analysis of series of blackouts: cascading failure, critical points, and self-organization,” Chaos, vol. 17(2), pp. 026103.1-13, 2007. [21] A. A. Moreira, J. S. Andrade, H. J. Herrmann, and J. O. Indekeu, “How to make a fragile network robust and vice versa,” Physical Review Letters, vol. 102, pp. 018701, 2009. [22] R. Cohen and S. Havlin, Complex networks: structure, robustness and function, Cambridge University Press, Cambridge, 2010. [23] J. Lorenza, S. Battiston and F. Schweitzer, “Systemic risk in a unifying framework for cascading processes on networks,” European Physical Journal B, vol. 71, pp. 441460, 2009.
