Edge-based-attack induced cascading failures on scale-free networks

Physica A 388 (2009) 1731–1737

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Edge-based-attack induced cascading failures on scale-free networks Jian-Wei Wang ∗ , Li-Li Rong Institute of Systems Engineering, Dalian University of Technology, 2 Ling Gong Road., Dalian 116024, Liaoning, PR China

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Article history: Received 3 October 2008 Received in revised form 26 December 2008 Available online 20 January 2009 PACS: 89.75.Hc 89.75.-k 89.75.Fb Keywords: Cascading failure Scale-free network BA network Attack Breakdown probability

a b s t r a c t Most previous existing works on cascading failures only focused on attacks on nodes rather than on edges. In this paper, we discuss the response of scale-free networks subject to two different attacks on edges during cascading propagation, i.e., edge removal by either the descending or ascending order of the loads. Adopting a cascading model with a breakdown probability p of an overload edge and the initial load (ki kj )α of an edge ij, where ki and kj are the degrees of the nodes connected by the edge ij and α is a tunable parameter, we investigate the effects of two attacks for the robustness of Barabási–Albert (BA) scalefree networks against cascading failures. In the case of α < 1, our investigation by the numerical simulations leads to a counterintuitive finding that BA scale-free networks are more sensitive to attacks on the edges with the lowest loads than the ones with the highest loads, not relating to the breakdown probability. In addition, the same effect of two attacks in the case of α = 1 may be useful in furthering studies on the control and defense of cascading failures in many real-life networks. We then confirm by the theoretical analysis these results observed in simulations. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Complex networks [1] such as the Internet, the electrical power grid, and the transportation networks, are an essential part of modern society. Network robustness [1–13] subject to random or intentional attacks has been one of the most central topics in network safety. Therefore, cascading failures on complex networks have been highly concerned and widely investigated. Cascading failures refer to the subsequent failure of other parts of a network induced by the failure of or attack on only a few nodes (or edges). It can happen in many infrastructure networks. Some famous accidents, such as the largest blackout in US history took place on 14 August 2003 [14], the Western North American blackouts in July and August 1996 [15,16], and the Internet collapse caused by congestion [5,17], are believed by some researchers to be typical examples of cascading failures. Some aspects of cascading failures in complex networks have been discussed in literature, including the cascade control and defense strategy [14,18–22], the model for describing cascade phenomena [23–29], the analytical calculation of capacity parameter [26,27,30–33], and so on. However, in all studies cited above, most previous works on cascading failures only consider attacks on nodes rather than on edges. Attacks on edges are as important for the network security as those on nodes, and therefore deserve a careful investigation. Following a recent work of Wang and Chen [26], we also assume the initial load of an edge ij to be (ki kj )α with ki and kj being the degrees of the nodes connected by the edge, where α is a tunable parameter and governs the strength of the edge load. Since there exist a certain degree monitoring and control in most real-life complex networks, not all overloaded edges will be removed from networks. Therefore, we propose a new concept, i.e., the breakdown probability of an overload

∗

Corresponding author. Tel.: +86 411 81258693. E-mail address: [email protected] (J.-W. Wang).

0378-4371/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2009.01.015

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Fig. 1. The scheme illustrates the load redistribution triggered by an edge-cut-based attack and the removal mechanism with the breakdown probability P (Lim + 4Lim ) of a neighboring edge im of the breakdown edge ij.

Fig. 2. Illustration of the relation between the removal probability P (L(ij)) and the load L(ij) of an edge ij.

edge, by which we describe the removal mechanism of an overload edge. Adopting two attacks on edges, we investigate cascading reaction behaviors on the Barabási—Albert (BA) [34] modeling networks. We numerically find some interesting and counterintuitive results in the cascading model, of which one unexpected finding is that attacking the edges with the lowest loads is more harmful than attacking on the ones with the highest loads in the case of α < 1. In addition, we verify these results by theoretical analysis. The rest of this paper is organized as follows: in Section 2, we describe the cascading model in detail. The effects of two attacks for network robustness are discussed based on BA networks in Section 3. In Section 4, the numerical simulations are verified by theoretical analysis. Finally, some summaries and conclusions are shown in Section 5. 2. The model Here we focus on cascading failures triggered by the removal of a single edge. If an edge has a relatively small load, its removal will not cause major changes in the balance of loads, and subsequent overload failures are unlikely to occur. However, when the load at an edge is relatively large, its removal is likely to affect significantly loads at other edges and possibly to start a sequence of overload failures and eventually a large drop in the network performance. We assume the initial load of an edge ij to be Lij = (ki kj )α with ki and kj being the degrees of the nodes connected by the edge. The additional Lim received by edge im after the collapse of an edge ij is proportional to its initial load, P load ∆P i.e., ∆Lim = Lij Lim /( a∈Γi Lia + b∈Γj Ljb ), where Γi and Γj represent the sets of the neighboring nodes of the node i and j, respectively. Since edge capacity on real-life networks is generally limited by cost, it is natural to assume that the capacity Cij of an edge ij is proportional to its initial load for simplicity: Cij = β Lij , where the constant β(≥ 1) is a tolerance parameter. In our study, to accord with the positive proportion correlation between the initial load of an edge ij and ki kj , we set α > 0. Fig. 1 illustrates the effect of an edge-cut-based attack for its neighboring edges. In most real-life networks, owing to a certain protection strategy, an overloaded edge is not always removed. Therefore, we propose a new concept of the breakdown probability of an overload edge to reflect the removal mechanism. Taking the limit of the protection capacity into account, we assume that an edge ij has a removal threshold, i.e., the removal probability P (L(ij)) of the overload edge ij is equal to 1 when L(ij) ≥ γ Cij , where L(ij) represents the load of the edge ij and γ (≥ 1) is a tunable parameter. The expression of the breakdown probability P (L(ij)) of an edge ij reads (See Fig. 2),

P (L(ij)) =

 0,   L(ij) − C

Cij > L(ij) ij

  γ Cij − Cij 1,

,

Cij ≤ L(ij) < γ Cij

(1)

γ Cij ≤ L(ij).

In our cascading model, the damage caused by attacking a single edge ij is quantified in terms of the avalanche size Sij , namely, the number of broken edges after the cascading process is over. It is evident that 0 ≤ Si ≤ Nedge − 1, where Nedge denotes the

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Fig. 3. Percentage Sattack of breakdown edges induced by two attacks, as a function of the parameter β when α < 1. Each curve corresponds to the average over 50 triggers and ten realizations of the network.

total number of the edge in a network. To make a meaningful comparison of the effects P of two attacks for network robustness against cascading failures, we adopt the normalized avalanche size, i.e., Sattack = ij∈A Sij /(EdgeA (Nedge − 1)). Here, A and EdgeA represents the set and the number of edges attacked, respectively. The higher the value β , the stronger the network robustness against cascading failures. Therefore, there should exist a critical threshold βc , i.e., the lowest value to avoid cascading failures. When β > βc , no cascading failure occurs and the system maintains its normal and efficient functioning; while for β < βc , Sattack suddenly increases from 0 and cascading failure emerges because the capacity of each edge is limited, propagating the whole or part network to stop working. 3. Analysis of the effects of two attacks We adopt two simple attacks in our cascading model. (1) Attack on the edges with the highest loads(HL): the targeted edge is selected from those with the highest loads. The removal rule is to select the edges in the descending order of the loads and then to remove them one by one starting from the edges with the highest loads (if some edges happen to have the same loads, we randomly choose one of them). In the heterogeneous networks, i.e., scale-free networks, the removal of the edges with the highest loads is more likely to trigger cascading failures in general. (2) Attack on the edges with the lowest loads(LL): this attack, rarely used to the real-life networks, is to select the edges in the ascending order of loads and then to remove them one by one starting from the edges with the lowest loads (if some edges happen to have the same loads, we randomly choose one of them). We consider cascade failures on Barabási—Albert (BA) [34] Qnetworks. Starting from m0 nodes, one node with m links is attached at each time step in such Q a way that P the probability i of being connected to the existing node i is proportional to the degree ki of that node, i.e., i = ki / j kj , where j runs over all existing nodes. In our work, the total network size is fixed as N = 5000 and the parameters are set to be m0 = m = 2. To obtain an effective estimate of two attacks, we focus on the relationship between the critical threshold βc and some parameters of our cascading model. For each attack, we choose 50 edges as the attacked ones, and each simulation result is obtained by averaging over experiments on 10 independent networks. According to the role of the tunable parameter α in adjusting the initial load of an edge, we investigate the effects of two attacks in three cases of α < 1, α = 1, and α > 1. Fig. 3 illustrates the normalized avalanche size Sattack after cascading failures of all attacked edges, as a function of the tolerance parameter β . For each case of four different values of the parameter γ , we estimate the critical threshold βc by

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Fig. 4. Comparison between the effects of two edges with the higher (edge ij) or lower loads (edge mn) for the neighboring edges.

Fig. 5. Comparison between the efficiencies of the HL and LL attacks in the case of α = 1.

using Sattack . Since the nodes (or edges) with the highest loads may easily trigger cascading failures, it is expected that the HL should be more harmful than the LL to destroy BA networks due to cascading failures. However, on the contrary, the attack on the edges with the lowest loads (LL) is clearly more destructive than the attack on the ones with the highest loads (HL) when α ≤ 0.8, as shown in four sub-figures, not relating to γ . We furthermore try to explain this counterintuitive phenomenon by adopting two sub-graphs of the network (see Fig. 4). Assume Fig. 4 to be two different parts of Internet or a power grid. When α = 0.5, we compare the local effects of two attacks on edges with the higher or lower loads for network robustness. In order to avoid the breakdown of the neighboring edges, we find that the lowest value of the capacity parameter β is 1.183 and 1.5 in two cases of the failures of the edges ij and mn in Fig. 4, respectively. Therefore, in this case the edge with the lower loads is more important in network safety than the one with the higher loads. In Fig. 3, one can see that the bigger the value α , the smaller the difference of the effects of two attacks. Thus a natural question arises: does there exist the value α at which the effects of two attacks are almost identical? For discussing this problem, we further explore the effects of two attacks for network robustness when α = 1. As shown in Fig. 5, it is easy to find that the obtained βc originating from two attacks in the same value γ are almost identical. Such result is also different from many previous studies on scale-free networks, at which the presence of a few edges with exceptionally large loads may make the network vulnerable to a cascade of overload failures capable of disrupting the network into small fragments. At α = 1, no difference phenomenon of two attacks by the numerical simulations will be verified by the latter theoretical analysis. In the case of α > 1, we also check network robustness under two attacks in Fig. 6. Apparently, as expected, the edges with the highest loads play an important role in the network security. In addition, contrasting to four sub-figures of Fig. 6, it is also found that the parameter γ for the LL has a greater impact than for the HL, e.g., γ = 1.4, Sattack under the LL close to zero. We furthermore explore the role of the parameter γ in cascading propagation. As shown in Fig. 7, switching of the efficiency of two attacks exists at the point of α = 1. 4. Theoretical analysis We now provide the theoretical prediction support for the numerical simulations. To facilitate analysis, we use a parameter p to denote the threshold of the breakdown probability of an edge. Based on the mechanism of the load redistribution in the cascading model, to avoid the occurrence of cascading failures for a neighboring edge im of the breakdown edge ij in Fig. 1, the following condition should be satisfied:

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Fig. 6. The scheme illustrates the comparison between two attacks in the case of α > 1.

Fig. 7. Relation between the critical threshold βc and the parameter γ .

 Lim + ∆Lim < Cim , Lim + ∆Lim − Cim < p,  γ Cim − Cim

γ =1 (2)

γ > 1.

According to the definitions of Lim , ∆Lim , and Cim , the above inequality (2) can be refined to:

 (ki km )α α α  P P ( k k ) + ( k k ) < β(ki km )α ,  i m i j   (ki ka )α + (kj kb )α   a∈Γi b∈Γj

γ =1

α

(ki km )α + (ki kj )α P (ki ka()kαi k+m )P (kj kb )α − β(ki km )α     a∈Γi b∈Γj   < p, α γ β(ki km ) − β(ki km )α

(3)

γ >1

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which can be simplified as:

 (ki kj )α  P P < β, 1 +    (kj kb )α (ki ka )α +   a∈Γi b∈Γj 1+      

P a∈Γi

γ =1

(ki kj )α P (ki ka )α + (kj kb )α b∈Γj

(4)

< β,

pγ − p + 1

γ > 1.

Here,

X

kαa =

a∈Γi

kmax X

α

ki P (k0 |ki )k0 ,

(5)

k0 =kmin

where P (k0 |ki ) is the conditional probability that a node of ki has a neighbor of k0 . Since the BA network has no degree–degree correlation, P (k0 |ki ) = k0 P (k0 )/hki. So, we have

X

kmax α X ki hkα+1 i k0 P (k0 )k0 = , hki hki k0 =k

(6)

kmax α X kj hkα+1 i k0 P (k0 )k0 kαb = kj = . hki hki k 0 =k

(7)

kαa = ki

a∈Γi

X b∈Γj

min

min

So, Eq. (4) can be reduced to,

(ki kj )α

  1+     

α+1 α+1 ki hk i

1+      

+

α+1 α+1 kj hk i

< β,

γ =1

hki hki (ki kj )α α+ 1 α+1 α+ 1 α+1 hk i k hk i k i

hki

+

(8)

j

hki

pγ − p + 1

< β,

γ > 1.

That is,

 1 hki   < β, 1 + α+1  k ki  h k i  + kαj  kα j i hki 1  1 + hkα+1 i ki + kj  α kα  k  j i   < β, pγ − p + 1

γ =1 (9)

γ > 1.

By noting that ki α

kj

+

kj α

ki

≥

2

(ki kj )

α−1

,

(10)

2

we can obtain

 α−1  hki(ki kj ) 2   1 + < β.   2hkα+1 i α−1 hki(k kj ) 2  1 + 2hki α+  1i    < β, pγ − p + 1

γ =1 (11)

γ > 1.

Based on Eq. (11), when α < 1, bigger βc can be obtained by attacking the edges with the lowest loads since ki kj has positive correlative with (ki kj )α when α > 0; when α = 1, βc has only relation with the hki and hk2 i; when α > 1, the edges with the highest loads play an important role in destroying networks. Due to not considering the degree distribution of the networks, our findings can be applied to all networks without degree–degree correlation.

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5. Conclusion The study of attacks on complex networks is important in order to identify robustness and vulnerability of real-life networks, which can be used either for protection of many infrastructure networks or for destruction of the spread of rumors and epidemic diseases. Assume the initial load of an edge ij to be (ki kj )α with ki and kj being the degrees of the nodes connected by the edge, where α is a tunable parameter. We investigate the cascading reaction behaviors under two attacks on BA networks, i.e., attacking on the edges with the highest loads and the ones with the lowest loads, respectively. The numerical simulations indicate that the effects of two attacks are closely relative with α , and not relative with the breakdown probability of an overload edge proposed in our cascading model. A counterintuitive result that attacking the edges with the lowest loads is harmful to disrupt BA networks than attacking the ones with the highest loads when α < 1 can provide new guidance in protecting networks to avoiding cascading failures. In addition, we also provide the theoretical analysis to verify the numerical simulations. In this paper, although we have focused on BA scale-free networks, we can see that the discussions can be applied to all networks with no degree–degree correlation. Our findings may be very helpful to avoiding cascadingfailure-induced disasters. Acknowledgement This work was supported by the National Natural Science Foundation of China under Grant nos. 70571011 and 70771016. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]

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