Physica A 387 (2008) 6671–6678

Contents lists available at ScienceDirect

Physica A

journal homepage: www.elsevier.com/locate/physa

Attack vulnerability of scale-free networks due to cascading failuresJianwei Wang a,∗, Lili Rong a, Liang Zhang a, Zhongzhi Zhang b

a Institute of System Engineering, Dalian University of Technology, 2 Ling Gong Rd., Dalian 116024, Liaoning, PR Chinab Department of Computer Science and Engineering, Fudan University, Shanghai 200433, PR China

a r t i c l e i n f o

Article history:Received 2 June 2008Received in revised form 8 August 2008Available online 5 September 2008

PACS:89.75.Hc89.75.-k89.75.Fb

Keywords:Cascading failureScale-free networkAttackCritical threshold

a b s t r a c t

In this paper, adopting the initial load of a node i to be akαi with ki being the degree of the

node i, we propose a cascadingmodel based on a load local redistribution rule and examinecascading failures on the typical network, i.e., the BA network with the scale-free property.We find that the BA scale-free network reaches the strongest robustness level in the caseof α = 1 and the robustness of the network has a positive correlation with the averagedegree 〈k〉, where the robustness is quantified by a transition fromnormal state to collapse.In addition, we further discuss the effects of two different attacks for the robustness againstcascading failures on our cascading model and find an interesting result, i.e., the effects oftwo different attacks, strongly depending to the value α. These results may be very helpfulfor real-life networks to avoid cascading-failure-induced disasters.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Security of complex networks in response to random failures or attacks has attracted a great deal of attention in recentyears [1–4]. The key factor prompting this research is the observation that in many growing networks some nodes evolve tobecome much more important than others [5,6]. From a global perspective, the important nodes are those whose removalmay either cause the network to disintegrated or severely limit the communication between the other nodes. A primeparadigm in this context is that of the scale-free networks, the degree distributions ofwhich followapower-law: P(k) ∼ k−γ ,where k is the number of links of a randomly chosen node in the network and γ is the scaling exponent. The work by Albertet al. [7] demonstrated that scale-free networks, such as the World Wide Web, Internet, protein networks, or ecologicalnetworks, possess the robust-yet-fragile property, in the sense that they are remarkable robustness against random failuresof nodes but, at the same time, very vulnerable to intentional attacks such as the removal of the most highly connectednodes. Since previous much studies have focused only on the static properties of the network showing that the removalof a group of nodes altogether can have important consequences, recently, cascading failures on complex networks havebeen highly concerned and widely investigated because of taking the redistribution dynamics of flows on the network intoaccount.

Cascading failures are common in most of the communication and/or transportation networks that are the basiccomponents of our lives and industry. Typical examples are several blackouts [8] in some countries and Internetcongestion [9]. Recently, attack vulnerability originating from cascading failures in complex networks has been extensivelyexplored and many valuable results have been found. Motter et al. [10] discussed cascade-based attacks on complexnetworks and demonstrated that the heterogeneity of the scale-free networks make them particularly vulnerable to attacks

∗ Corresponding author. Tel.: +86 411 81258693.E-mail address:wdut@yahoo.cn (J. Wang).

0378-4371/$ – see front matter© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2008.08.037

6672 J. Wang et al. / Physica A 387 (2008) 6671–6678

in that a large-scale cascademay be triggered by disabling a single key node. Zhao et al. [11] investigated attack vulnerabilityof scale-free networks due to cascading failure and obtained a theoretical estimate for the phase transition point and providenumerical support. Crucitti et al. [12] proposed the model for cascading failures in complex networks and showed that thebreakdown of a single node is sufficient to collapse the efficiency of the entire system if the node is among the ones withlargest load.Wang et al. [13] investigated universal robustness characteristic of weighted networks against cascading failureandobtained the optimal parameter value in themodel against cascading failures. In addition,many cascading failuremodelshad been presented, such as the sandpile model [14,15], the ORNL-PSerc-Alaska (OPA) model [16,17]to study blackoutdynamics in the power transmission grid, the CASCADE model [18,19] used to examine power transmission system criticalloading and power tails in probability distributions of blackout size, etc. In most studies, the load on a node was generallyestimated by its degree or betweenness and the load redistribution after overloaded nodes are removed from the networksare forwarded either following the shortest path or next-nearest-neighbor search strategy. These load redistribution rulesallow each node to have thewhole network’s topological information or at least two order neighbor information, whichmaybe practical for small or medium size networks but not for very large networks in some real networks such as the Internetor WWW.

Because such strategies with local information are favored in cases where there is a heavy communication cost tosearching the network in real time, in this paper, we propose a cascading model based on a load local redistribution rule.Assume the initial load of a node i to be Li = akα

i , where a and α are tunable parameters, governing the strength of theinitial load, and ki is the degree of the node i. For a typical network, i.e., the Barabási–Albert (BA) network [5] with scale-free property, we find the strongest robustness level against cascading failures at a specific parameter α. The robustnessis quantified by the critical threshold of the node’s capacity to handle the load, at which a phase transition occurs fromnormal state to collapse. The correlation between the robustness and the average degree of a network is further explored.In addition, the numerical simulations are also verified by the theoretical analysis. Adopting two attack strategies includingattacking the nodes with the biggest degree and the ones with the smallest degree, we explore the effects of two attacksfor the robustness against cascading failures and draw some valuable results. These results should be useful in furtheringstudies in the important area of network security.

The rest of this paper is organized as follows: in Section 2, we describe the cascading model in detail. The robustnessof BA networks are analyzed in Section 3. In Section 4, we investigate the effects of two attacks for the robustness of BAnetworks against cascading failures. Finally, some summaries and conclusions are shown in Section 5.

2. The model

Here we focus on cascades triggered by the removal of a single node. If a node has a relatively small load, its removal willnot cause major changes in the balance of loads, and subsequent overload failures are unlikely to occur. However, when theload at a node is relatively large, its removal is likely to affect significantly loads at other nodes and possibly starts a sequenceof overload failures and eventually a large drop in the network performance such as those observed in real systems, like theInternet or the electrical power grids.

Inspired by the above process of cascading failures, we propose a cascading model under the load local redistributionrule. In our model, the main differences with respect to previous models are as follows:

(1) For simplicity, adopt that initial load Lj of each node j in the network is a function of its degree kj and is defined as:

Lj = akαj (1)

where a and α are tunable parameters in our study, which control the strength of the initial load of the node j. Thisassumption is reasonable, e.g., traffic flow of every node on traffic network usually has a certain correlation with itsdegree;

(2) The load on the broken node iwill be redistributed to its neighboring node j, according to the preferential probability:∏j

=akα

j∑m∈Γi

akαm

=kαj∑

m∈Γi

kαm

(2)

where Γi represents the set of all neighboring nodes of the broken node i.According to the rule of (2), the additional load ∆Lji received by the node j is proportional to its initial load, i.e.,

∆Lji = Likαj∑

m∈Γi

kαm

. (3)

Following previous models addressing cascading problems [13,20,21], each node j in the network has a capacity threshold,which is the maximum flow that the node can transmit. Since the node capacity on real-life networks is generally limitedby cost, it is natural to assume that the capacity Cj of the node j is proportional to its initial load for simplicity: Cj =

T ∗ Lj, j = 1, 2, 3, . . . ,N , where the constant T (≥1) is the tolerance parameter characterizing the tolerance of the network

J. Wang et al. / Physica A 387 (2008) 6671–6678 6673

Fig. 1. Load redistribution after removing the node i.

[10,13,20,21]. Because every node has a limited capacity to handle the load, so for the node j, if Lj +∆Lji > Cj, then the nodejwill be broken and induce further the redistribution of the load Lj + ∆Lji and potentially further other nodes breaking (seeFig. 1).

To measure the robustness level of the whole networks against cascading failures, we remove every node in a networkone by one and calculate the corresponding results, e.g., removing the node i and calculating CFi (here CFi denotes theavalanche size, i.e., the number of broken nodes, induced by removing i) after the cascading process is over. Apparently,here 0 ≤ CFi ≤ N − 1. To quantify the robustness of the whole network, we adopt the normalized avalanche size, i.e.,

CFN =

∑i∈N

CFi

N(N − 1)(4)

where the summation over all the avalanche sizes by removing each node initially at each time.In our cascading model, it is evident that the value T will decide to the robustness level against cascading failure.

Therefore, inspired by Wang et al. [13], to quantify the robustness, we propose a new measure, i.e., the critical thresholdTc . The corresponding principle is that when T > Tc , no cascading failure occurs and the system maintains its normaland efficient functioning, since every node in the network has capacity to handle the extra load redistributed; while forT < Tc , CFN suddenly increases from 0 and cascading failure emerges because the capacity of each node is limited, sweepingthe whole or part network to stop working. Hence Tc is the least value of protection strength to avoid cascading failure.Apparently, the lower the value of Tc , the stronger the robustness of the network against cascading failure.

3. Analysis of the critical threshold Tc

Since the network structure plays an important role in the robustness of the network and the scale-free (SF) property isubiquitous in nature and human society, it is natural and important to consider the robustness against cascading failures onSF networks. For simplicity, we use thewell-knownBarabási–Albert (BA) SF networkmodel [5] as the physical infrastructureon top of which a cascading process takes place. BA networks can be constructed as follow. Starting fromm0 nodes, one nodewith m links is attached at each time step in such a way that 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 totalnetwork size is fixed asN = 5000 and the parameters are set to bem0 = 3,m = 2 (hence the average degree of the networkis 〈k〉 = 2m = 4 [5]).

According to our cascading model, we investigate the relationship between some parameters and the critical thresholdTc . Firstly, we focus on the effect of the parameter α for CFN . The simulation result is the average one of 20 independentnetworks.

As shown in Fig. 2, When α = 1, in the case of the same T , all CFN are minimums. Since the value CFN decide to the Tc ,a natural question arises: is α = 1 the least value of protection strength to avoid cascading failure, i.e., the optimal valuefor enhancing the robustness of the network against cascading failure? For investigating this problem, further numericalsimulation are shown in Fig. 3.

Based on Fig. 3, on the one hand, in sub-figure (a), all values on the curve α = 1 are smaller than ones on other curves inthe case of the same T , which has been verified by Fig. 2; on the other hand, in sub-figure (b), Tc is also minimal value whenα = 1. Based on the aforementioned two ways, we can draw a conclusion that BA networks reach the strongest robustnesslevel against cascading failure at α = 1.

In the above analysis, we only take the fixed average degree 〈k〉 into account,which is 4. Next,we focus on the relationshipbetween the different 〈k〉 and the critical threshold Tc (see Fig. 4).

From sub-figure (a) and sub-figure (b), we can see that Tc is a negative correlation with 〈k〉, i.e., the bigger the value of〈k〉, the smaller the critical threshold Tc related with the robustness of the network against cascading failure. So, in order

6674 J. Wang et al. / Physica A 387 (2008) 6671–6678

Fig. 2. CFN as a function of the parameter α.

Fig. 3. Relation between parameter α and Tc .

Fig. 4. CFN as a function of the parameter T for different 〈k〉.

to avoid cascading failure, it is the efficient way to increase the number of edges on networks without taking the price intoaccount.

For better understanding the observed universal cascading phenomenon and verifying the simulation results, we alsoprovide some theoretical analysis.

J. Wang et al. / Physica A 387 (2008) 6671–6678 6675

Based on the mechanism of load redistribution in the cascading model, to avoid the emergence of cascading failure, thefollowing condition should be satisfied:

Lj + ∆Lji < Cj. (5)

According to the definitions of Lj and ∆Lji, the above inequality (5) can be refined to:

akαj + akα

i

akαj∑

m∈Γi

akαm

< Takαj (6)

which can be simplified as

1 +kαi∑

m∈Γi

kαm

< T (7)

Here∑

m∈Γikαm =

∑kmaxk′=kmin

kiP(k′|ki)k′α , where P(k′

|ki) is the conditional probability that a node of ki has a neighbor of k′.Since the BA network has no degree–degree correlation, P(k′

|ki) = k′P(k′)/〈k〉. So, we have

∑m∈Γi

kαm = ki

kmax∑k′=kmin

k′P(k′)k′α

〈k〉=

ki〈kα+1〉

〈k〉. (8)

So (7) can be expressed by:

1 +kαi

ki〈kα+1〉〈k〉

= 1 +kα−1i 〈k〉〈kα+1〉

< T . (9)

From the above inequality, the critical threshold Tc can be obtained by considering the ranges of α < 1 and α = 1, α > 1,respectively:

Tc =

1 +kα−1min 〈k〉〈kα+1〉

, α < 1

1 +〈k〉〈k2〉

, α = 1

1 +kα−1max 〈k〉〈kα+1〉

, α > 1

(10)

where kmin and kmax are the minimum and maximum none degrees. To find the minimum value of Tc , we consider Tc − 1 inthe case of α < 1 in Eq. (10):

Tc − 1 =kα−1min 〈k〉〈kα+1〉

=kα−1min 〈k〉

1N

N∑i=1

kα+1i

=〈k〉

1N

N∑i=1

k2i(

kikmin

)α−1>

〈k〉

1N

N∑i=1

k2i

=〈k〉〈k2〉

. (11)

Hence, Tc(α < 1) > Tc(α = 1). Similarly, we can get Tc(α > 1) > Tc(α = 1). Apparently, the lowest value of atα = 1 indicates that the system possesses the universal strongest robustness, independent of the degree distribution of thenetwork.

In the case of α = 1, for obtaining the value Tc , we firstly calculate the value 〈k2〉. For the BA network with finite size, itsdegree distribution is P(k) = 2m2k−3, here m = kmin based on the evolving mechanism of the BA network. Hence we getthe expression of P(k), i.e., P(k) = 2k2mink

−3. Thus, 〈k2〉 in the BA network can be obtained by

〈k2〉 =

∫ kmax

kmin

P(k)k2dk =

∫ kmax

kmin

2k2mink−3k2dk

= 2k2min

∫ kmax

kmin

k−1dk = 2k2min(ln kmax − ln kmin). (12)

6676 J. Wang et al. / Physica A 387 (2008) 6671–6678

Fig. 5. Comparison of analytic results and simulations.

kmax can be calculated by∫

∞

kmaxP(k)dk = 1/N , i.e.,

∫∞

kmax2m2k−3dk = 1/N ⇒ kmax =

√Nkmin. So, 〈k2〉 =

2k2min(ln(√Nkmin) − ln kmin) = k2min lnN . Then we get

Tc(α = 1) = 1 +〈k〉〈k2〉

= 1 +〈k〉

k2min lnN. (13)

Due to 〈k〉 = 2kmin, Eq. (12) can be simplified as

Tc(α = 1) = 1 +2

kmin lnN= 1 +

2m lnN

. (14)

From Eq. (13), we can see that whenm ≥ 2, i.e., the average degree 〈k〉 ≥ 4, the critical threshold Tc has negative correlationwith the average degree 〈k〉, i.e., the bigger the value of 〈k〉, the smaller the critical threshold Tc .

The comparison between analytic results and simulation results is shown in Fig. 5. FromFig. 5,we can see that the curve ofanalytic results is in good agreementwith one of simulation results except for the difference in the case of smaller 〈k〉, whichmainly results from one approximation in the analysis. For the BA network with the finite size, the exponent of power-lawdegree distribution is higher than −3. Hence the use of P(k) k−3 leads to the difference in comparison.

4. Correlation between Tc and the different attack strategies

Here we mainly focus on the effects of two attack strategies for the critical threshold Tc , i.e., removing nodes by eitherthe descending or ascending order of the node degree. To quantify the attack-based robustness of the whole network, weadopt the normalized avalanche size, i.e.,

CFattack =

∑i∈A

CFi

NA(N − 1)(15)

where A and NA represents the set and the number of nodes attacked, respectively.In the case of α = 1, to better explore the effects of two different attacks for the critical threshold Tc , we select the nodes

in the descending or ascending order of degrees in the network, and then remove nodes one by one starting from the nodewith the biggest or smallest degree. For two attacks, we choose 25 nodes as the objects attacked in the BA network withN = 5000 and 〈k〉 = 4, respectively. Each curve is obtained by averaging over experiments on 20 independent networks.Fig. 6 shows CFattack as a function of the tolerance parameter T for two attack strategies on BA networks.

As shown in Fig. 6, it is easy to find that two curves are not the same under two attacks, although the difference betweenthem is small. The number of broken nodes induced by attacking bigger degree nodes is slightly more than the other one.However, obtaining Tc originating from two attacks is almost identical, which is about 1.17.

Compared with the previous results, i.e., the robust-yet-fragile property of scale-free networks on random failures andintentional attacks of nodes, the numerical simulations obtain an interesting and totally different result, i.e., two attacks onBA networks can not impact on the critical threshold Tc when α = 1. Actually, based on Eq. (10), we can see that Tc has onlyrelated with 〈k〉 > and 〈k2〉 but not the nodes with the biggest degree or the smallest degree on BA networks.

For the same Tc under two attacks, we will further explain it by the another angle, i.e., the load redistribution rule andthe structure characteristic of BA networks. Based on Eq. (2), when α = 1, it is evident that both the extra load received bythe initial load of every node on BA networks and its neighboring nodes are proportional to their degrees. Since cascadingfailures are triggered by a single broken node, it is sufficient that we only take the neighboring nodes of the broken node into

J. Wang et al. / Physica A 387 (2008) 6671–6678 6677

Fig. 6. CFN as a function of the tolerance parameter T for the different attack strategies.

Fig. 7. Effect attack for removing the nodes with the smallest average degree of its neighbor nodes.

account. To better understand the universal cascading phenomenon, we investigate the relation between Tc and the degreeof neighboring nodes of the broken node from some theoretical analysis. Due to the load local preferential redistributionand α = 1, to avoid the emergence of cascading failure, the following condition should be satisfied∑

j∈Γi

[T (kj) − kj] > ki (16)

which can be simplified as∑j∈Γi

kj

ki>

1T − 1

. (17)

So, by analyzing carefully the inequality (17), we draw a conclusion that cascading failures will be triggered in the case thatthe average degree of neighboring nodes of the broken node is not bigger than 1/(T − 1). Fig. 7 illustrates that attacking onthe node j is more likely to lead to cascading failures than attacking on the node i based on Eq (17).

Since the BA network has no degree–degree correlation, which leads to the analytic results Eq. (10), the average degreeof neighboring nodes of every node on the BA network is almost the average degree 〈k〉 of all nodes, whether the nodes withthe bigger or the smaller degree. This also leads to the conclusion that two Tcs induced by two attacks are almost identical.However, due to the bigger initial load of the bigger degree node, once it leads to the failure of its neighboring nodes, thenumber of broken nodes is more than one induced by the smaller degree node. Hence, the difference between two curvesemergent in Fig. 6.

In addition, taking the case of α < 1 and α > 1 into account, we investigate the effect of two attacks for the criticalthreshold Tc on BAnetworks. Fig. 8 shows CFattack as a function of the tolerance parameter T for two different attack strategiesin the case of α < 1 and α > 1 for BA networks with size N = 5000 and average degree 〈k〉 = 4. For two attacks, we alsochoose 25 nodes as broken ones, respectively. Each curve is obtained by averaging over experiments on 20 independentnetworks.

Based on Eq. (10) and the simulation results in Fig. 8, we can draw two conclusions that: attacking the node with thesmaller degree are more likely to lead to cascading failures than attacking ones with the bigger degree when α < 1; while,

6678 J. Wang et al. / Physica A 387 (2008) 6671–6678

Fig. 8. Comparison of two attacks in the case of α < 1 and α > 1.

one the contrary in the case ofα > 1. Our results enable a possible implementation of predicting and preventingmechanismfor cascading failures in scale-free networks.

5. Conclusion

In conclusion, we propose a cascading model and investigate the cascading reaction behaviors on BA networks withrespect to small node-based initial attacks. We find that various parameters play an important role in robustness againstcascading failure. By the analysis of the critical threshold Tc , we obtain a theoretical estimate for the phase-transition pointand provided a numerical check. The numerical simulations and the analytic results both show that BA networks display thestrongest robustness against cascading failure when α = 1 and Tc has a negative correlation with the average degree 〈k〉in the case of the fixed size of networks. In addition, we draw some interesting and counterintuitive conclusions, which aremuch different from the previous results: on one hand, when α = 1, attacks have no impact on the robustness of networks;on the other hand, when α < 1, as unexpected, the attack on the nodes with smallest degree is more harmful than the otherone. The important role of the nodes with the smaller degree in cascading failures gives us a implication, i.e., not ignoringthe effect of thus nodes for the network’s robustness against cascading failures. In summary, our work may have practicalvalue for controlling various cascading-failure-induced disasters in the real world.

Acknowledgments

Weare grateful toDr. YongguiWang for his valuable comments and suggestions. Thisworkwas supported by theNationalNatural Science Foundation of China under Grant nos. 70571011 and 70771016.

References

[1] R. Cohen, K. Erez, D.B. Avraham, S. Havlin, Phys. Rev. Lett. 85 (2000) 4626.[2] A.X. Valente, A. Sarkar, H.A. Stone, Phys. Rev. Lett. 92 (2004) 118702.[3] R. Cohen, K. Erez, D.B. Avraham, S. Havlin, Phys. Rev. Lett. 86 (2001) 3682.[4] B. Bollobás, O. Riordan, Internet Math. 1 (2003).[5] A.-L. Barabási, R. Albert, Science 286 (1999).[6] L.A.N. Amaral, A. Scala, M. Barthwlwmy, H.E. Stanley, Proc. Natl. Acad. Sci. USA 97 (2000) 11149.[7] R. Albert, H. Jeong, A.-L. Barabási, nature 406 (2002).[8] R. Albert, I. Albert, G.L. Nakarado, Phys. Rev. E 69 (2004) 025103.[9] K.-I. Goh, B. Kahng, D. Kim, Phys. Rev. Lett. 88 (2002) 108701.

[10] A.E. Motter, Y.C. Lai, Phys. Rev. E 66 (2002) 065102.[11] L. Zhao, K. Park, Y.C. Lai, Phys. Rev. E 70 (2004) 035101(R).[12] P. Crucitti, V. Latora, M. Marchiori, Phys. Rev. E 69 (2004) 045104.[13] W.X. Wang, G.R. Chen, Phys. Rev. E 77 (2008) 026101.[14] Z. Olami, H.J.S. Feder, K. Christensen, Phys. Rev. Lett. 68 (1992).[15] K.I. Goh, D.S. Lee, B. Kahng, D. Kim, Phys. Rev. Lett. 91 (2003) 148701.[16] I. Dobson, J. Chen, J.S. Thorp, et al. Proceedings of 35th Hawaii International Conference on System Sciences, 2002.[17] B.A. Carreras, V.E. Lynch, I. Dobson, D.E. Newman, Chaos 12 (4) (2002).[18] I. Dobson, B.A. Carreras, D.E. Newman, Proceedings of 35th Hawaii International Conference on System Sciences, 2003.[19] I. Dobson, B.A. Carreras, D.E. Newman, Probab. Engrg. Inform. Sci. 19 (1) (2005).[20] A.E. Motter, Phys. Rev. Lett. 93 (2004) 098701.[21] Z.X. Wu, G. Peng, W.X. Wang, S. Chan, E.E.M. Wong, Stat. Mech. (2008) P05013.