Robustness of Internet under targeted attack: A cascadingfailure perspective

Jianwei Wang n, Chen Jiang, Jianfei QianSchool of Business Administration, Northeastern University, Shenyang 110819, PR China

a r t i c l e i n f o

Article history:Received 1 May 2012Received in revised form27 April 2013Accepted 24 August 2013Available online 17 September 2013

Keywords:Cascading failureRobustnessAttackInternet

a b s t r a c t

In many networks, there exist some heterogeneous nodes, for example the hosts and the routers in theInternet, and the users and the supply-grid stations in the power grid. In the previous studies, however,few cascading models were constructed in the heterogeneous networks. Considering two types of nodes,we propose a new cascading edge model and investigate the cascading dynamic behaviors in theInternet. Compared with an extending Barabási–Albert (BA) network with two types of nodes, we findthat the Internet displays the stronger robust level under two targeted attacks and propose someeffective methods to protect these networks. The cascading model in the Internet can be extensivelyapplied to other real-life networks, and may provide a new perspective to study the network safety.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

In recent years, owing to the importance of the network safety inour daily life, the question of the robustness of complex networks hasattracted a large amount of interests from many researchers (Lin,2007; Bossardt et al., 2007; Wu et al., 2007a,b, 2011; Holme et al.,2002; Xia and Hill, 2008). In particular, there has been the extensiveeffort to study and understand the Internet robustness, which hasbeen one of the most central topics in the network safety. Animportant conclusion in this context is that the Internet displays thestronger robustness against random failure, but shows the exceptionalvulnerability against intentional attack (Albert et al., 2000). Since theearlier studies mainly focus on the static properties of a network(Cohen et al., 2000, 2001), recently cascading failures induced by theredistribution dynamics of the load on a network have been highlyconcerned and widely investigated. Cascading failures are common inreal-life systems and can occur not only in the Internet but also inmany other infrastructure networks. Typical examples include severalblackouts in some countries, e.g., the largest blackout in US historytook place on 14 August 2003 and the Western North Americanblackouts in July and August 1996, and the Internet collapse caused bythe submarine earthquake near Taiwan in December 2006.

A number of important aspects of cascading failures have beendiscussed in the literature and many valuable results have been found,including the robustness of interdependent networks (Vespignaniet al., 2010; Buldyrev et al., 2010; Parshani et al., 2010; Huang et al.,

2011; Shao et al., 2011), the cascading control and defense strategies(Wang and Rong, 2009a,b; Yang et al., 2009; Wang et al., 2010;Simonsen et al., 2008; Motter, 2004; Ash and Newth, 2007), themodels for describing the cascading phenomena (Crucitti et al., 2004;Wang and Chen, 2008; Wang and Xu, 2004; Wang and Rong, 2009c;Lehmann and Bernasconi, 2010; Goh et al., 2001; Sandro et al., 2008),and cascading failures triggered by intentional attacks (Motter and Lai,2002; Wang et al., 2008; Wang and Rong, 2008; Zhao et al., 2004,2005; Ricard et al., 2008). These studies not only construct cascadingmodels from single networks but also pay close attention to cascadingmodels from coupled networks. However, in all studies cited above, afew works explore cascading failures in the Internet with two types ofnodes, i.e., the routers and the hosts. Therefore, in the Internet, how toconstruct a cascading model with heterogeneous nodes and investi-gate its robustness against cascading failures is a significant topic.

To this end, the aim of this paper is to construct a new cascadingmodel and investigate the roles of different edges in the cascadingpropagation in the Internet. Considering two types of nodes, weconstruct a new cascading model and compare the effects of differentways of attacking edges in the Internet. We numerically find someinteresting and important results, such as, targeted attacks are notalways effective and the Internet is more robust than Barabasi andAlbert (1999) (BA) networks under targeted attacks. In addition, wecarefully analyze these results by the local topological structure of twokinds of networks. Our findings may be useful in further studies onhow to build attack-robust networks and increase the robustness ofreal-life networks against cascading failures.

The rest of this paper is organized as follows: in Section 2, wedescribe the cascading model in detail. In Section 3, we analyzethe robustness of the Internet under two targeted attacks. Finally,some summaries and conclusions are shown in Section 4.

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Journal of Network and Computer Applications

1084-8045/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jnca.2013.08.007

n Corresponding author. Tel.: þ86 024 83672631.E-mail address: jwwang@mail.neu.edu.cn (J. Wang).

Journal of Network and Computer Applications 40 (2014) 97–104

2. The model

In general, the Internet contains two types of nodes, the routersand the computers, which play different roles in the function. Thecomputers are the terminal users, while the cables between therouters can transport the load. We can simply use a two-layernetwork to represent the Internet (see Fig. 1), in which the routersand the computers lie in the top layer and the bottom layer,respectively. When the cable between two routers fails owing tosome reasons, the load on it will be redistributed to other cables,which may further lead to the malfunction of other cables andeventually trigger the larger cascading failures in the wholeInternet. According to the cascading evolving process mentionedabove, a cascading model should include three aspects: how todefine the initial load on an edge, how to assign the capacity of anedge, and how to redistribute the load on it after an edge fails.Therefore, considering the different roles of the routers and thecomputers in the cascading propagation and three aspects relatedwith cascading failures, we propose a new cascading model.

Next, we briefly introduce our cascading edge model. Firstly,we give a new method to assign the initial load on an edge. In theInternet, the load on the cable between two routers A and B isgenerally decided by the number of the routers and the computersconnected with A and B. Therefore, considering the effects of therouters in the R layer and the computers in the C layer on theinitial load on an edge, we assume the initial load Lij of edge ij to be

Lij ¼ δf Rðki;R-R; kj;R-RÞf Cðki;R-C ; kj;R-CÞ ð1Þwhere ki;R-R and ki;R-C is the degrees of node i connecting withthe routers in the R layer and the computers in the C layer,respectively, and δ is a tunable parameter, governing the strengthof the initial load on an edge. Two functions fR and fC represent theeffects of the routers and the computers connected with routersi and j on the initial load on edge ij, respectively. Our main aim isto investigate the robustness of the Internet under targeted attacks

against cascading failures. Therefore, we assume

f Rðki;R-R; kj;R-RÞ ¼ δ1ðki;R-Rkj;R-RÞα ð2Þ

f Cðki;R-C ; kj;R-CÞ ¼ δ2ðki;R-Ckj;R-CÞβ ð3Þ

of which δ1, δ2, α, and β are tunable parameters. The definitions ofEqs. (1) and (2) are supported by empirical evidences of realnetworks (Wang and Rong, 2009b; Wang and Chen, 2008; Barratet al., 2004). Moreover, Holme et al. (2002) shows that thebetweenness of an edge has a positive correlation with theproduct form of node degrees at both ends of the edge. Specially,in a single network with one type of nodes, many cascadingmodels are constructed according to the rules above. In this sense,our assumption about the initial load on the edge is in accordancewith the previous models. By analyzing the definition of the initialload on an edge, we can see that the distribution of the initial loadis decided by parameters δ, δ1, δ2, α, β, and the network topologystructure. Therefore, we mainly focus on the correlation betweenthese parameters and the effects of targeted attacks.

Since the capacity of an edge in real-life networks is generallylimited by cost, it is natural to assume that the capacity Cij of edgeij is proportional to its initial load for simplicity:

Cij ¼ ð1þγÞLij ð4Þ

where γð40Þ is a uniform capacity parameter (Fig. 2).Considering that the edge with the higher capacity has generally

the stronger capacity to handle the extra load, we assume topreferentially redistribute the load along those higher-capacityedges (Wang and Rong, 2008, 2009a,b; Wang and Chen, 2008).Thus, the load on a failed edge ij will be preferentially redistributedto its neighboring edges in the R layer (see Fig. 3 for illustration).The additional load ΔLim received by edge im is proportional to its

Fig. 1. The scheme demonstrates that the Internet with the routers and the computers can be represented by two layers networks.

J. Wang et al. / Journal of Network and Computer Applications 40 (2014) 97–10498

capacity, i.e.,

ΔLim ¼ LijCim

∑aAΓiCiaþ∑bAΓj

Cjbð5Þ

where Γi and Γj are the sets of the neighboring nodes of nodes i andj, respectively. For edge im received the additional load ΔLim,if LimþΔLim4Cim, then it will be broken and the load on it has tobe transferred to its neighboring edges. This load redistribution mayfurther lead to the cascading propagation in the entire network.

Here we focus on cascading failures triggered by attacking asingle edge. In a real-life network, an attack on a particular edge isdefined as an event that disables or removes this edge from thenetwork. We use seij, srnij , and scnij to denote the avalanche sizes ofedges, and nodes in the R layer and the C layer induced byremoving edge ij, respectively. It is evident that 0rseijrE�1,0rsrnij rNr , and 0rscnij rNc , where E, Nr, and Nc represents thenumber of edges, and nodes in the R layer and the C layer in thenetwork, respectively. Therefore, without loss of generality, weadopt the normalized avalanche size S of edges and the normal-ized avalanche sizes SRN and SCN of nodes in the R layer and the Clayer to quantify the network robustness, i.e.,

S¼ 1EA

∑ijAA

seijE�1

ð6Þ

SRN ¼ 1EA

∑ijAA

srnijNr

ð7Þ

SCN ¼ 1EA

∑ijAA

scnijNc

ð8Þ

where A and EA represent the set and the number of edges attacked,respectively.

3. The analysis of the robustness of the Internet

To better investigate the robustness of the Internet againstcascading failures and propose the effective protection strategies,we focus on two simple targeted attacks in our cascading model.The first one is to attack the edges with the highest load (HL). Theremoval rule is to select the edges in the descending order of theload and then to remove them one by one starting from the edgeswith the highest load (if some edges happen to have the sameload, we randomly choose one of them). The second one is toattack the edges with the lowest load (LL). The removal rule is toselect the edges in the ascending order of load and then to removethem one by one starting from the edges with the lowest load(if some edges happen to have the same load, we randomly chooseone of them). The LL is rarely applied to the real-life networks,because ones generally think that the HL is easier to lead to thecascading propagation on a global scale in the whole network.

As an example we consider the structure of the Internet at thelevel of autonomous systems with 22 963 nodes and 48 436 edges.The data are collected by Mark Newman on July 22, 2006(Network data from Mark Newman's, 2006). We use autonomoussystems as the routers in the R layer in our cascading model, andgenerate the computers in the C layer. For the node i in the R layer,we simply set ki;R-R ¼ ki;R-C , i.e., if the degree of node i in the Rlayer is ki;R-R, we will generate ki;R-R nodes as the computersin the C layer. The Internet by extending the computers consistsof 119 835 nodes and 145 208 edges. For simplicity, we setδ¼ δ1 ¼ δ2 ¼ 1 in numerical simulations. To better compare theeffects of two targeted attacks in the Internet, we firstly calculatethe order of the initial load on edges when α¼ β¼ 0:1, and findthat the smallest value of the load in all edges consists of 138edges. Therefore, we choose 138 edges as the attacked objects inthe LL. While we only choose 10 edges in the HL, considering thedifferences among the edges with the highest load. In simulations,each curve is averaged over 10 realizations.

Firstly, we perform numerical simulations with 0:03rαr0:75and 0:07rβr0:85 in Fig. 4. We can observe that the curves of thefailed edges in the R layer display threshold-like behaviors, i.e.,a phase transition from normal state to collapse, which can beexplained by the role of the parameter γ. In many previous studieson cascading failures (Wang and Rong, 2008, 2009a,b,c; Wang andChen, 2008; Wang et al., 2008), quantifying the network robust-ness by this phase transition point has been widely applied.To better compare two attacks, we also adopt this phrase transi-tion point to quantify the robustness of the Internet againstcascading failures, i.e., the critical threshold γc . Apparently, thebigger the value γc , the more efficient the targeted attack. In Fig. 4,when 0:03rαr0:15 and 0:07rβr0:25, both the critical thresh-old γc and the larger avalanche sizes in the comparison betweentwo attacks show that the LL is relatively easy to trigger cascadingfailures than the HL. This result means that protecting the edgeswith the lowest load is more important than protecting the oneswith the highest load. In other words, the vulnerability in theInternet is not originated from attacking the edges with thehighest load. According to the different cases of the load distribu-tion in the Internet, our finding has an important implication thatit can provide guidance in protecting some edges selected effec-tively to avoid cascading-failure-induced disasters. Additionally, inthe ranges of 0:20rαr0:35 and 0:30rβr0:45, we can see that

Fig. 2. The scheme demonstrates the correlation between the initial load Lij on anedge ij in the R layer and two kinds of different degrees of nodes i and j. The nodesin the R layer and in the C layer correspond to the routers and the computers,respectively.

Fig. 3. Illustration of the local preferential redistribution of the load among routersin the R layer triggered by an edge-cut-based attack.

J. Wang et al. / Journal of Network and Computer Applications 40 (2014) 97–104 99

the LL is clearly more destructive than the HL only according to thecritical threshold γc . While in the ranges of the larger avalanchesizes, there exists a SS point. When γoSS, the HL is relatively easyto trigger cascading failures than the LL, while in the case of γ4SS,the result is on the contrary. Moreover, we also compare twoattacks in the ranges of 0:40rαr0:75 and 0:50rβr0:85, andfind that the HL is easier to trigger cascading failures. By numericalsimulations, we know the effects of the targeted attacks on theInternet against cascading failures and how to protect the Internetby allocating protective resources effectively.

In Fig. 5, we compare the number of the failed routers and thefailed computers induced by two targeted attacks, of which thelegends and other parameters are the same as Fig. 4. We also findthat the bigger cascades can be easier to be triggered by the LLwithin a certain range of parameters. And as the values of α and βincrease, the edges with the highest load play more and moreimportant roles in the cascading propagation.

In Fig. 6 we further explain the above phenomena by theproportion SC/C between the total capacities of all neighboringedges and the capacity of each edge in the Internet. According tothe definition of the parameter SC/C, in general, attacking the edgewith the lower SC/C is easier to trigger the malfunction of itsneighboring edges, and eventually lead to a larger cascadingpropagation in the whole network. In calculating the value ofSC/C, if some edges have the same capacity, SC/C is obtained byaveraging over these edges. The dotted line is used to look for suchedges that are easier to trigger the breakdown of its neighboringedges. According to the above analysis, it is known that attackingthe edges below the dotted line is prone to lead to cascadingfailures than attacking the ones above the dotted line. When γ ¼ 1,the value C of an edge is equal to its initial load. Thus, the edges inthe left square and in the right square represent the ones with thelower load and the higher load, respectively. Therefore, we can seethat, in the ranges of 0:03rαr0:30 and 0:07rβr0:40 attacking

the edges with the lower load is relatively easy to trigger cascadingfailures than attacking the ones with the higher load, which canexplain the phenomena in Figs. 4 and 5. The green dotted squarerepresents the more stable region, in which the failures of theedges cannot be likely to trigger the cascading propagation.The solid arrow represents that the bigger the values of α and β,the more efficient the HL. By two targeted attacks, we find thecorrelations between the effects of two attacks and the parametersin our model. Then a natural question arises: how is the robust-ness of the Internet compared with other networks with thesimilar structure?

The Internet is a typical scale-free network, the degree dis-tribution of which follows a power-law: PðkÞ � k� γ , where k is thenumber of links of a randomly chosen node in the network and γ isthe scaling exponent. To better compare the robustness of theInternet with other networks with the similar structure, we selectBarabasi and Albert (1999) (BA) network with the scale-freeproperty. In numerical simulations, the total network size is fixedas N¼22 963 and the average degree is ⟨k⟩¼ 4. We adopt a BAnetwork as the routers in the R layer in our cascading model, andalso set ki;R-R ¼ ki;R-C . Thus, the extending BA network consists of114 815 nodes and 137 778 edges. The legends and the parametersare same as the studies in the Internet. In Fig. 7, by the comparisonamong the evaluated γc induced by two targeted attacks, we find,in ranges of 0:03rαr0:45 and 0:07rβr0:55, that the biggercascades can be easier to be triggered by the LL than the HL.In addition, as α and β increase, according to γc the differencesbetween the LL and the HL is smaller and smaller, similar to Fig. 4.While in the ranges of 0:45rαr0:55 and 0:55rβr0:65, theeffectiveness of two targeted attacks in the Internet is not obvious.When 0:55rα and 0:65rβ, the order of the disruption degree isHL4LL. By labeling the region in Fig. 8, we can observe thatattacking what edge is prone to lead to the cascading propagation.We further compare the failed edges of Figs. 4 and 7 in Fig. 9, and

Fig. 4. Demonstration of the comparison of the failed edges between two targeted attacks which depends on the initial load of an edge in the Internet in the ranges of0:03rαr0:75 and 0:07rβr0:85.

J. Wang et al. / Journal of Network and Computer Applications 40 (2014) 97–104100

surprisingly find that the robustness of the Internet under thesame attack is stronger than that of the extending BA network.To better address this question, according to the critical thresholdγc obtained by the minimal value γ when So0:001 in Fig. 10, weclearly compare the effectiveness of two targeted attacks in theInternet and the extending BA network. When αþβ¼ 0:88 in theInternet and αþβ¼ 1:26 in the extending BA networks, we findthat the effectiveness of two attacks is almost identical. Inaddition, two curves of the LL show that the robustness of theInternet and the extending BA networks has a positive correlationwith αþβ (β�α¼ 0:1 in simulations) in 0:1rαþβr1:6. Whiletwo curves of the HL show the robustness of two kinds ofnetworks has a negative correlations with αþβ (β�α¼ 0:1 insimulations) in 0:1rαþβr1:6.

According to the comparison between two targeted attacks, weobtain that the Internet displays the stronger robust level than theextending BA networks. Compared with the general scale-freenetworks, our results suggest that the Internet with the self-organization evolution is not vulnerable, which is different frommany previous studies (Bossardt et al., 2007; Holme et al., 2002;Albert et al., 2000), i.e., a scale-free network shows the robust-yet-fragile property.

4. Conclusion

In summary, a cascade in the Internet, triggered by two targetedattacks on a single edge, but spreading to the entire network, is one

Fig. 5. Illustration of the failed nodes in the R layer and in the C layer induced by two targeted attacks.

Fig. 6. Demonstration of the proportion SC/C between the total capacities of all neighboring edges of and the capacity of each edge in the Internet.

J. Wang et al. / Journal of Network and Computer Applications 40 (2014) 97–104 101

of the intriguing problems in many areas. Different from mostprevious studies on cascading failures, in the Internet we considertwo types of nodes and propose a new method to assign the initialload of an edge. Our main goal is to investigate how to effectivelyprotect the Internet under targeted attacks. Our findings show thatthe Internet displays the stronger robust level against cascading

failures compared with BA scale-free networks. In addition, bystudying the effectiveness of attack strategies, we obtain the moreeffective methods to protect the Internet. These results may bevery useful for improving the robustness of the Internet andavoiding various cascading-failure-induced disasters in real-lifenetworks.

Fig. 7. Illustration of the comparison between two targeted attacks in BA scale-free networks with N¼22 963 and the average degree ⟨k⟩¼ 4.

Fig. 8. Demonstration of the proportion SC/C between the total capacities of all neighboring edges of and the capacity of each edge in BA scale-free networks.

J. Wang et al. / Journal of Network and Computer Applications 40 (2014) 97–104102

Acknowledgments

This work was supported by the National Natural Science Founda-tion of China under Grant no. 71101022, the Fundamental ResearchFunds for the Central Universities under Grant no. N110406003, andthe Program for New Century Excellent Talents in University underGrant no. NCET-12-0100.

References

Albert R, Jeong H, Barabási A-L. Attack and error tolerance in complex networks.Nature 2000;406(July):378–82.

Ash J, Newth D. Optimizing complex networks for resilience against cascadingfailure. Physica A 2007;380(July):673–83.

Barabasi AL, Albert R. Emergence of scaling in random networks. Science 1999;286(October (5439)):509–12.

Barrat A, Barthelemy M, Pastor-Satorras R, Vespignani A. The architecture ofcomplex weighted networks. Proceedings of the National Academy of Sciencesof the United States of America 2004;101(March (11)):3747–52.

Bossardt M, Dübendorfer T, Plattner B. Enhanced Internet security by a distributedtraffic control service based on traffic ownership. Journal of Network andComputer Applications 2007;30(August (3)):841–57.

Buldyrev SV, Parshani R, Paul G, Stanley HE, Havlin S. Catastrophic cascade offailures in interdependent networks. Nature 2010;464(April):1025–8.

Cohen R, Erez K, ben Avraham D, Havlin S. Resilience of the Internet to randombreakdowns. Physical Review Letters 2000;85(November (21)):4626–8.

Cohen R, Erez K, ben Avraham D, Havlin S. Breakdown of the Internet underintentional attack. Physical Review Letters 2001;86(April (16)):3682–5.

Crucitti P, Latora V, Marchiori M. Model for cascading failures in complex networks.Physical Review E 2004;69(April):045104.

Goh K-I, Kahng B, Kim D. Universal behavior of load distribution in scale-freenetworks. Physical Review Letters 2001;87(December (27)):278701.

Holme P, Kim BJ, Yoon CN, Han SK. Attack vulnerability of complex networks.Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Interdisci-plinary Topics 2002;65(May (5)):056 109.

Huang XQ, Gao JX, Buldyrev SV, Havlin S, Stanley HE. Robustness of interdependentnetworks under targeted attack. Physical Review E 2011;83(June):065101(R).

Lehmann J, Bernasconi J. Stochastic load-redistribution model for cascading failurepropagation. Physical Review E 2010;81(March):031129.

Lin Y-K. Reliability evaluation for an information network with node failure undercost constraint. IEEE Transactions on Systems, Man and Cybernetics, Part A:Systems and Humans 2007;37(March (2)):180–8.

Motter AE. Cascade control and defense in complex networks. Physical ReviewLetters 2004;93(August):098701.

Motter AE, Lai YC. Cascade-based attacks on complex networks. Physical Review E2002;66(December):065102(R).

Network data from Mark Newman's home page, ⟨http://www-personal.umich.edu/�mejn/netdata/⟩, 2006.

Parshani R, Buldyrev SV, Havlin S. Interdependent networks: reducing the couplingstrength leads to a change from a first to second order percolation transition.Physical Review Letters 2010;105(July):048701.

Ricard VS, Martí RC, Bernat CM, Valverde S. Robustness of the European power gridsunder intentional attack. Physical Review E 2008;77(February):026102.

Sandro M, Jesús G-G, Vito L, Moreno Y. Scaling breakdown in flow fluctuations oncomplex networks. Physical Review Letters 2008;100(May (20)):208701.

Shao J, Buldyrev SV, Havlin S, Stanley HE. Cascade of failures in coupled networksystems with multiple support-dependence relations. Physical Review E2011;83(March):036116.

Simonsen I, Buzna L, Peters K, Bornholdt S, Helbing D. Transient dynamicsincreasing network vulnerability to cascading failures. Physical Review Letters2008;100(May):218701.

Vespignani A. The fragility of interdependency. Nature 2010;464(April):984–5.

Fig. 9. Illustration of the comparison of the failed edges between the Internet and BA scale-free networks under two targeted attacks.

Fig. 10. Illustration of the comparison between two targeted attacks in the Internetand BA networks.

J. Wang et al. / Journal of Network and Computer Applications 40 (2014) 97–104 103

Wang WX, Chen GR. Universal robustness characteristic of weighted networksagainst cascading failure. Physical Review E 2008;77(February):026101.

Wang JW, Rong LL. Effect attack on scale-free networks due to cascading failures.Chinese Physics Letters 2008;25(October (10)):3826–9.

Wang JW, Rong LL. Cascade-based attack vulnerability on the US power grid. SafetyScience 2009a;47(December (10)):1332–6.

Wang JW, Rong LL. Robustness of the western United States power grid under edgeattack strategies due to cascading failures. Safety Science 2009b;47(December(10)):1332–6.

Wang JW, Rong LL. A model for cascading failures in scale-free networks with abreakdown probability. Physica A 2009c;388(January):1289–98.

Wang XF, Xu J. Cascading failures in coupled map lattices. Physical Review E2004;70(November):056113.

Wang JW, Rong LL, Zhang L, Zhang ZZ. Attack vulnerability of scale-free networksdue to cascading failures. Physica A 2008;387(September (26)):6671–8.

Wang WX, Yang R, Lai YC. Cascade of elimination and emergence of purecooperation in coevolutionary games on networks. Physical Review E 2010;81(March):035102(R).

Wu J, Deng HZ, Tan YJ, Li Y. A robustness model of complex networks with tunableattack information parameter. Chinese Physics Letters 2007a;24(July (7)):2138–41.

Wu J, Tan YJ, Deng HZ, Zhu DZ. Vulnerability of complex networks underintentional attack with incomplete information. Journal of Physics A2007b;40(March (11)):2665–71.

Wu J, Barahona M, Tan YJ, Deng HZ. Spectral measure of structural robustness incomplex networks. IEEE Transactions on Systems, Man and Cybernetics, Part A:Systems and Humans 2011;99(January):1–9.

Xia YX, Hill DJ. Attack vulnerability of complex communication networks. IEEETransactions on Circuits and Systems II: Express Briefs 2008;55(January (1)):65–9.

Yang R, Wang WX, Lai YC, Chen GR. Optimal weighting scheme for suppressingcascades and traffic congestion in complex networks. Physical Review E2009;79(February):026112.

Zhao L, Park K, Lai YC. Attack vulnerability of scale-free networks due to cascadingbreakdown. Physical Review E 2004;70(September (3)):035101(R).

Zhao L, Park K, Lai YC, Ye N. Tolerance of scale-free networks against attack-inducedcascades. Physical Review E 2005;72(August):025104(R).

J. Wang et al. / Journal of Network and Computer Applications 40 (2014) 97–104104