PHYSICAL REVIEW E 85, 016112 (2012)

Percolation of partially interdependent networks under targeted attack

Gaogao Dong,1,* Jianxi Gao,2,3,† Lixin Tian,1 Ruijin Du,1,4 and Yinghuan He1

1Nonlinear Scientific for Research Center, Faculty of Science, Jiangsu University, Zhenjiang, 212013, China2Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China

3Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA4College of Mathematics Science, Chongqing Normal University, Chongqing, 401331, China

(Received 18 July 2011; revised manuscript received 2 December 2011; published 23 January 2012)

We study a system composed of two partially interdependent networks; when nodes in one network fail, theycause dependent nodes in the other network to also fail. In this paper, the percolation of partially interdependentnetworks under targeted attack is analyzed. We apply a general technique that maps a targeted-attack problemin interdependent networks to a random-attack problem in a transformed pair of interdependent networks. Weillustrate our analytical solutions for two examples: (i) the probability for each node to fail is proportional to itsdegree, and (ii) each node has the same probability to fail in the initial time. We find the following: (i) For anytargeted-attack problem, for the case of weak coupling, the system shows a second order phase transition, and forthe strong coupling, the system shows a first order phase transition. (ii) For any coupling strength, when the highdegree nodes have higher probability to fail, the system becomes more vulnerable. (iii) There exists a criticalcoupling strength, and when the coupling strength is greater than the critical coupling strength, the system showsa first order transition; otherwise, the system shows a second order transition.

DOI: 10.1103/PhysRevE.85.016112 PACS number(s): 89.75.Hc, 64.60.ah, 89.75.Fb

I. INTRODUCTION

Complex networks exist in many different areas in the realworld and have been studied in the past 15 years. However,almost all researchers have been focused on properties of asingle network that does not interact with or depend on othernetworks [1–11]. Such situations rarely, if ever, occur in reality[12–16]. In 2010, Buldyrev et al. [12] developed a theoreticalframework for studying the process of cascading failures infully interdependent networks caused by random initial failureof nodes. Surprisingly, they found a first order percolationtransition and that a broader degree distribution increased thevulnerability of interdependent networks to random failure, incontrast to the behavior of a single network. Recently, fiveimportant generalizations of the basic model [13–20] havebeen proposed sequentially. (i) Parshani et al. [13] presenteda theoretical framework for studying the case of partiallyinterdependent networks. Their findings showed that reducingthe coupling strength lead to a change from a first to secondorder percolation transition. (ii) Because in the real world anetwork is not always attacked randomly, Huang et al. [14]investigated the robustness of fully interdependent networksunder targeted attack. The result implied that interdependentnetworks are difficult to defend. (iii) In real scenarios, theassumption that one node in a network depends only onone node in the other network is not valid. Shao et al.[17] investigated a framework to study the percolation oftwo interdependent networks with multiple support-dependentrelations. (iv) Hu et al. [18] studied the percolation of a pairof coupled networks with both interdependency links andconnectivity links. They found unusual discontinuous changesfrom second order to first order transition as a function of

*gago999@126.com†jianxi.gao@gmail.com

the dependency coupling between the two networks. (v) Inthe real word, for more than two networks coupled together,Gao et al. [15,19,20] proposed a framework to study therobustness of a network of networks (NON). Their resultsshowed that for a treelike Erdos-Renyi (ER) NON the robust-ness decreases with the number of networks and for a looplikeER NON the giant component is independent of the numberof networks. However, for real scenarios, two infrastructuresare always partially coupled together [21,22], such as energyand communications, power stations and transportation, etc.,and they might be attacked intentionally on high degree nodes.Understanding the robustness due to partial interdependencyand under targeted attack is one of the major challenges fordesigning resilient infrastructures.

Here we develop a generalized framework to study the per-colation of partially interdependent networks under targetedattack. We further develop a general technique [14] that mapsthe targeted-attack problem in interdependent networks to therandom-attack problem in a transformed pair of interdependentnetworks. We find the following: (i) For any targeted-attackproblem, for the case of weak coupling, the system showsa second order phase transition, and for strong coupling,the system shows a first order phase transition. (ii) For anycoupling strength, when the high degree nodes have a largerprobability to fail, the system becomes more vulnerable.(iii) There exists a critical coupling strength, and when thecoupling strength is greater than the critical coupling strength,the system shows a first order transition; otherwise, thesystem shows a second order transition. In the followingtwo examples, the critical coupling strength can be explicitlyderived analytically: (i) the probability for each node to failis proportional to its degree, and (ii) each node has the sameprobability to fail in the initial time. Although case (ii) wassolved in Ref. [15], we present here a more general case whereboth interdependent networks are initially attacked randomly.

016112-11539-3755/2012/85(1)/016112(7) ©2012 American Physical Society

DONG, GAO, TIAN, DU, AND HE PHYSICAL REVIEW E 85, 016112 (2012)

II. THE MODEL

In this model, there are two networks, A and B, withthe number of nodes NA and NB , and within each network,the nodes are connected with degree distributions PA(k) andPB(k), respectively. We suppose that the average degree ofnetwork A is a and the average degree of network B is b.In addition, a fraction qA of A nodes depends on the nodesin network B, and a fraction qB of B nodes depends onthe nodes in network A. That is, if node Ai of network A

depends on node Bj of network B and Bj depends on nodeAs of network A, then s = i, which satisfies the no-feedbackcondition [19]. Consequently, when nodes in one network fail,the interdependent nodes in the other network also fail, andwe suppose that only the nodes in the giant component remainfunctional, which leads to further failure in the first network.This dynamic process leads to a cascade of failures. In order tostudy the cascading failure under targeted attack, we apply thegeneral technique that a targeted-attack problem in networkscan be mapped to a random-attack problem [14,23]. A valueWα(ki) is assigned to each node, which presents the probabilitythat a node i with ki links becomes inactive by targeted attack.We focus on the family of functions [24]

Wα(ki) = kαi∑N

i=1 kαi

, −∞ < α < +∞. (1)

When α > 0, nodes with a higher degree are more vulnerableto the targeted attack, while for α < 0, nodes with a lowerdegree have a higher probability to fail. For α = 0, all thenodes in a network have the same probability to fail, which isequivalent to the case of random attack.

Without loss of generality, we begin by studying thegenerating function and the giant component of network A

after a targeted attack, which can be directly applied to networkB. Next we study the generating functions of network A at eachiteration step.

(i) The generating function of network A is defined as

GA0(x) =∑

k

PA(k)xk. (2)

The generating function of the associated branching process isGA1(x) = G′

A0(x)/G′A0(1) [12,13,25,26]. The average degree

of network A is defined as a = k = ∑k PA(k)k.

(ii) We intentionally remove 1 − p1 fraction of nodes fromnetwork A according to Eq. (1) and remove the links betweenthe removed nodes. Thus, we obtain that the generatingfunction of the nodes left in network A is [14,26,27]

GAb(x) =∑

k

Pp1A (k)xk = 1

p1

∑k

PA(k)hkα

1 xk, (3)

where the new degree distribution of the remaining p1 fractionof nodes P

p1A (k) ≡ 1

p1PA(k)hkα

1 , and h1 satisfies

p1 = Gα(h1) ≡∑

k

PA(k)hkα

1 , h1 ≡ G−1α (p1). (4)

The average degree of the remaining nodes in network A inthis step is k(p1) = ∑

k Pp1A (k)k.

(iii) We remove the links between the removal nodes and theremaining nodes. Thus we obtain that the generating functionof the network composed by the remaining nodes is [27]

GAc(x) = GAb(1 − p1 + p1x), (5)

where p1 is the fraction of the original links that connect tothe nodes that remain, which satisfies

p1 = p1NAk(p1)

NAk=

∑k PA(k)khkα

1∑k PA(k)k

. (6)

Then we can find the equivalent network A′ with generatingfunction GA0(x), such that after a fraction 1 − p1 of nodesis randomly removed, the new generating function of nodesleft in A′ is the same as GAc(x). By solving the equationGA0(1 − p1 + p1x) = GAc(x) and Eq. (5), we can get

GA0(x) = GAb

(1 − p1

p1+ p1

p1x

). (7)

And the generating function of the associated branchingprocess GA1(x) = G′

A0(x)/G′A0(1).

(iv) Thus, the targeted-attack problem on network A canbe mapped to the random-attack problem on network A′.For network A, a 1 − p1 fraction of nodes in network A isintentionally removed according to Eq. (1), and the fraction ofnodes that belongs to the giant component is [14,27,28]

pA(p1) = 1 − GA0[1 − p1(1 − fA)], (8)

where fA ≡ fA(p1) satisfies a transcendental equation,

fA = GA1[1 − p1(1 − fA)]. (9)

For network B, a 1 − p2 fraction of nodes in network B isintentionally removed according to Eq. (1), and the fraction ofnodes that belongs to the giant component pB(p2) is similarto Eq. (8), but p1 changes to p2 and A changes to B.

According to the definition of the fraction of nodes thatbelongs to the giant component, we perform the dynamic ofcascading failures as follows: Initially, the 1 − p1 and 1 − p2

fractions of nodes are intentionally removed from networkA and network B, respectively. The remaining fraction ofnetwork A nodes after an initial removal of 1 − p1 is ψ ′

1 = p1,and the remaining fraction of network B nodes after an initialremoval of 1 − p2 is φ′

0 = p2. The remaining functional partof network A contains a fraction ψ1 = ψ ′

1pA(ψ ′1) of network

nodes. Accordingly, for the same reason, the remainingfraction of network B is φ′

1 = p2{1 − qB[1 − pA(ψ ′1)p1]}, and

the fraction of nodes in the giant component of network B

is φ1 = φ′1pB(φ′

1). Then the sequence, ψn and φn, of giantcomponents and the sequence, ψ ′

n and φ′n, of the remaining

fractions of nodes at each stage of the cascading failures areconstructed as follows:

ψ ′1 = p1, ψ1 = ψ ′

1pA(ψ ′1), φ′

0 = p2,

φ′1 = p2{1 − qB[1 − pA(ψ ′

1)p1]}, φ1 = φ′1pB(φ′

1),

ψ ′2 = p1{1 − qA[1 − pB(φ′

1)p2]}, ψ2 = ψ ′2pA(ψ ′

2),

φ′2 = p2{1 − qB[1 − pA(ψ ′

2)p1]}, φ2 = φ′2pB(φ′

2), (10)

· · ·ψ ′

n = p1{1 − qA[1 − pB(φ′n−1)p2]}, ψn = ψ ′

npA(ψ ′n),

φ′n = p2{1 − qB[1 − pA(ψ ′

n)p1]}, φn = φ′npB(φ′

n).

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PERCOLATION OF PARTIALLY INTERDEPENDENT . . . PHYSICAL REVIEW E 85, 016112 (2012)

1 2 3 4 5 60.55

0.6

0.65

0.7

0.75

0.8

n

ψn,

φ n simulation of ψn

simulation of φn

theory of ψn

theory of φn

(a)

0 20 400

0.2

0.4

0.6

0.8

n

ψn

theory of ψn

simulation of ψn

(b)

FIG. 1. (a) The giant component of both networks A and B, ψn

and φn, after time n cascading failures for the case when a = b =3, p1 = 0.8, p2 = 0.9 > pc

2, qA = 0.45, α = 1, and qB = 0.15. Thesimulation results show excellent agreement with the theory, system(10). All estimates are the results of averaging over 40 realizations.(b) The giant component of network A, ψn, after time n cascadingfailures for the case when a = b = 3, p1 = 0.9, qA = 0.65, qB = 0.7,α = 0, and p2 = 0.6726 < pc

2 = 0.673. The simulation results showexcellent agreement with the theory, system (10). In both (a) and (b),NA = NB = 2 × 105.

Figure 1 shows the giant components ψn and φn as func-tions of time step n for different values of a = b, p1,p2, qA, qB , and α. The simulation results show excel-lent agreement with the theory, system (10). Figure 1(a)shows that a finite giant component exists for p2 > pc

2, andFig. 1(b) shows for the case when p2 < pc

2, the two networkscollapse.

Next, we study the steady state of system (10) af-ter the cascading failures, which can be represented byψ ′

n,φ′n at the limit of n → ∞. The limit must satisfy

the equations ψ ′n = ψ ′

n+1,φ′n = φ′

n+1 since eventually theclusters stop fragmenting and the fractions of randomlyremoved nodes at steps n and n + 1 are equal. Denotingψ ′

n = x, φ′n = y, we arrive at a system of two symmetric

equations:

x = p1{1 − qA[1 − pB(y)p2]},(11)

y = p2{ 1 − qB[1 − pA(x)p1]}.

III. ANALYTICAL SOLUTION

In this section we present two examples that can beexplicitly solved analytically: (i) α = 1 and (ii) α = 0 fortwo interdependent ER networks. Case (ii) is similar tothat of Parshani et al. [13] but more general. For theER [29,30] networks, characterized by the Poisson degreedistribution, GA0(x) = GA1(x) = exp[a(x − 1)], GB0(x) =GB1(x) = exp[b(x − 1)].

(i) For the case of α = 1, substituting α = 1 intoEqs. (3)–(7), we obtain that GAb(x), GAc(x), and GA0(x)can be represented by GA0(x), which reflects the mappingfrom a targeted-attack problem to random-attack problem.Then we get GA0(x) = GA1(x) = exp[ah2

1(x − 1)], GB0(y) =GB1(y) = exp[bh2

2(y − 1)]. Thus, from Eq. (9) we obtain

fA =exp[ −ah2

1x(1 − fA)], fB =exp

[ − bh22y(1 − fB)

].

(12)

Substituting Eqs. (8), (9), and (11) into Eqs. (12), by eliminat-ing x and y, we obtain

fA = e−ap1h21(1−fA)[1−qA+p2qA(1−fB )],

(13)fB = e−bp2h

22(1−fB )[1−qB+p1qB (1−fA)].

According to the definition of ψ∞ = pA(x)x and φ∞ =pB(y)y, we obtain the giant component of networks A and B,respectively, at the end of the cascading failure as

ψ∞ = p1(1 − fA)[1 − qA + p2qA(1 − fB)],(14)

φ∞ = p2(1 − fB)[1 − qB + p1qB(1 − fA)].

Solving the Eqs. (13), we obtain fA and fB , and then weobtain ψ∞ and φ∞ by substituting fA and fB into Eqs. (14).

The numerical simulation results of system (14) are shownin Fig. 2. As shown in Fig. 2, for fixed a, b, and qB , thereexists a critical pc

2; when p2 < pc2, φ∞ = 0, and when p2 > pc

2,φ∞ > 0. For the weak coupling case, i.e., when qA is small(qA = 0.1 in Fig. 2), φ∞(pc

2) = 0, which shows a second orderphase transition, and the transition threshold is defined aspII . For strong coupling, i.e., when qA is large (qA = 0.7 inFig. 2), φ∞(pc

2) > 0, which represents a first order percolation

0 0.5 10

0.2

0.4

0.6

p2

φ ∞

qA=0.1

qA=0.7

(a)

0 0.5 10

0.2

0.4

0.6

0.8

1

p2

φ ∞

qA=0.1

qA=0.7

(b)

0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1

1−p2c

1−q A

Second orderCritical lineFirst orderp =1.0

p =0.9

p =0.8

p =0.7

p =0.6

(c)

FIG. 2. (a) The giant component φ∞ of network B as a function of the initial attack on network B, 1 − p2, when p1 = 0.7, a = 3, b = 4,qB = 0.7, and α = 1 for two different qA. (b) The giant component φ∞ of network B as a function of the initial attack on network B, 1 − p2,when p1 = 0.9, a = 3, b = 4, qB = 0.7, and α = 1 for two different qA. For the weak coupling strength (qA = 0.1), the system shows a secondorder phase transition, and for the strong coupling strength (qA = 0.7), the system shows a first order phase transition. From (a) and (b), wefind that the changes in the critical threshold depend on the changes in p1. (c) The coupling strength 1 − qA as a function of 1 − pc

2 for differentvalues of the remaining fraction of nodes after the initial attack on network A, p1, when a = 3, b = 4, qB = 0.7. For each p1, 1 − qA as afunction of 1 − pc

2 is divided into two regions by an open circle. The dash-dotted curve above an open circle represents the second order phasetransition, and the solid curve below the open circle represents the first order phase transition. All the circles are connected to form a criticalline, which represents 1 − qc

A as a function of 1 − pc2. It also shows that qc

A increases as p1 increases.

016112-3

DONG, GAO, TIAN, DU, AND HE PHYSICAL REVIEW E 85, 016112 (2012)

phase transition, and the transition threshold is defined as pI .Figures 2(a) and 2(b) indicate that there exists a critical qc

A,which corresponds to the condition when pI = pII ; whenqA < qc

A, the system shows a second order phase transition,and when qA > qc

A, the system shows a first order phasetransition. Furthermore, Figs. 2(a) and 2(b) indicate that thecritical threshold changes with p1, i.e., qc

A also changes withp1. In Fig. 2(c), we studied by numerical simulation 1 − qA

as a function of 1 − pc2 for different values of p1 when a = 3,

b = 4, qB = 0.7. As shown in Fig. 2(c), for each p1, thereexists a critical qc

A (open circles), which corresponds to thecondition pI = pII . Moreover, qc

A increases as p1 increases,which is represented by the curve connecting the circles inFig. 2(c) and indicates that the two networks become morerobust as qA decreases.

Next, we study the transition threshold pI and pII an-alytically when a = b = k, p1 = p2 = p, qA = qB = q. Inthis case, from Eqs. (13) and (14), we obtain that the giantcomponents of networks A and B at the end of the cascadingfailure ψ∞ = φ∞ satisfy

φ∞ = p(1 − e−kh2φ∞

)[1 − q + pq

(1 − e−kh2φ∞

)], (15)

and f ≡ fA = fB satisfies

f = e−kph2(1−f )[1−q+pq(1−f )], (16)

where h = ln p/k + 1. The condition for the first ordertransition (p = pI ) is

1=f [kpIh2(1 − q) + 2k(pI )2qh2(1 − f )], 0 � f < 1.

(17)

And solving Eq. (16) for f → 1 yields the condition for thesecond order transition (p = pII ),

kpII (1 − q)h2 = 1. (18)

The analysis of Eqs. (17) and (18) shows the first ordertransition at p = pI occurs for networks with strong coupling(q > qc), whereas the second order transition at p = pII

occurs for networks with weak coupling (q < qc). Thisbehavior is shown in Fig. 3, where the solid curves show the

0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

1−pI,1−pII

1−q

Second orderCritcial lineFirst order

k = 10k = 8

k = 5k = 4

k = 3

FIG. 3. The coupling strength 1 − q = 1 − qA = 1 − qB as afunction of the first order and second order phase transition thresholds,1 − pI ,pII for different values of average degree k = a = b whenα = 1. The curve connecting the circles shows the critical line, belowwhich the system shows a first order phase transition and above whichthe system shows a second order phase transition. The simulation ofthe critical line agrees well with the theory, Eq. (20).

case of first order phase transition and the dash-dotted curvesshow the case of second order phase transition. The criticalvalue of qc (and pc) for which the phase transition changesfrom first order to second order is obtained when the conditionsfor both the first and second order transitions are satisfiedsimultaneously. Applying both conditions, Eqs. (17) and (18),we obtain

k

[1 + ln

(1 − qc

2qc

) /k

]2

= 2qc

(1 − qc)2. (19)

Solving Eq. (19), we obtain qc, and then we can get pc by

pc = 1 − qc

2qc

. (20)

(ii) For the case when α = 0, W0 = 1/N , the targeted-attack problem is equivalent to the random-attack problemstudied in Ref. [13]. For the case of two ER [29,30] networkswith average degrees a and b, we can easily get pA(x) =1 − fA, pB(y) = 1 − fB from the Eqs. (8) and (9), and system(11) becomes

x = p1[1 − qA + p2qA(1 − fB)],(21)

y = p2[1 − qB + p1qB(1 − fA)].

According to Eqs. (9) and (21), fA, fB satisfy

fA = e−ap1(1−fA)[1−qA+p2qA(1−fB )],(22)

fB = e−bp2(1−fB )[1−qB+p1qB (1−fA)].

Thus, we obtain the fraction of nodes in the giant componentsof networks A and B at the end of the cascading process,

ψ∞ = p1(1 − fA)[1 − qA + p2qA(1 − fB)],(23)

φ∞ = p2(1 − fB)[1 − qB + p1qB(1 − fA)].

Our framework is equivalent to Ref. [13] when p2 = 1. Indetail, when p2 = 1, Eqs. (22) are the same as Eqs. (7)in Ref. [13]. Here we study the more general case wherep2 < 1.

For the case α = 0, numerical simulation results of system(23) are shown in Fig. 4. For given a = b, qB , p1, there existsa critical pc

2; when p2 < pc2, φ∞ = 0, and when p2 > pc

2,φ∞ > 0. For weak coupling, i.e., when qA is small (qA = 0.1in Fig. 4), φ∞(pc

2) = 0, representing a second order phasetransition, and the percolation threshold is defined as pII .For strong coupling, i.e., when qA is large (e.g., qA = 0.65in Fig. 4), φ∞(pc

2) > 0, representing a first order phasetransition, and the percolation threshold is defined as pI .Similar to the case when α = 1, Fig. 4 indicates that thereexists a critical qc

A, which corresponds to the condition whenpI = pII . When qA < qc

A, the system shows a second orderphase transition, and when qA > qc

A, the system shows a firstorder phase transition. Furthermore, Fig. 4 indicates that thecritical threshold changes with p1, i.e., qc

A also changes withp1. In Fig. 4(c), we investigate, using numerical calculations,1 − qA as a function of 1 − pc

2 for different values of p1 whena = b = 3, qB = 0.7. As shown in Fig. 4(c), for each p1, thereexists a critical qc

A (open circles), which corresponds to thecondition pI = pII . Moreover, qc

A increases as p1 increases,which is represented by the curve with circles in Fig. 4(c) and

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PERCOLATION OF PARTIALLY INTERDEPENDENT . . . PHYSICAL REVIEW E 85, 016112 (2012)

0 0.5 10

0.2

0.4

0.6

0.8

p2

φ ∞

qA=0.1

qA=0.65

(a)

0 0.5 10

0.2

0.4

0.6

0.8

p2

φ ∞

qA=0.1

qA=0.65

(b)

0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1

1−p2c

1−q A

Second orderCritical lineFirst order

p =0.5

p =0.6

p =0.7

p =1

p =0.8

p =0.9

(c)

FIG. 4. (a) The giant component φ∞ of network B as a function of the initial attack on network B, 1 − p2, when p1 = 0.7, a = b = 3,qB = 0.7, and α = 0 for two different qA. (b) The giant component φ∞ of network B as a function of the initial attack on network B, 1 − p2,when p1 = 0.9, a = b = 3, qB = 0.7, and α = 0 for two different qA. For the weak coupling strength (qA = 0.1), the system shows a secondorder phase transition, and for the strong coupling strength (qA = 0.65), the system shows a first order phase transition. (c) The couplingstrength 1 − qA as a function of 1 − pc

2 for different values of the remaining fraction of nodes after the initial attack on network A, p1, whena = b = 3, qB = 0.7. For each p1, 1 − qA as a function of 1 − pc

2 is divided into two regions by open circles. The dash-dotted curve above anopen circle represents the second order phase transition, and the solid curve below an open circle represents the first order phase transition. Allthe circles are connected to form a critical line, which represents 1 − qc

A as a function of 1 − pc2. It also shows that qc

A increases as p1 increases.

indicates that the two networks become more robust as qA

decreases.By substituting a = b = k, p1 = p2 = p, qA = qB = q

into Eqs. (22) and (23), we obtain that the giant components ofnetworks A and B at the end of the cascading failure ψ∞ = φ∞satisfy

φ∞ = p(1 − e−kφ∞ )[1 − q + pq(1 − e−kφ∞ )], (24)

and f ≡ fA = fB satisfies

f = ekp(f −1)[1−q+pq(1−f )], 0 � f < 1. (25)

Thus we obtain the condition for the first order transition(p = pI ),

1 = f [kpI (1 − q) + 2k(pI )2q(1 − f )]. (26)

Solving Eq. (25) for f → 1 yields the condition for the secondorder transition (p = pII ),

kpII (1 − q) = 1. (27)

Similar to the case of α = 1, the analysis of Eqs. (26) and(27) shows that the first order transition at p = pI occurs for

0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

1−pI,1−pII

1−q

Second orderCritical lineFirst order

k = 3

k = 4

k = 5

k = 8

k = 10

FIG. 5. The coupling strength 1 − q = 1 − qA = 1 − qB as afunction of the first order and second order phase transition thresholds,1 − pI ,1 − pII , for different values of average degree k = a = b

when α = 0. The curve with circles shows the critical line, belowwhich the system shows a first order phase transition, and above whichthe system shows a second order phase transition. The simulation ofthe critical line agrees well with the theory, Eq. (28).

networks with strong coupling (q > qc), whereas the secondorder transition at p = pII occurs for networks with weakcoupling (q < qc). This behavior is shown in Fig. 5, wherethe solid curves show the case of first order phase transition,and the dashed-dotted curves show the case of second orderphase transition. The critical values of qc (and pc) for whichthe phase transition changes from first order to second orderare obtained when the conditions for both the first and secondorder transitions are satisfied simultaneously. Applying bothconditions Eqs. (26) and (27), we obtain

qc = k + 1 − √2k + 1

k, pc =

√2k + 1 + 1

2k. (28)

IV. NUMERICAL SOLUTIONS OF THEGENERAL CASE

Our theoretical study can be applied to any case of α. Inthis section, we investigate the solutions for the general casesof α. Figure 6 shows the giant component φ∞ of networkB as a function of the initial attack on network B, 1 − p2

0 0.5 10

0.2

0.4

0.6

0.8

p2

φ ∞

qA=0.1

qA=0.8

(a)

0 0.5 10

0.2

0.4

0.6

0.8

1

p2

φ ∞

qA=0.1

qA=0.8

(b)

FIG. 6. (a) The giant component φ∞ of network B as a function ofthe initial attack on network B, 1 − p2, when p1 = 0.8, a = 3, b = 4,qB = 0.7, and α = 2 for two different qA. (b) The giant component φ∞of network B as a function of the initial attack on network B, 1 − p2,when p1 = 0.8, a = 3, b = 4, qB = 0.7, and α = −1 for two differentqA. For weak coupling strength (qA = 0.1), the system shows a secondorder phase transition, and for strong coupling strength (qA = 0.8),the system shows a first order phase transition.

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DONG, GAO, TIAN, DU, AND HE PHYSICAL REVIEW E 85, 016112 (2012)

0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1−p2c

1−q A

Second orderCritical lineFirst order

α=2

α=−3

α=1

α=0

α=−0.5α=−2

α=−1

FIG. 7. The coupling strength 1 − qA as a function of 1 − pc2 for

different values of α when a = 3, b = 4, qB = 0.7, and p1 = 0.8.The curve connecting the circles shows the critical line, below whichthe system shows a first order phase transition and above which thesystem shows a second order phase transition.

for α = 2 [Fig. 6(a)] and α = −1 [Fig. 6(b)]. For given a,b, qB , p1, there exists a critical pc

2; when p2 < pc2, φ∞ = 0,

and when p2 > pc2, φ∞ > 0. For weak coupling, i.e., when

qA is small (qA = 0.1 in Fig. 6), φ∞(pc2) = 0, which shows a

second order phase transition. For strong coupling, i.e., whenqA is large (qA = 0.8 in Fig. 6), φ∞(pc

2) > 0, which shows afirst order phase transition. Figure 6 indicates that there existsa critical qc

A; when qA < qcA, it shows a second order phase

transition, and when qA > qcA, the system shows a first order

phase transition. Furthermore, Fig. 6 indicates that the criticalthreshold changes with α, i.e., qc

A also changes with α.In Fig. 7, we investigate the numerical simulation of 1 −

qA as a function of 1 − pc2 for different values of α when

a = 3, b = 4, p1 = 0.8, qB = 0.7. As shown in Fig. 7, foreach α, there exists a critical qc

A (circles). Moreover, 1 − qcA

increases as α increases, which is represented by the curve

connecting the circles in Fig. 7 and indicates that the twonetworks becomes more robust as α decreases.

V. CONCLUSIONS

In summary, we developed a framework for studying thepercolation of two partially interdependent ER networks undertargeted attack for the cases of a high degrees of attack, α = 1,and a random attack, α = 0. For any value of α, the systemshows a second order phase transition when q is small, and afirst order phase transition when q is large. We find the criticalqc and critical threshold pc when the percolation of the systemchanges from first to second order for the cases when α = 1and α = 0. Moreover, we find that when α increases, i.e., thehigh degree nodes have a larger probability to fail, the systembecomes more vulnerable.

ACKNOWLEDGMENTS

L.T. was supported by the National Natural ScienceFoundation of China (Grants No. 91010011, No. 71073072,and No. 51007032), the Natural Science Foundation of JiangsuProvince (Grant No. 2007098), and the National NaturalScience (Youth) Foundation of China (Grant No. 10801140).G.D. acknowledges support from the China Scholarship Fund(Grant No. 2011832326). R.D. was supported by the GraduateInnovative Foundation of Jiangsu Province CX10B 272Zand the Youth Foundation of Chongqing Normal University(Grant No. 10XLQ001). J.G. thanks the doctoral visitingscholar program of SJTU, the Shanghai Key Basic ResearchProject (Grant No. 09JC1408000), and the National NaturalScience Foundation of China (Grant No. 61004088) forsupport.

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