k-core percolation on interdependent and interconnected multiplex networks
aa r X i v : . [ phy s i c s . s o c - ph ] J a n epl draft k -core percolation on interdependent and interconnected multi-plex networks Kexian Zheng , Ying Liu , Yang Wang and Wei Wang
School of Computer Science, Southwest Petroleum University, Chengdu , China Cybersecurity Research Institute, Sichuan University, Chengdu , China
PACS – Structures and organization in complex systems
PACS – Dynamics of social systems
PACS – Complex systems
Abstract – Many real-world networks are coupled together to maintain their normal functions.Here we study the robustness of multiplex networks with interdependent and interconnected linksunder k -core percolation, where a node fails when it connects to a threshold of less than k neigh-bors. By deriving the self-consistency equations, we solve the key quantities of interests such asthe critical threshold and size of the giant component analytically and validate the theoreticalresults with numerical simulations. We find a rich phase transition phenomenon as we tune theinter-layer coupling strength. Specifically speaking, in the ER-ER multiplex networks, with theincrease of coupling strength, the size of the giant component in each layer first undergoes afirst-order transition and then a second-order transition and finally a first-order transition. Thisis due to the nature of inter-layer links with both connectivity and dependency simultaneously.The system is more robust if the dependency on the initial robust network is strong and morevulnerable if the dependency on the initial attacked network is strong. These effects are evenamplified in the cascading process. When applying our model to the SF-SF multiplex networks,the type of transition changes. The system undergoes a first-order phase transition first only whenthe two layers’ mutually coupling is very strong and a second-order transition in other conditions. Introduction. –
Most real-world networks are notisolated but are coupled with other systems to implementtheir functions well, which can be described by the multi-layer network [1–6]. A typical example is the transporta-tion network composed of airlines and railways, where dif-ferent transportation ways support each other to maintainthe traffic flow of people [7, 8]. Another example is theinterdependent communication and power grid network,where a failure of a tiny fraction of nodes in one networkinduces the block down of the whole system [9]. The mul-tilayer network approach has proved to be very successfulin many fields of network science, such as resilience androbustness [10], spreading dynamics [11–16], synchroniza-tion [17, 18].Real complex systems are frequently under random orintentional attacks, such as natural disasters of hurricanes,earthquakes, power outages, Internet router failure, and (a)
E-mail: [email protected] (b)
E-mail: [email protected] terrorist attacks [19, 20]. These damages may cruciallychange or even destroy the structure and function of thenetwork. Understanding the cascading induced by an ini-tial failure is a critical question in the study of complexsystems. Researches on network robustness is an activetopic in network science [5, 21–24], where the robustnessquantifies how resilient the network is under perturba-tions [25]. In studying the robustness of networks, theordinary percolation model is usually used [22]. In the or-dinary percolation approach, initially, a fraction of 1 − p of nodes are removed, which may disconnect more nodesfrom the largest connected components. If 1 − p is largeenough, then at a critical value p c , the whole network col-lapses, and there are only negligible small clusters and iso-lated nodes. The size of the largest connected componentin the remaining network is served as the order parameterthat measures the robustness of the network structure andis used for studying the phase transition behaviors [26]. Anatural generalization of the ordinary percolation is thep-1exian Zheng et al. k -core percolation [27]. The k -core is a highly intercon-nected part of the network, with each node in the k -corehas at least k neighbors [28]. It is obtained by removingthe nodes with less than k neighbors. If there appear newnodes with degree less than k , keep removing them untilno further removal is possible. The k -core percolation im-plies the emergence of a giant k -core at a threshold of aproportion of nodes removed at random, which displays anentirely different phase transition from the ordinary per-colation [29]. These methods are generalized to study therobustness of multilayer networks [9, 30, 31].Studies on the robustness of multilayer networks focuson two types of networks distinguished by their inter-layercoupling nature, which are the interdependent networksand the interconnected networks. In the interdependentnetworks, the presence of a node in one layer dependson a node’s presence in the other layers. The removalof nodes on one layer leads to isolated nodes or small clus-ters in the same layer and leads to the removal of theirdependent nodes in the other layers. Extensive studieshave found that interdependency among layers makes thesystem more vulnerable [9, 32–37]. While in the inter-connected networks, the connectivity links coupling thetwo layers provide an additional connection for the nodes,making the system significantly more robust [38–40]. Be-sides, scientists studied the coupled networks with bothinterdependent and interconnected links, where the twokinds of links have competing effects of either decreasingor increasing the robustness of the system [41,42]. Under-standing how the interdependency and interconnectivityimpact the system’s robustness is the main challenge indesigning resilient infrastructures.In some real-world scenarios, the coupling of networksmay be interdependent as well as interconnected simulta-neously. For example, in the social networks, an individualmay have connections in different social media correspond-ing to other layers. To be active in the network, the ac-tor should have a certain amount of effective connectionswithin a layer or have connections in different layers. Theindividual’s failure in one network leads to the failure ofthe individual in the other network because it is the sameactor. In this case, the coupling of different social networksis interdependent and interconnected simultaneously. An-other example is the coupled financial systems, where in-dividual banks are considered nodes. Their economic in-terplays such as credit, derivatives, foreign exchange, andsecurities are represented by links and grouped into lay-ers by different types [43]. A bank’s business on one layermay support this bank’s business in another layer due toits reputation in the first business. Still, the failure in onebusiness may lead to the bankruptcy of the financial orga-nization, and thus its activities in all layers are disabled.In this paper, we study the robustness of coupled net-works in which the inter-layer links have a nature of bothdependency and connectivity simultaneously. Using the k -core percolation approach, we analyze the phase tran-sition under random attacks, which agrees well with the simulation results. We find that in the ER-ER multiplexnetworks (i.e., each subnetwork is an Erd¨os-R´enyi (ER)network), with the increase of the asymmetrical couplingstrength q a and q b between two layers, the interdependentand interconnected network first undergoes a first-orderphase transition, and then a seconde order transition andfinally a first-order transition again. Meanwhile, as theinter-layer links have a competing effect of either increas-ing the robustness due to the interconnectivity nature ordecreasing the robustness due to the interdependency na-ture, we find that in a relatively large parameter ranges,the coupled network are more vulnerable when the cou-pling strength becomes strong, which means the depen-dency of links dominates in the cascading process. Specif-ically speaking, when the coupling strength q a is fixed, therobustness decreases with q b . While q b is fixed, there aretwo stages. When q b is within 0 .
6, the robustness increasewith q a , which means the connectivity of inter-layer linksdominates in the cascading process. If q b is above 0 . q a . Finally, westudy the coupled SF-SF multiplex networks (i.e., eachsubnetwork is scale-free (SF) network) and find that thephase transition phenomena are different from those of thecoupled ER-ER networks.The rest of the paper is organized as follows. In themodel section, we describe the cascading model in the in-terdependent and interconnected networks. In the theoryanalysis section, we analyze the k -core percolation processin the coupled networks and derive the percolation thresh-old. In the simulation results section we demonstrate thesimulation results and analyze the phase transition in thecoupled ER-ER networks and SF-SF networks. A conclu-sion is given finally. The model. –
Consider a system composed of twouncorrelated random networks A and B with the samenumber of nodes N with distribution P ( k a ) and P ( k b ) re-spectively. A fraction q a of nodes in layer A depend onand are connected by nodes in layer B, which means fora node i in layer A when its dependent node in layer B isfunctional, it provides one connection (one degree) to node i , but if the dependent node in layer B fails, node i in layerA also fails. Thus the inter-layer link can be considered asdirectional with both dependency and connectivity. Simi-larly, a fraction q b of nodes in layer B depend on and areconnected by nodes in layer A. We assume that each nodein a layer depends on and is connected by at most onenode in the other layer. q a and q b determines the couplingstrength.Initially, a fraction 1 − p of randomly chosen nodes innetwork A are removed, along with their edges. The nodesthat are dependent on the removed nodes in layer B arealso removed. In the classical k -core percolation, nodes inlayer A with fewer neighbors than φ a , the local threshold,are removed, along with all the nodes in layer B dependenton them. Then in layer B, nodes with fewer neighbors than φ b are removed and the nodes in layer A are dependent onp-2 -core percolation on interdependent and interconnected multiplex networks Fig. 1: (Color online) Schematic representation of the cascad-ing model. The dashed lines represent the inter-layer linksdirected from a node in one network to a node that dependson it. (a) Initially, the grey node is attacked, and the red nodethat is dependent on it is also removed. (b) k -core pruningin network A, where the yellow node with neighbors few than φ a is removed. If there are nodes in network B dependent onthe yellow nodes, also remove them. (c) k -core pruning in net-work B, where the yellow node with neighbors few than φ b areremoved. (d) The red node in network A that is dependenton the yellow nodes in network B is removed. Here φ a = 3, φ b = 3. them. This cascade process continues until a steady-stateis reached. The system is either completely fragmentedor a mutually connected giant k -core appears, where k =( φ a , φ b ) [30]. Note that in the process of obtaining the k -core, a node i in layer A that is dependent and connectedby a node in layer B needs only ( φ a −
1) neighbors tobe in the k -core, because the inter-layer link provides aconnection to i . If a node in layer A has no inter-layerlink, then it must have at least φ a neighbors within layerA to be in the k -core. It is similar for nodes in layer B.The cascading process is demonstrated in Fig. 1. Theoretical Analysis. –
In this section, we studythe size of the giant k -core connected cluster and phasetransition in the final sate. When the system reaches thefinal state in the cascading process, let f a be the probabil-ity that a given end of a randomly selected edge is the rootof an infinite ( φ a − φ a − φ a − φ a − φ b -coreof network B. Similarly, f b is the probability that a givenend of a randomly selected edge is the root of an infinite( φ b − f a can be expressed as f a = p [(1 − q a ) ∞ X k a = φ a − P ( k a +1)( k a +1) h k a i k a X j = φ a − C jk a f ja (1 − f a ) k a − j + q a ∞ X k a = φ a − P ( k a +1)( k a +1) h k a i k a X j = φ a − C jk a f ja (1 − f a ) k a − j ψ b ( f b )] , (1) where p is the probability that a node is not removed ini-tially, and C jk a = k a ! / ( k a − j )! j !. The term P ( k a + 1)( k a +1) / h k a i is the probability that an end of an randomly cho-sen edge has k a out-going edges except the chosen edge,and C jk a f ja (1 − f a ) k a − j is the probability that if a givenend of an edge has k a children, then exactly j of themare the roots of ( φ a − − q a ), where these nodes should have noless than φ a neighbors within network A. The second typeis the nodes with inter-layer links, corresponding to theterm with coefficient q a , where these nodes should haveno less than φ a − φ b -core of network B. The term ψ b ( f b )is the probability of a node in the φ b -core of network B inthe steady state, where it is expressed as ψ b ( f b ) = (1 − q b ) ∞ X k b = φ b P ( k b ) k b X j = φ b C jk b f jb (1 − f b ) k b − j + q b ∞ X k b = φ b − P ( k b ) k b X k b = φ b − C jk b f jb (1 − f b ) k b − j . (2) In Eq. (2), the two terms on the right respectively rep-resent the probability of nodes without or with inter-layerlinks in the φ b -core in network B.Similarly, we can obtain the equation of f b as f b =(1 − q b ) ∞ X k b = φ b − P ( k b +1)( k b +1) h k b i k b X j = φ b − C jk b f jb (1 − f b ) k b − j + pq b ∞ X k b = φ b − P ( k b +1)( k b +1) h k b i k b X j = φ b − C jk b f jb (1 − f b ) k b − j ψ a ( f a )(3) In the second term on the right of the equation, themultiplier p represents the probability that an end of theedge is occupied (not removed initially). ψ a ( f a ) is theprobability that a randomly chosen node belongs to the φ a -core in network A, which is ψ a ( f a ) = (1 − q a ) ∞ X k a = φ a P ( k a ) k a X j = φ a C jk a f ja (1 − f a ) k a − j + q a ∞ X k a = φ a − P ( k a ) k a X k a = φ a − C jk a f ja (1 − f a ) k a − j . (4) For any given value of p , the f a and f b can be solvedfrom Eqs. (1) and (3) using the Newton’s method [35] aftergiving appropriate initial values.p-3exian Zheng et al. fb f a fa=fa(fb)fb=fb(fa) fb f a fa=fa(fb)fb=fb(fa) (b)(a) p=0.515 p=0.8 Fig. 2: Graphical solution of Eqs. (1) and (3). (a) At p = p c =0 . f a ( f b ) and f b ( f a ) tangentially touch eachother at the critical point. (b) When p > p c , the two curveshave two intersections and the larger one corresponds to thesolution of Eqs. (1) and (3) where the giant k -core exists. Theother parameters are set as φ a = φ b = 3, q a = q b = 0 .
8, andthe mean degree h k a i = h k b i = 7. We denote M Aφ a ( p ) and M Bφ b ( p ) as the probability of arandomly chosen node in network A or B belongs to themutually connected ( φ a , φ b )-core, which satisfy M Aφ a ( p ) = p [(1 − q a ) ∞ X k a = φ a P ( k a ) k a X j = φ a C jk a f ja (1 − f a ) k a − j + q a ∞ X k a = φ a − P ( k a ) k a X j = φ a − C jk a f ja (1 − f a ) k a − j ψ b ( f b )] , (5) and M Bφ b ( p ) = (1 − q b ) ∞ X k b = φ b P ( k b ) k b X j = φ b C jk b f jb (1 − f b ) k b − j + pq b ∞ X k b = φ b − P ( k b ) k b X j = φ b − C jk b f jb (1 − f b ) k b − j ψ a ( f a ) . (6) The solutions of Eqs. (1) and (3) can be graphically rep-resented on a f a , f b plane [32]. For small values of p , theEqs. (1) and (3) has the trivial solution of f a = f b = 0,which implies the absence of k -core in the system. As p increases, at a critical value p = p c , the mutually con-nected giant k -core appears. In this case, the two curves f a = f a ( f b ) and f b = f b ( f a ) tangentially touch each otherat a point, and meet the condition df a df b · df b df a = 1 , (7)which implies a first-order transition at the touchingpoint as shown in Fig. 2 (a). When p > p c , shown inFig. 2 (b), the two curves will always have non-zero in-tersections. The larger value of f a and f b is the physicalsolutions of Eqs. (1) and (3) where the giant k -core ex-ists. For the smaller one, as it is under the value at thethe threshold, there is no giant k -core and this solution isphysically meaningless. Simulation results. – p N O I p M φ * * ( p ) A B 0 0.2 0.4 0.6 0.8 1 p p (c) (d)(b)(a) q a =0.2 q a =0.5 Fig. 3: Size of the k -core of network A M Aφ a ( p ) and networkB M Bφ b ( p ) and the number of iterations in obtaining f a and f b as a function of p . (a) Size of k -core as a function of p,where q a = 0 . q b = 0 .
5. (b) Size of k -core as a functionof p , where q a = 0 . q b = 0 .
5. (c) NOI as a function of p , where q a = 0 . q b = 0 .
5. (d) NOI as a function of p ,where q a = 0 . q b = 0 .
5. The solid lines are theoreticalpredictions and the symbols are simulation results, which agreevery well.
ER-ER multiplex networks.
We construct the inter-dependent and interconnected ER-ER multiplex networks,where the network size is N a = N b = 10000 and the de-gree distribution of each network is P ( k ) = e − λ λ k k ! , and λ a = λ b = 7 is the average degree. We set φ a = φ b = 3as the local threshold to obtain k -core. The k -core per-colation process starts by randomly removing a fraction1 − p of nodes in network A, and the pruning process con-tinues until the steady state is reached. Fig. 3 shows thegiant connected cluster size of the k -core of network Aand network B, and the number of iterations (NOI) in ob-taining f a and f b as a function of p . It can be seen fromFigs.3 (a) and (b) that when the coupling strength is weak,for q a = 0 .
2, both M Aφ a ( p ) and M Bφ b ( p ) undergo a discon-tinuous first-order transition. As the coupling strengthbecomes strong, for q a = 0 .
5, the transition becomes acontinuous second-order transition. Increasing the num-ber of inter-layer links makes the system more robust andmore controllable. The number of iterations in obtaining f a and f b from Eqs. (1) and (3) reaches a maximum at thepercolation threshold p c , as shown in Figs. 3 (c) and (d).We demonstrate the phase diagram of the percolationin network A as a function of the coupling strength q a and q b in Fig. 4. The two types of phase transition isseparated by the black lines. To determine the boundaryof the first-order and second-order transition, we use thefollowing method [42]. Consider p increases from 0 to 1.p-4 -core percolation on interdependent and interconnected multiplex networks Fig. 4: Phase diagram of p c as a function of q a and q b . Theblack line separates the region of first-order transition andsecond-order transition. At the percolation threshold p = p c , there is a first-orderor second-order transition. Define F a ( p, f a , f b , ψ b ) as theright side of Eq. (1) and F b ( p, f a , f b , ψ a ) as the right sideof Eq. (3). When p is approaching the second-order tran-sition threshold p II c , f a , f b , ψ a , ψ b →
0. Do the Taylorexpansion of F a , we obtain f a = F ′ a ( p II c , , , f a + 12 F ′′ a ( p II c , , , f a + O ( f a ) . (8)Dividing f a on both sides of Eq. (8) comes to1 = F ′ a ( p II c , , ,
0) + 12 F ′′ a ( p II c , , , f a + O ( f a ) . (9)As f a →
0, we neglect the second and third terms on theright side of Eq. 9 and get 1 = F ′ a ( p II c , , , p II c can be calcu-lated. As for the case when p is approaching the first-ordertransition threshold p I c , there is a jump for f a , f b , ψ a , ψ b from 0 to non-zero values. The non-trivial solutions f a c and f b c at this point satisfies ∂F a ( p, f a , f b , ψ b ) ∂f a | f a = f ac ,f b = f bc ,p = p I c = 1 . (10)and ∂F b ( p, f a , f b , ψ b ) ∂f b | f a = f ac ,f b = f bc ,p = p I c = 1 . (11)Combing Eqs. (1)-(4) and (10)-(11), the threshold for first-order transition p I c can be solved. At the boundary, p I c = p II c = p c , where p c satisfies both conditions for determiningthe first-order threshold and second-order threshold. Take1 = F ′ a ( p c , , ,
0) into Eq. (8), we obtain12 F ′′ a ( p c , , , f a + O ( f a ) = 0 . (12)As f a and f b has non-trivial solution at the first-ordertransition threshold, then F ′′ a ( p c , , ,
0) = 0 . (13)By solving Eq. 13, the boundary of the first-orderthreshold and second-order threshold can be obtained.From Fig. 4 it can be seen that when q a < . p c increases with q b and there is only first-order transition.When q a > . p c also increases with q b , and the phasetransition is a second-order then followed by a first-ordertransition. As for q b , when it is smaller than 0 .
6, with theincrease of q a , the p c decreases, which reflects an increasedsystem robustness. When q b > .
6, with the increase of q a , p c increases first and then decrease a little bit. The abovephenomena can be explained as follows. For a fixed q a , theincrease of q b means stronger dependency and connectivityof network B on network A. As network A is initially un-der random attack, the larger dependency, the more nodesin network B will be removed at the first step. This willcause larger removed nodes in network A and B in thefollowing cascading process, thus making the system morevulnerable. While for a fixed q b not too large, with the in-crease of q a , the more nodes in network A are dependenton and connected by nodes from network B. As initially allnodes in network B are functional, the increasing depen-dency and connectivity on network B makes more nodesin network A receive one more connection from network Bthus making network A more robust, which also increasethe robustness of the system. We can consider this simplyas increasing dependency on the initially attacked networkA makes the system more vulnerable, while increasing de-pendency on the network B makes system more robust.We compare the phase transitions under different com-binations of coupling strength q a and q b , as shown inFig. 5. In Figs. 5 (a) and (b), it can be seen that in-creasing the coupling strength q a reduces the percolationthreshold p c and makes the network more robust when q b is relatively small. This is because network B is weaklyimpacted by network A, and more nodes in network A areconnected by inter-layer links from B with the increaseof q a . In this case the network B helps to maintain therobustness of network A. But if network B is impactedstrongly by network A, corresponding to the large q b asshown in (c) and (d), the damaged network B further re-duces the robustness of network A as q a increases. If wecompare Figs. 5 (a) and (c), as well as (b) and (d), we canfind that no matter how large q a is, the system becomesmore vulnerable when the dependency of network B onnetwork A q b increases. These results are consistent withour analysis on Fig. 4.Finally we focus on the case of symmetrical couplingof q a = q b = q on the robustness of network A. FromFig. 6 (a), we can see that when q increases from 0 to 0 . p c decreases, and the network A undergoesa first-order transition first and then becomes a second-order transition. The increase of inter-layer links boostthe robustness of the system. When q increases from 0 . .
0, the threshold p c begin to increase, and the phasep-5exian Zheng et al. p M φ * * ( p ) q a = 0.2 ,q b = 0.2 A B 0 0.2 0.4 0.6 0.8 1 p q a = 0.8 ,q b = 0.2 A B0 0.2 0.4 0.6 0.8 1 p M φ * * ( p ) q a = 0.2 ,q b = 1 A B 0 0.2 0.4 0.6 0.8 1 p q a = 0.8 ,q b = 1 A B (a) (b)(c) (d)
Fig. 5: Size of the giant connected component of k -core innetwork A and network B as a function of p . (a) NetworkA undergoes a first-order transition at q a = 0 . q b = 0 . p c = 0 .
42. (b) Network A undergoes a second-order transitionat q a = 0 . q b = 0 . p c = 0 .
16 (c) Both networks undergo afirst-order transition at q a = 0 . q b = 1 . p c = 0 .
54. (d) Bothnetworks undergo a first-order transition at q a = 0 . q b = 1 . p c = 0 .
57. The solid lines are theoretical predictions and thesymbols are simulation results.Fig. 6: Size of the giant connected component of k -core innetwork A (a) and network B (b) as a function of removedfraction p and the symmetrical coupling strength q . In networkA, the sequence of phase transition is first-order, second-orderand first-order with the increase of q . While in network B, thephase transition is second-order and then first-order. (c) p c asa function of q . There is an optimal q o at which the network isthe most robustness. The p c is obtained by simulations fromEqs.(1)-(6). transition changes to the first-order transition again. Theincrease of inter-layer links reduce the robustness of thesystem when the coupling becomes strong. As for networkB, the increase of q makes the transition changes fromsecond-order to first-order, which implies the dependencyon the initial attacked network makes the system morevulnerable. There is an optimal coupling strength q o , atwhich p c is the smallest and the network A is the mostrobustness. This implies a mediate coupling strength bestbalance the effects of dependency and connectivity. Fig. 7: Phase diagram of p c as a function of q a and q b fornetwork A. The black line separates the region of the first-order transition and second-order transition. In most of thediagrams, network A undergoes a first-order transition. Whenboth q a and q b are above 0 .
95, the network undergoes a first-order transition. N a = N b = 10000, k min = 2, k max = 50, λ = 2 . φ a = φ b = 3. SF-SF multiplex networks.
Next we construct twonetworks that both follow the power law degree distri-bution p ( k ) ∼ k − λ , and connect them by inter-layer links,where λ is the degree exponent. Fig.7 shows the phase di-agram of k-core percolation transition of network A. It canbe seen that the percolation under most of the couplingstrength q a and q b is second-order, which is quite differentfrom that of the ER networks. This is because in the ERnetworks, a large number of nodes have a degree aroundthe mean degree thus are above the local threshold. Whenthere are inter-layer links, the nodes are much impactedby the interdependency and interconnectivity. For the ERnetworks, the strong dependency on the initially robustnetwork B, corresponding to large q a and small q b , makesthe system undergo a second-order transition, comparedto that of the first-order transition in the single ER net-works [35]. While for the SF networks, a large number ofnodes have a small degree and will be pruned in the k -corepercolation process. Thus the system is less impacted bythe inter-layer coupling. The coupled network undergoesa second-order transition, which is similar to that of thesingle SF networks, corresponding to q a = q b = 0. Whenthe coupling is very strong, corresponding to large q a and q b , the system undergoes a first-order transition. Conclusion. –
In this work we have studied the ro-bustness of the coupled networks by k -core percolation,where the inter-layer links with the nature of interdepen-dency and interconnectivity either increase or decrease therobustness of the system under different coupling strength.As the coupling is directed, the interdependency and in-terconnectivity have competing effects in impacting therobustness. We find that the strong dependency on thep-6 -core percolation on interdependent and interconnected multiplex networksinitial attacked network leads to a more vulnerable cou-pled system, while the strong dependency on the initialrobust network leads to a more robust system. When in-creasing the mutual coupling strength of both layers, thesystem undergoes a first-order transition first, and thena second-order transition, followed by a first-order tran-sition. There is an optimal coupling strength where thepercolation threshold is the lowest, and the system is themost robust. When applied our model on the coupled SFnetworks, the phase transition is different from that of thecoupled ER due to the local structural characteristics ofthe networks. As the SF networks are robust under ran-dom attack, using the SF networks as the coupled networkbetter increases the robustness of an initially attacked ERnetwork under some coupling strength. ∗ ∗ ∗ This work is supported by the National NaturalScience Foundation of China (Nos. 61802321 and61903266), Sichuan Science and Technology Program(Nos. 2020YJ0125 and 2020YJ0048), China PostdoctoralScience Special Foundation (No. 2019T120829), Fun-damental Research Funds for the Central Universities,and the Southwest Petroleum University Innovation Base(No.642).
REFERENCES[1]
Boccaletti S., Bianconi G., Criado R., del GenioC. I., G´omez-Garde˜nes J., Romance M., Sendi´na-Nadal I., Wang Z. and
Zanin M. , Physics Reports , (2014) 1.[2] Kivel¨a M., Arenas A., Barthelemy M., GleesonJ. P., Moreno Y. and
Porter M. A. , Journal of Com-plex Networks , (2014) 203.[3] De Domenico M., Sol´e-Ribalta A., Cozzo E.,Kivel¨a M., Moreno Y., Porter M. A., G´omez S. and
Arenas A. , Physical Review X , (2013) 041022.[4] G´omez S., D´ıaz-Guilera A., G´omez-Garde˜nes J.,P´erez-Vicente C. J., Moreno Y. and
Arenas A. , Physical Review Letters , (2013) 028701.[5] Gao J., Buldyrev S. V., Stanley H. E. and
HavlinS. , Nature Physics , (2012) 40.[6] Gao J., Li D. and
Havlin S. , National Science Review , (2014) 346.[7] Barth´elemy M. , Physics Reports , (2011) 1.[8] Halu A., Mukherjee S. and
Bianconi G. , Physical Re-view E , (2014) 012806.[9] Buldyrev S. V., Parshani R., Paul G., StanleyH. E. and
Havlin S. , Nature , (2010) 1025.[10] Gao J., Buldyrev S. V., Havlin S. and
StanleyH. E. , Physical Review Letters , (2011) 195701.[11] Wang W., Tang M., Yang H., Do Y., Lai Y.-C. and
Lee G. , Scientific Reports , (2014) 5097.[12] Cozzo E., Ba˜nos R. A., Meloni S. and
Moreno Y. , Physical Review E , (2013) 050801.[13] De Domenico M., Granell C., Porter M. A. and
Arenas A. , Nature Physics , (2016) 901. [14] De Arruda G. F., Cozzo E., Peixoto T. P., Ro-drigues F. A. and
Moreno Y. , Physical Review X , (2017) 011014.[15] Jalili M. and
Perc M. , Journal of Complex Networks , (2017) 665.[16] Wang W., Cai M. and
Zheng M. , Physica A: StatisticalMechanics and its Applications , (2018) 121.[17] Del Genio C. I., G´omez-Garde˜nes J., BonamassaI. and
Boccaletti S. , Science Advances , (2016)e1601679.[18] Zhuang J., Cao J., Tang L., Xia Y. and
Perc M. , IEEE Transactions on Systems, Man, and Cybernetics:Systems , (2020) 4807.[19] Huang X., Gao J., Buldyrev S. V., Havlin S. and
Stanley H. E. , Physical Review E , (2011) 065101.[20] Gao J., Barzel B. and
Barab´asi A. L. , Nature , (2016) 307.[21] Albert R., Jeong H. and
Barab´asi A. L. , Nature , (2000) 378.[22] Callaway D. S., Newman M. E. J., Strogatz S. H. and
Watts D. J. , Physical Review Letters , (2000)5468.[23] Wang W.-X. and
Chen G. , Physical Review E , (2008)026101.[24] Yang Y., Nishikawa T. and
Motter A. E. , Science , (2017) 6365.[25] Schneider C. M., Moreira A. A., Andrade J. S.,Havlin S. and
Herrmann H. J. , Proceedings of the Na-tional Academy of Sciences , (2011) 3838.[26] Feng L., Monterola C. P. and
Hu Y. , New Journal ofPhysics , (2015) 063025.[27] Dorogovtsev S. N., Goltsev A. V. and
MendesJ. F. F. , Physical Review Letters , (2006) 040601.[28] Seidman S. B. , Social Networks , (1983) 269.[29] Goltsev A. V., Dorogovtsev S. N. and
MendesJ. F. F. , Physical Review E , (2006) 056101.[30] Azimi-Tafreshi N., G´omez-Garde˜nes J. and
Doro-govtsev S. N. , Physical Review E , (2014) 032816.[31] Radicchi F. , Nature Physics , (2015) 597.[32] Parshani R., Buldyrev S. V. and
Havlin S. , PhysicalReview Letters , (2010) 048701.[33] Gao J., Buldyrev S. V., Havlin S. and
StanleyH. E. , Physical Review Letters , (2011) 195701.[34] Baxter G. J., Dorogovtsev S. N., Goltsev A. V. and
Mendes J. F. F. , Physical Review Letters , (2012) 248701.[35] Yuan X., Dai Y., Stanley H. E. and
Havlin S. , Phys-ical Review E , (2016) 062302.[36] Panduranga N. K., Gao J., Yuan X., Stanley H. E. and
Havlin S. , Physical Review E , (2017) 032317.[37] Liu X., Pan L., Stanley H. E. and
Gao J. , PhysicalReview E , (2019) 012312.[38] Leicht E. A. and
D’Souza R. M. , arXiv preprint arXiv , (2009) 0894. https://arxiv.org/abs/0907.0894 [39] De Domenico M., Sol´e-Ribalta A., G´omez S. and
Arenas A. , Proceedings of the National Academy of Sci-ences , (2014) 8351.[40] Gross B., Sanhedrai H., Shekhtman L. and
HavlinS. , Physical Review E , (2020) 022316.[41] Hu Y., Ksherim B., Cohen R. and
Havlin S. , PhysicalReview E , (2011) 066116. p-7exian Zheng et al. [42] Cao Y.-Y., Liu R.-R., Jia C.-X. and
Wang B.-H. , Communications in Nonlinear Science and NumericalSimulation , (2020) 105492.[43] Poledna S., Molina-Borboa J. L., Mart´ınez-Jaramillo S., Van Der Leij M. and
Thurner S. , Jour-nal of Financial Stability , (2015) 70.(2015) 70.