Cascading failures on interdependent networks with multiple dependency links and cliques

Accepted Manuscript Cascading failures on interdependent networks with multiple dependency links and cliques Xin Su, Jinming Ma, Ning Chen, Xuzhen Zhu

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S0378-4371(19)30515-1 https://doi.org/10.1016/j.physa.2019.04.143 PHYSA 20907

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Physica A

Received date : 3 February 2019 Revised date : 24 March 2019 Please cite this article as: X. Su, J. Ma, N. Chen et al., Cascading failures on interdependent networks with multiple dependency links and cliques, Physica A (2019), https://doi.org/10.1016/j.physa.2019.04.143 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Highlights (for review)

Highlights 1. Propose a cascading failure model on interdependent networks. 2. The system always undergoes a first order phase transition when the dependency relations are small. 3. Increasing the dependency and clique size, the robustness of the system increase.

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Cascading failures on interdependent networks with multiple dependency links and cliques Xin Su 1 , Jinming Ma 2 , Ning Chen 3 and Xuzhen Zhu 2∗ 1. School of Economics and Management, Beijing University of Posts and Telecommunications, Beijing, China 100876, China 2. State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing, 100876, China 3. School of Urban Transportation, Beijing University of Technology, Beijing, 100124, China

Abstract Cascading failures on interdependent networks have attracted much attention in recent years. In this paper, we study a cascading failure model on interdependent networks with multiple dependency relations and cliques, in which a clique is dependent on multiple cliques in other network and all nodes in the same clique are survive or fail together. Through a percolation theory, we find that the system always undergoes a first order phase transition when the dependency relations are small. The robustness of the system increases with increasing the number of multiple dependency relations between two networks and the size of cliques. The theory can well predict the numerical simulations. Keywords: Complex networks, cascading failure, interdependent networks 1. Introduction The real-world systems are composed by some subsystems, and each of them provides some services to support the function of the system [1, 2, 3, 4, 5, 6]. For instance, the power network provides the electric power for the Internet, and the Internet can transmit the information to control the power network. Such functional dependency between different subsystems may induce the cascading ∗

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Preprint submitted to Physica A

March 24, 2019

failures when the failures happened in one subsystem. To model how cascading failures on real-world systems, scholars proposed some interesting models [7, 8]. Among those models, the most famous one is proposed by Buldyrev and his colleagues in 2010 [1]. In their mathematical model, they assumed that the function of two networks are dependent on each other. Specifically, every node in a network is dependent on a node in the other network, and vice visa. After an initial failure in one network, those nodes which do not connect to the giant connected cluster do not work, and further cause their dependency nodes fail. As a result, the cascading failure happens in the system until no further nodes fail. Through percolation theory, Buldyrev et al. found that the system undergoes a first order phase transition, which is different from the second order phase transition on a single network. Following this work, many scholars investigated the dynamics, such as epidemic spreading [9, 10, 11, 12, 13, 14, 15, 16, 17, 18], social contagions [19, 19], game [20, 21], synchronization [22, 23, 24], and other dynamics [25] on interdependent networks and multiplex networks. Researchers found that the topology [3, 26, 27, 28, 29, 30, 31] of the network markedly affect the dynamics on networks, especially on the cascading failures. Shao et al. [32] studied the cascading failures on interdependent networks with multiple support-dependence relations, and found that the system undergoes a second order phase transition when the system has a large number of support relations. Recently, Wang et al. [33] investigated the effects of group dependency on cascading failures, and found that the system always undergoes a first-order phase transition. However, Wang et al. [34] found that the group dependency on information spreading dynamics induces a double transition. In this paper, we will study a novel cascading failure model on interdependent networks with two characters: (i) multiple support-dependence relations and (ii) cliques (or called as group dependency). Through extensive numerical simulations and a generalized percolation theory, we find that the system always undergoes a first order phase transition when the support relations are small. The robustness of the system increases with the growing of the number of dependency links and clique size. In section 2, we describe the cascading failure model, and present the theory in section 3. In section 4, we perform extensive numerical simulations. Finally, we draw conclusions in Sec. 5.

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2. Model descriptions Here, we describe the cascading failure on two interdependent networks with multiple dependency edges. We consider two-layered networks A0 and B0 of sizes NA0 and NB0 following degree distribution PA0 (kA0 ) and PB0 (kB0 ), respectively. Layer A0 is compose of regular nodes, and the other layer B0 is compose of cliques. We assume that the functions of all nodes in the same clique are dependent on each other. Then, we randomly divide all nodes in network B0 into cliques with the same sizes m. We aggregate a clique as a new node and all the related links in a clique will be merged onto it. Two cliques are connected by a link if and only if there is at least one link between regular nodes in the two cliques. As a result, layer B0 is aggregated a new network B of sizes NB . We do nothing with network A0 , but we can think it as a clique network with clique size m = 1, and denote as A (A = A0 ) with sizes NA (NA = NA0 ). Then, two networks of cliques, A and B are constructed. To get multiple dependent networks, we assume that every clique from network A randomly depends on n cliques in network B, and every clique from network B randomly depends on n cliques in network A. That means if all the dependent cliques of a clique in network A fail, then this clique will also fail, and vise versa. Then, we describe the cascading failure on multiple interdependent networks with cliques. Initially, we randomly remove a fraction of pA cliques in networks A and pB cliques in networks B to initiate the iterative cascading process. A clique in layer X ∈ {A, B} is functional when (i) it connects to the giant connected cluster of layer X ∈ {A, B}. (ii) at least one of its dependent cliques is functional. Note that all regular nodes in a clique are survive or fail together. At each iterative step, we remove the nonfunctional cliques until all cliques are functional. When no more cliques fail, networks A and B reach their final steady state. All regular nodes within cliques belonging to the giant component will survive together in the original regular node networks. The giant clique cluster size is the number of nodes in the original system after the cascading process. 3. Theoretical analysis In this section, we will theoretically derive the giant component of the cliques networks and the giant clique cluster of the original networks. In theory, we assume that the network is large, sparse and local tree like. In networks A and B, the size of a clique is the number of regular nodes in this clique. Denoting PA0 (kA0 ) and PB0 (kB0 ) as the the degree distributions of A0 and B0 , respectively. That is, 3

PA0 (kA0 ) (or PB0 (kB0 )) is the probability that randomly select a regular node in network A0 (or B0 ) with a degree of kA0 (or kB0 ). Similarly, we denote PA (kA ) (or PB (kB )) as the probability that randomly select a clique in network A (or B) with degree kA (or kB ). Since the size of the clique in network A is 1, we have PA (kA ) = PA0 (kA0 ). For a clique with m randomly selected nodes from network B0 , using the generating function [33], the generating function of PB (kB ) meets [∑ ]m ∑ kB kB0 PB (kB )x = PB0 (kB0 )x . (1) kB

kB0

We denote tA as the probability that a randomly chosen link in network A leads to the giant component at one of its end. Similarly, tB is defined for network B. For a dependency link between clique a with degree kA in network A and clique b in B, ukA is defined as the probability that this link from a leads to the giant component, and ukB is defined similarly for a dependency link from B to A. Therefore, we randomly choose a link l in network A and reach clique a at one of its ends. For clique a to be part of the giant component, its functional conditions are: (i) it isn’t removed with probability 1 − pA , and (ii) it must connect with the giant component of network A. (iii) at least one of its dependent cliques connects to the giant component of network B. Therefore, the probability that a randomly chosen link in network A leads to the giant component at one of its end is { } ∑ PA (kA )kA tA = (1 − pA ) [1 − (1 − tA )kA −1 ][1 − (1 − ukA )n ] , (2) ⟨k A⟩ k A

where PA (kA )kA / ⟨kA ⟩ is the probability that a random selected link connecting with a clique a which has a degree of kA , 1 − (1 − tA )kA −1 the probability that at least one of the other kA −1 links of clique a (other than the one first chosen) leads to the giant component in network A, 1−(1−ukA )n denotes the probability that at least one of the n dependency links of a leads to the giant component. Similarly, we can get the self-consistent equation of tB as } { ∑ PB (kB )kB kB −1 n [1 − (1 − tB ) ][1 − (1 − ukB ) ] . (3) tB = (1 − pB ) ⟨kB ⟩ k B

The conditional probability PAB (kB |kA ) is that given a clique a in A with degree kA , the probability that any of its dependent clique b in B is of degree kB . 4

Similar definitions carry over to network B for PAB (kA |kB ). For the probability ukA and ukB , we have { } ∑ ukA = (1 − pB ) PAB (kB |kA )[1 − (1 − tB )kB ] , k

ukB = (1 − pA )

{ B ∑ kA

}

(4)

PAB (kA |kB )[1 − (1 − tA )kA ] .

where 1 − (1 − tB )kB is the probability that at least one of the n dependency neighbors (in network B) of clique a (which has degree kA ) is in the giant component. Accordingly, the probability PA∞ that a randomly chosen clique in network A belongs to the giant component is calculated as } { ∑ PA (kA )[1 − (1 − tA )kA ][1 − (1 − ukA )n ] . (5) PA∞ = (1 − pA ) kA

Similarly, we have

PB∞ = (1 − pB )

{

∑ kB

}

PB (kB )[1 − (1 − tB )kB ][1 − (1 − ukB )n ] .

(6)

We further define PA∞0 is the probability that a randomly chosen regular node belongs to the giant clique cluster in network A0 . Similarly, PB∞0 is defined for network B0 . Due to A = A0 , we have PA∞0 = PA∞ . In network B, since all cliques have same m nodes, we also have PB∞0 = PB∞ . To locate the percolation threshold of network A, we first transform the Eqs. (2) and (3) into tA = FA (pA , tB ), (7) and tB = FB (pA , tA ),

(8)

respectively. Usually, at pA = pcA , the two functions tA = FA (pcA , tB ) and tB = FB (pcA , tA ) meet tangentially with each other, and fulfill ∂FA (pcA , tB ) ∂FB (pcA , tA ) = 1. ∂tB tA

(9)

So, we can numerically solve the Eq. (9) to get the threshold of network A. Similarity, we can use the same methods to get the threshold of pcB in network B. 5

4. Numerical results In this section, we perform the cascading failures on artificial interdependent networks with multiple dependency links and cliques. In simulations, we set NA = NB = N and pA = pB = p. In simulations, we compute the average value over 1000 dependent numerical simulations. In Fig. 1, we present the giant clique cluster of networks A0 and B0 on ER-ER networks, in which degree distributions are PA0 (kA0 ) = e−⟨kA0 ⟩ ⟨kA0 ⟩kA0 /kA0 ! and PB0 (kB0 ) = e−⟨kB0 ⟩ ⟨kB0 ⟩kB0 /kB0 !, where average degree ⟨kA0 ⟩ = ⟨kB0 ⟩ = 3. For a given value of m, such as m = 1, the robustness of the interdependent network increases with n. Specifically, PA∞0 and PB∞0 increases with n and p as shown in Figs. 1(a)–(b). In addition, the percolation threshold pc increases with n. When we fix the value of m, the more interdependency links between two layers, the more robustness of the network. That is to say, PA∞0 and PB∞0 increases with m, and the percolation threshold pc increases with m. Note that PA∞0 and PB∞0 decreases discontinuously with p at any values of m and n. Our suggested theory agree well with the numerical simulations. The differences near the threshold point are caused by the effects of network size and dynamical correlations. Thus, in what follows, we only present the theoretical results. In Fig. 2, we study the giant clique cluster sizes PA∞0 and PB∞0 of networks A0 and B0 in plane (p, n) in detail. We find that PA∞0 and PB∞0 increases with n, and decrease discontinuously with p. The plane is divided into two regions. When p ≤ pc , there is a giant clique cluster in the system; otherwise when p > pc , the giant clique cluster disappears. We further investigate the heterogeneity of degree distribution of the interdependent networks in Fig. 3. In simulations, we set the degree distribution of networks A0 and B0 are power-law, in which PA0 (kA0 ) ∼ k −γA0 and PB0 (kB0 ) ∼ k −γB0 , and denote as SF-SF interdependent networks. We set ⟨kA0 ⟩ = ⟨kB0 ⟩ = 3 and γA0 = γB0 = γ = 2.7. Similar with the results presented in Fig. 1, we find that PA∞0 and PB∞0 decrease discontinuously with p regardless of the values of m and n, and increase with m and n. Again, our theory can well predict the numerical simulations. We finally present PA∞0 and PB∞0 on SF-SF networks in Fig. 4. The plane is divided into two regions. There is a finite values of PA∞0 and PB∞0 when p ≤ pc , otherwise, no finite connected clusters exist. In addition, PA∞0 and PB∞0 always decrease discontinuously with p.

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Figure 1: (Color online) Cascading failures on ER-ER networks when the size of cliques are m in network B. Fixing m = 1, the giant clique cluster of networks A0 (a) and B0 (b) versus the remove probability p with different dependency links n. When we set n = 4, the giant clique cluster of networks A0 (c) and B0 (d) versus the remove probability p with different clique size m. The symbols are the numerical simulation results, and lines are the corresponding theoretical predictions. Other parameters are set to be N = 10000, ⟨kA0 ⟩ = ⟨kB0 ⟩ = 3.

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Figure 2: (Color online) In ER-ER networks, when fixing m = 2, colore represents the the giant clique cluster of networks A0 (a) and B0 (b) in plane (p, n). The result is obtained by theory, and the dotted lines are critical remove probability pc , which is obtained by solving Eq. (9). Other parameters are set to be ⟨kA0 ⟩ = ⟨kB0 ⟩ = 3. 5. Conclusions In this paper, we investigated the cascading failures on interdependent networks, in which we assumed that (i) there are multiple dependency links between two networks, and (ii) cliques in which all nodes in the same clique are dependent on each other. We developed a percolation theory to study the giant clique cluster of the original system, and found that the system always undergoes a first order phase transition when the dependency links are small. In addition, the robustness of the network increases with the number of dependency links and size of cliques. Our theory can well predict the giant clique cluster. Our results may help us have a better understanding the cascading failures on interdependent networks. 6. Acknowledgements This work was supported by the National Natural Science Foundation of China (No.61602048) and the National Key R&D Program of China (No.2017YFC0803903). [1] S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, S. Havlin, Catastrophic cascade of failures in interdependent networks, Nature 464 (7291) (2010) 1025–1028. 8

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Figure 3: (Color online) Cascading failures on SF-SF networks when the size of cliques are m in network B. Fixing m = 1, the giant clique cluster of networks A0 (a) and B0 (b) versus the remove probability p with different dependency links n. When we set n = 4, the giant clique cluster of networks A0 (c) and B0 (d) versus the remove probability p with different clique size m. The symbols are the numerical simulation results, and lines are the corresponding theoretical predictions. Other parameters are set to be N = 10000, ⟨kA0 ⟩ = ⟨kB0 ⟩ = 3 and degree exponent γ = 2.7.

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