Estimation of Robustness of Interdependent Networks against Failure of Nodes

Srinjoy Chattopadhyay∗ and Huaiyu Dai† Department of Electrical and Computer Engineering, North Carolina State University. Email: ∗[email protected], †[email protected]

Abstract—We consider a partially interdependent network and from a and branching process point of develop mathematical equations relating the fractional size of the view, based on the classical work [6]. This method of analysis connected of the network, surviving the cascading is more amenable to analysis and has been used in some failure, to the intra-layer distribution of the nodes. We show that these system equations can be mathematically analyzed recent works like [7] to obtain system equations modeling and closed form expressions for the metrics of robustness can the . Using this technique of analysis, we be obtained for the Erdos-Renyi (ER) model of have been able to obtain relationships between the robustness generation. We have described the application of our analysis of interdependent networks and its structural parameters for technique to networks with general degree distributions. In our various network models and attack models which, to the analysis, we consider the two extremes of the attack model: randomized attack, where nodes are attacked at random without best of our knowledge, do not exist in literature. The work any knowledge of intra-layer degrees and perfect targeted attack, presented in this paper is in two broad domains. Firstly, we where nodes are attacked based on the strict descending order of have applied our technique of analysis on a general framework their intra-layer degrees. Our results can enable researchers to of partially interdependent network to mathematically obtain gain a better understanding of the robustness of interdependent the robustness against randomized node failure for networks networks. Index Terms—Interdependent networks, Critical fraction, net- generated as ER graphs. Furthermore, we have been able to work robustness. obtain closed form expressions for the critical fraction of nodes I.INTRODUCTION and phase transitions points, which are important properties Over the past decade, inter-connectivity of systems has [3], [7] governing the variation of robustness of networks emerged as a prominent avenue for research in a wide variety for different strengths of attack. We have also discussed the of fields. Throughout the spectrum of cyber and physical application of the analysis techniques developed here to real systems, we can observe a steadfast transition to interde- world networks with known . Secondly, we pendent systems: smart power distribution grids which have have considered the case of a targeted node failure model, inter-coupled power distribution and communication networks where the attacker selects nodes in the order of their intra- [1] being the prime example in this case. The increased layer degrees and developed theoretical approximations for the functionality of interdependent systems however comes at corresponding robustness for any general network. the price of enhanced sensitivity to node failures due to the The remainder of this paper is organized as follows. Section phenomenon of cascading failure, where failure of a fraction II presents the system model and system equations which have of nodes initiates a recursive cascade of failures between the been analyzed for randomized attack in Section III and tar- different layers of the network. geted attack in Section IV. Section V presents a comparison of A better understanding of the cascading failure phenomenon the theoretically derived results with the simulations. Finally, has been a prime focus of research work in this area. The tradi- we conclude and indicate possible future work in Section VI. tional technique of analysis in this field utilizes the generating function of the intra-layer node degree distribution. Using II.SYSTEM MODEL this approach, researchers have obtained equations modeling We consider a system of partially interdependent network the cascading failure for various scenarios like completely comprising two layers A and B with N nodes each. The interdependent networks with one-to-one interdependence [1], layers consist of two types of nodes: autonomous nodes (X one-to-many interdependence [2] and partial interdependence in Fig. 1), which do not require support from nodes of the [3], [4]. Some works have also considered the case of targeted other layer; and interdependent nodes (Y in Fig. 1), which attack [5]. Due to the inherent complexity of the generating are dependent on the other layer nodes for their survival. function approach, closed form results relating the robustness Let the fraction of interdependent nodes be denoted by q, to the structural parameters of the network were only obtained where they are chosen randomly so that all nodes have the for a small subset of networks, particularly, completely inter- same probability (q) of being interdependent. The two layers dependent networks with identically distributed layers. are constructed as random graphs with small average degrees In this work1, we have analyzed interdependent networks λA and λB respectively (λA, λB  N). This construction 1This work was supported in part by National Science Foundation under ensures the locally tree-like property [6] of the layers, which Grants CNS-1016260, ECCS-1307949 and EARS-1444009. is necessary for the formulation of the system equations networks as will be shown in Section III-C. To formulate the system equations, we need to define node (ψA and ψB) and edge (pA and pB) percolation probabilities denoting the probability that a randomly chosen node (or edge) belongs to the MCC. The survival of any node in the network depends on three conditions [6], [7], [8]: 1) the node survives the initial attack; 2) the node belongs to the largest connected component in its layer; and 3) if the node is interdependent in nature, its support node in the other layer should also satisfy Fig. 1. System model for partially interdependent network. the above two conditions. On the basis of these conditions, we have obtained percolation theory [6], [7] based self-referencing modeling cascading failure. The interdependency between the system equations, the details of which can be obtained from two layers is taken to be one-to-one and bidirectional, which the report [9]: has been widely adopted in relevant literature as a first critical X X X ψA = (1 − q) f2(kA) + q f2(kA)f2(kB), (1) step towards understanding the operation and key phenomena kA kA kB of interdependent networks. Thus all interdependency links X X X correspond to an interdependent pair of nodes, (a, b) with pA = (1 − q) f1(kA) + q f1(kA)f2(kB), (2) a ∈ A and b ∈ B, which depend on each other for their kA kA kB kl k −1 survival. The interdependency links are assumed to be inde- where f1(kl) = ηP [kl] [1 − (1 − pl) l ] and f2(kl) = λl k pendent with respect to (w.r.t.) the intra-layer degrees of the ηP [kl][1 − (1 − pl) l ], and kl is the intra-layer degree of a nodes, i.e. inter-links are constructed without the knowledge node in layer l with mean degree λl. Here f1 (or f2) can of the graph structure. be understood as the probability that a randomly chosen edge In this model, we are interested in studying the effect of the (or node) is connected to a node of (is of) degree kl and failure of a fraction (1 − η) of nodes from both layers. For the belongs to the MCC. Thus the summation of these individual first part of our analysis we will assume that the attack strategy contributions of edges and nodes gives the edge (pl) and node is random in nature, i.e. the attacker (or the mother nature) fails percolation (ψl) probabilities. Note here that ψ is nothing all nodes with the same probability. Subsequently in Section but the fractional size of the MCC which is our metric IV, we will consider a targeted attack scenario and investigate quantifying the robustness. The system equations for layer B the corresponding robustness. Our main interest is to relate the can be symmetrically obtained. Similar system equations exist robustness of the network to its structural parameters (λ, q). in literature [7] and thus we have omitted the details of the The fractional size of the mutually connected component derivation of these system equations. In this work, we have (MCC) of the network which survives the cascading failure theoretically analyzed these system equations for networks is classically [1]–[5] taken to be the metric for robustness. generated as ER graphs to derive expressions describing the It is known from works like [1], [5] that for completely network robustness (ψA and ψB), the critical node fraction interdependent networks, the of the network (ηc), and percolation phase transition points (qc) in terms of robustness with varying attack strength (η) is of first order, structural parameters (q, λ). i.e. there exists a minimum η = ηc which would lead to a III.ANALYSIS:RANDOM ATTACK η < η non-zero MCC size in the steady state and for all c We start our analysis considering random node failure, and the average size of the MCC is arbitrarily close to 0. In consider the case when the network layers are generated other words, the variation of robustness with attack strength is according to the ER model. We analyze the system equations η = η discontinuous at c. Furthermore, partially interdependent for layer A (analysis for layer B is symmetrical) in parts by networks [3], [4] undergo a phase transition from first order focusing on the individual terms. The details of the analysis (discontinuous) to second order (continuous) as the fraction of is included in the report [9] which gives us: interdependent nodes (q) decreases below a certain threshold   {A} 1 ∂ X q kA defined as the phase transition point ( c). In this work, we F1 = η 1 + P [kA](1 − pA) , (3) λA ∂pA intend to obtain closed form expressions for these phenomena kA on networks with known degree distributions by formulating   {A} X kA system equations for our particular network and attack models F2 = η 1 − P [kA](1 − pA) , (4) and mathematically analyzing these equations for networks kA generated as the celebrated ER graphs. ER graphs have been {A} P {A} P where F = f1(A) and F = f2(A). It can predominantly examined [1], [5] as a first step in network 1 kA 1 kA be seen that the term of interest here is the expression: F science research to obtain insights, thanks to their amenability 3 , (P P [k](1−p)k), and its partial derivative with respect to p. to analysis. Although such theoretical network structures are k The intra-layer degrees of the nodes in ER graphs is Poisson rarely encountered in the practical world, we strongly believe distributed, i.e. P [k] = e−λ/λkk!, where λ denotes the mean that their analysis is still important since the qualitative node degree. It has been shown in the report [9] that F = features of many of our results below are valid for general 3 e−λp. −λp 2 −λ?p 2 REMARK 1 It can be observed that ∂F3(λ, p)/∂p = −λe , ψ = p ≈ η (1 − e ) , (10) which upon substitution into (3),(4) gives F (λ, p) = F (λ, p). 1/2 1 2 2.4554 Thus from (1) and (2), we get ψ = p for ER graphs, which η ≈ . (11) c λ? is a very interesting result that proves ψ and p can be used −λ p interchangeably as a metric for network robustness. Taking x , e A . the approximation error can be written as: Using the simplifications of F3(λ, p), the system equations r+1 2r r 4r (1)–(2) can be written as: fe(r) = x + 2x r+1 − x − x − x r+1 . (12) −λApA  ψA = η(1 − q) 1 − e It has been shown in report [9] that limr→1 fe(r) = 0. Further 2 −λApA  −λB pB  graphical analysis gives us that f (r) becomes very close to +η q 1 − e 1 − e = pA. (5) e 0 for values of r > 0.5. Thus for practical purposes, we can We present some interesting results below that help shed light consider the harmonic approximation to be an adequate mea- on the resilience of interdependent networks against random sure for approximate network performance for interdependent node failure. networks generated as non-identically distributed ER graphs. A. Completely Interdependent Networks B. Partially Interdependent Networks The system equations for this case (q = 1) can be written as: The analysis for partially interdependent networks is signif- ψ = η2(1 − e−λAp)(1 − e−λB p) = p. (6) icantly more complicated due to the quadratic nature of the system equations. Thus we simplify the problem by focusing Thus for any strength of attack η, we can obtain the robustness on the case of λA = λB, i.e. identically distributed layers. of the network (ψ) by solving (6). It is known from literature Under these conditions, the system equations can be written [1] that completely interdependent networks exhibit a first- as: order phase transition, wherein ψ abruptly falls to 0 when η ψ = (1 − q)η(1 − e−λp) + qη2(1 − e−λp)2 = p. (13) falls below a particular threshold ηc (critical fraction). THEOREM III.1 The robustness (ψc = pc) of an interdepen- The phase transition properties of partially interdependent dent ER network with non-identically distributed layers at the networks is much more interesting as compared to completely critical point (η = ηc) can be obtained by solving the following interdependent networks. This is motivated by the observation shown in literature [7], which indicates that phase transitions equations: f log f f log f A A + B B = 1, (7) in partially interdependent networks is not exclusively first fA − 1 fB − 1 order in nature. In such systems, as the coupling between the r fB = fA, (8) two layers of the network (q) is decreased, the phase transition order changes from first order (for higher values of q), where where f e−λApc and f e−λB pc , λ and λ are the A , B , A B the robustness abruptly falls to 0 after a certain threshold, to mean degrees of the two layers, and r denotes the ratio r = second order, where the robustness smoothly falls down to 0. λ /λ . The critical fraction of nodes (η ) is then given by: B A c Let this phase transition point be represented by q . It should −1/2 c ηc = [λAfA(1 − fB) + λBfB(1 − fA)] . (9) be noted here that for partially interdependent networks, the

Sketch of Proof. Note that the critical point (ηc) is the min- critical fraction of nodes (ηc) is only defined in the range of imum value of η for which the slopes of the functions q for which the phase transition is first order in nature. To 2 −λAp −λB p g1(p) , η (1 − e )(1 − e ) and g2(p) , p are equal. resolve this issue and to promote a unified analysis of such This idea has been used to derive the equations (7)-(9), the problems, we have defined the term critical strength of attack details of which can be obtained from [9].  (η0+) as the minimum value of η which results in a non-zero solution to the system equation (13). Let the non-zero solution COROLLARY III.2 For identically distributed ER networks, p to (13) at η = η0+ be represented by p0+. Thus for q > qc, ηc = 2.4554/λ, and ψc = 1.2864/λ. when the phase transition of the system is first order, η0+ = ηc Proof. Here (7) simplifies to f log f = (f − 1)/2, which can and p0+ = pc, whereas when the phase transition is second be solved numerically to yield f = 0.28467. Thus ψc = pc = q < q p η −1 order ( c) gradually goes to 0 with decreasing , i.e. − log f/λ, and ηc = [2λf(1 − f)] .  p0+ → 0 at η = η0+. The result in the corollary is the same as that in literature THEOREM III.3 For a partially interdependent ER network [1]. Thus it establishes the equivalence of our method of with identically distributed layers, if q < qc, the critical analysis with existing ones. We have furthermore shown that strength of attack is given by: the metrics of robustness (ψ vs η) of such non-identically −1 η0+ = [λ(1 − q)] , (14) distributed networks can be approximated by an equivalent identically distributed network whose mean intra-layer degree where λ is the mean intra-layer degree, q is the fraction of ? λ is the harmonic mean of λA and λB. This analysis is also interdependent nodes, and qc is the phase transition point. included in the report [9], by which the network robustness Sketch of Proof. Let us re-write the system equation under the (ψ) and critical fraction of nodes (ηc) can be approximated by critical strength of attack condition (η = η0+, p = p0+) using the following equations: the idea that p0+ → 0: 1 the value and derivative of the function F3(λ, p) is known 0.9 to us, we can write the system equations (1)–(2) in terms of 0.8

0.7 F3. Thus for the theoretical estimation of the performance of 0.6 real world networks whose degree distribution is known to

0+ 0.5 η 0.4 us, we can numerically estimate the values of the function 0.3 F3(λ, p) and its derivatives at the required values of p and 0.2 thereafter numerically solve the system equations modeling 0.1

0 0 0.2 0.4 0.6 0.8 1 the cascading failure: (1)–(2). Note that the system equations q are derived under the assumption that the network layers are Fig. 2. Comparison between the numerical solutions (markers) and theoret- locally tree-like. Random graphs have this property as long as ically derived formulas (solid) for η0+ and qc. The three cases shown here: correspond to mean intra-layer degrees λ = [2, 6, 20]. The dotted vertical the network is sparse [6]. However, real world networks would lines correspond to the theoretically obtained values of qc given by (16). usually not follow such strict mathematical properties and thus

 −λp0+ −λp0+ 2  the system equations derived in these cases are just theoretical 1−e 2 (1−e ) lim (1−q)η0+ +qη0+ = 1. (15) approximations to the actual performance. The comparison of p0+→0 p p 0+ 0+ this theoretically predicted performance and simulation results Evaluating the above limit, we get the required result.  will be presented in Section V. THEOREM III.4 For a partially interdependent ER network with identically distributed layers, the phase transition point IV. ANALYSIS:TARGETED ATTACK (qc) is given by: √ λ + 1 − 2λ + 1 The underlying assumption in the analysis presented in Section q = , (16) c λ III is that the attacker randomly fails nodes. However, attacks where λ is the mean intra-layer degree. on the nodes with high connectivity would undoubtedly be a better strategy from the perspective of disrupting the network. Sketch of Proof. Note that the critical point solution to (13) Previously, we have considered the case where the attacker below the phase transition point is p = 0 since the network has no knowledge about the node degrees and thus attacks phase transition is of second order. For q > q , the phase c nodes randomly. In this section, we analyze the other extreme transition is of first order and thus (13) has a non-zero finite situation: where the attacker has complete intra-layer degree solution. We have used this idea to analytically derive the information and attacks nodes in the strict descending order phase transition point for networks generated as ER graphs, of the degrees. We present the analysis of completely inter- the details of which can be obtained from [9].  dependent networks with identically distributed layers. For a Comparison of the numerical solutions to the system equa- particular instance of network generation, we index the nodes tions with the theoretically derived formulas for both η and 0+ in layers A and B in the increasing order of their intra-layer q is presented in Fig. 2. We can clearly observe from the c degrees. Let this index be represented by i: 1 ≤ i ≤ N, where figure that when the network phase transition is second order, N is the number of nodes in each layer. Let k{i} (or k{i}) be i.e. q < q , (14) accurately follows the numerical solution. The A B c the intra-layer degree of the ith node in the layer A ( or B) of phase transition points (q ) where the transition order changes c the network. Let sequence {ρ} be a permutation of the number is also very accurately presented in Fig. 2. However when the set {1, 2, ··· ,N}, where N is the number of nodes in each phase transition order of the network is first order, we cannot layer, ρ denotes the ith element of {ρ}, and the indices of any perform any simplification of the quadratic system equations i interdependent pair of nodes is represented by (i, ρ ). Thus ρ, since p = p is finite and positive. Thus for q > q , we i 0+ c c representing an arbitrary ordering the nodes, can completely have no choice but to numerically solve the system equations specify the one-to-one interdependence structure. (13) to theoretically estimate ηc. As previously, the initial attack leads to the failure of a C. Applications: Real World Networks fraction (1 − η) of the nodes in the network. Due to the Although the analytical results presented in the previous sec- perfectly targeted attack model, all top (1 − η)N nodes in tion have been derived for networks constructed as ER graphs, both layers fail due to the initial attack. It should be noted the qualitative features of our results hold for random graphs at this point that the metrics of network robustness ψ and with arbitrary degree distributions as well. Note that the essen- p are defined in a mean sense over all instances of network tial step, which leads to the formulation of the mathematical generation with specified intra-layer degree distribution of the equations relating robustness (ψ) to the structural parameters nodes. Thus for the purpose of obtaining system equations for P k (q, λ), was the simplification: F3(λ, p) = k P [k](1 − p) = ψ and p with respect to indexed nodes, we need to take an e−λp. For networks with arbitrary degree distribution, such expectation over all instances for a specific degree distribution. simplifications would not exist in most cases. However, if we Due to the identical distribution of the two layers of the know (or estimate) the degree distribution of the nodes in network and the simultaneous attack on both layers, ψ and the networks, we can numerically estimate the values of the p would be the same for both layers. Thus we will drop function F3(λ, p) and its derivative with respect to p. From the subscript denoting layer from the terms ψl and pl for the discussion in the previous section, it is evident that if simplicity. Similar to the previous case, we define functions {l} {l} G1 (i) and G2 (i) to simplify the representation of the 1 system equations and obtain: 0.9  N  0.8 1 X {A} {B} 0.7 ψ = E Iηg (i) · ηg (ρi) , (17) N 2 2 0.6 i=1 0.5 ψ  N  0.4 1 X {A} {B} 0.3 p = E I g (i) · ηg (ρ ) , (18) N η 1 2 i 0.2 i=1 0.1 0 0 0.2 0.4 0.6 0.8 1 {l} kl(i) k (i)−1 {l} where g (i) = [1 − (1 − p) l ] and g (i) = η 1 λl 2 [1 − (1 − p)kl(i)], and k (i) is the intra-layer degree of the ith Fig. 3. Completely interdependent networks under randomized attack. The l three sets of plots represent three cases with ‘o’: λ = 10,‘ ’: λ = 5, l λ E[·] B  B node in layer having mean degree of l. Here represents and ‘’:λB = 3; λA = 3 for all three cases. the expectation over all instances of network generation for a 1 specific degree distribution of the nodes and Iη is an indicator 0.9 function representing the targeted removal of the top (1−η)N 0.8 0.7 fraction of the indices i, i.e. Iη = 1 if i ∈ (1, ηN). 0.6

0.5 THEOREM IV.1 The robustness of an interdependent network ψ 0.4 with identically distributed layers against targeted attack can 0.3 be approximated by the two inequalities: 0.2 ψ ≥ η2[1 − (1 − p)λ][1 − (1 − p)λη ], (19) 0.1 0 0 0.2 0.4 0.6 0.8 1 ηN η 1 XΛi p ≥ η[1 − (1 − p)λ][1 − (1 − p)λη −1] , (20) Fig. 4. Partially interdependent networks under randomized attack. The three N λ i=1 sets of plots represent three cases with ‘o’ : q = 0.1,‘’: q = 0.5, and ‘’: q = 0.9; λ = 4 for all three cases. {i} where Λi = E[k ] denotes the order statistics of the intra- layer degree (k{i}) of the node with the ith lowest degree, λ for the simulation results, represented by the markers with is the intra-layer degree, and λη , E[ΛiIη] where Iη is an dashed lines in Figs. 3-6, is on the order of hours, whereas indicator function representing the targeted removal of (1 − the theoretical approximations, represented by the solid lines, η)N nodes of highest degrees. can be obtained on the order of seconds. This demonstrates a key aspect of our work; fast approximation of robustness Sketch of Proof. To prove this result, we use the concavity of without expensive simulations. We present a brief discussion the function f(x) = 1 − (1 − p)x for p ∈ [0, 1] and invoke on the simulation figures next. the Jensen’s Inequality on the system equations (17)-(18).The details of this can be obtained from [9].  • Fig. 3 gives the comparison between the simulation results and theoretical estimation for completely interde- We have used (19),(20) to design a numerical method based pendent networks with non-identically distributed layers. algorithm to estimate the value of ψ and p for particular η It can be easily verified from Fig. 3 that the analytical and λ. The order statistics [10] for the degrees (Λi) can be expressions very closely fit the simulation results for all estimated for random graph generators or real world networks three cases. This figure thus indicates that the robustness if the degree distribution is known or can be estimated. For the of interdependent networks can be very closely approxi- ER model, the intra-layer degrees follow Poisson distribution mated by theoretical expressions instead of computation- whose order statistics can be readily obtained [10]. ally intensive simulations. This idea is fundamental to this V. RESULTS AND DISCUSSIONS piece of work where the ultimate goal is to relate network We have designed a test bench to simulate the mechanism robustness to the structural parameters of the network. of cascading failure in interdependent networks initiated by • Fig. 4 gives the same comparison as above for the case random or targeted removal of a certain fraction (1 − η) of of partially interdependent networks. nodes. The robustness is depicted by plotting the fractional size • From Fig. 3 and Fig. 4, we can also compare the the- of the mutually connected component in the steady state (ψ), oretical estimation and simulation results for the critical for various strengths of attacks (η). The designed algorithm fraction of nodes (ηc), which is defined as the maximum obtains the largest connected component in each layer at each attack strength or equivalently the minimum value of η stage of the recursive cascading failure and outputs the steady which results in a non-zero MCC size in the steady state. state fractional size of this component. We have considered For the case of completely interdependent networks, ηc networks where the constituent layers of size N = 50000 is estimated by (11) and gives the following numerical are generated by the Erdos-Renyi model from the NetworkX results: ηc = [0.7294, 0.8092, 0.9047] for λB = [10, 5, 3]. Python library [11] and the figures represent the average results We can clearly observe from the figure that the the- of 200 independent runs (10 network construction instances oretical estimates are quite accurate. For the case of with 20 node removal instances each). The execution time partially interdependent networks, ηc (or technically η0+ 1 preserved even for the case of real world networks and thus 0.9 the relationships obtained here are vital towards understanding 0.8 the effect of structural parameters of network on its robustness 0.7

0.6 against node failures. Another reason depicting the importance

0.5 ψ of such a theoretical framework for robustness estimation is 0.4 that these can be used to predict the robustness of a network 0.3

0.2 given its structural parameters without resorting to brute force 0.1 simulations. The drawback of such simulations is that for 0 0 0.2 0.4 0.6 0.8 1 many real world networks, they might be computationally very η Fig. 5. Targeted attack on completely interdependent ER networks with mean expensive. Furthermore simulations require the knowledge of degrees of ‘’: λ = 4, and ‘o’: λ = 8. the exact architecture of the constituent layers of the network,

1 which can be unknown in many cases. For the case of 0.9 targeted attack, we have developed numerical lower bounds 0.8 for the performance metrics for any general network. One 0.7

0.6 of the main aspects on which we have not focused is the

0.5 ψ impact of interdependency structures on the network, which 0.4 we have considered in this work to be designed randomly 0.3

0.2 without the knowledge of the intra-layer degree distribution. 0.1 In future, we plan to generalize our current analysis and 0 0 0.2 0.4 0.6 0.8 1 also explore similar estimation problems for more practical η network models like Scale-Free graphs. We feel that our results Fig. 6. Real world networks with randomized attack. The three sets of plots would enable the academia to gain a better understanding of represent three cases with ‘o’: q = 0.1,‘’: q = 0.5, and ‘’: q = 0.9. robustness of interdependent networks against node failures as discussed previously) can be estimated by (14) when and its relationship with the structural parameters. the phase transition order is second order. For this case REFERENCES (λ = 4), the phase transition point is given by (16), which [1] S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, and S. Havlin, can be calculated to be qc = 0.5. Thus for the first two “Catastrophic cascade of failures in interdependent networks,” Nature, cases (q = 0.1, 0.5), the theoretical values of η0+ are vol. 464, no. 7291, pp. 1025–1028, 2010. 0.277 and 0.5 respectively which is also very accurate. [2] O. Yagan,˘ D. Qian, J. Zhang, and D. Cochran, “Optimal allocation • The mathematical approximations for the case of targeted of interconnecting links in cyber-physical systems: Interdependence, cascading failures, and robustness,” IEEE Transactions on Parallel and attack has been presented in Fig. 5. It can be observed that Distributed Systems, vol. 23, no. 9, pp. 1708–1720, 2012. the gap between the simulated and theoretically predicted [3] G. Ranjan and Z. Zhang, “How to glue a robust smart-grid?: a finite- performance is larger than the case of randomized attack for interdependent network robustness,” in Proceedings of the Seventh Annual Workshop on Cyber Security and Information but still the approximations are close enough for us to Intelligence Research. ACM, 2011, p. 22. gain a better qualitative understanding of the impact of [4] Roni Parshani, Sergey V Buldyrev, and Shlomo Havlin, “Interdependent structural parameters on the robustness (ψ) and even networks: reducing the coupling strength leads to a change from a first to second order percolation transition,” Physical review letters, vol. 105, obtain quantitative approximations for these metrics. no. 4, pp. 048701, 2010. • For testing our method of analysis on real world net- [5] X. Huang, J. Gao, S.V. Buldyrev, S. Havlin, and H.E. Stanley, “Robust- works, we have used the Gnutella peer-to-peer net- ness of interdependent networks under targeted attack,” Physical Review E, vol. 83, no. 6, pp. 065101, 2011. work dataset available from [12]. The interdependent [6] L.A. Braunstein, Z. Wu, Y. Chen, S.V. Buldyrev, T. Kalisky, S. Sreeni- network comprises two identical layers of this peer-to- vasan, R. Cohen, E. Lopez, S. Havlin, and H.E. Stanley, “Optimal path peer network and the interdependency has been assigned and minimal spanning trees in random weighted networks,” Interna- tional Journal of Bifurcation and Chaos, 2007. randomly. It can be clearly seen from Fig. 6 that the [7] L.D. Valdez, P.A. Macri, H.E. Stanley, and L.A. Braunstein, “Triple theoretical estimation of the performance follows the point in correlated interdependent networks,” Physical Review E, vol. simulated performance reasonably accurately. 88, no. 5, pp. 050803, 2013. [8] S. Chattopadhyay and H. Dai, “Towards optimal link patterns for VI.CONCLUSIONAND FUTURE WORK robustness of interdependent networks against cascading failures,” in IEEE Globecom 2015, Design and Next Gen. Network I, 2015. Our aim in this work is to estimate the robustness of interde- [9] S. Chattopadhyay and H. Dai, “Estimation of robustness of interdepen- pendent networks against failure of nodes and relate it to the dent networks against node failures,” Technical Report, Dept. of ECE, structural parameters of network. Furthermore for networks NCSU, 2016, http://www4.ncsu.edu/ ∼hdai/Interdependent TP.pdf . [10] H.A. David and H.N. Nagaraja, Order statistics, Wiley Online Library, generated as ER (Erdos-Renyi) graphs, we have been able to 1970. mathematically derive closed form expressions for the critical [11] A. A. Hagberg et al., “Exploring network structure, dynamics, and fraction of node (ηc) and phase transition points (qc), the function using ,” Proceedings of the 7th Python in Science Conference (SciPy2008), 2008. existence of which was known in literature [4]. Although [12] J. Leskovec, J. Kleinberg, and C. Faloutsos, “Graph evolution: Den- such theoretical results may not accurately apply to real world sification and shrinking diameters,” ACM Transactions on Knowledge networks due to the inherent simplicity of the ER model, we Discovery from Data (TKDD), vol. 1, no. 1, pp. 2, 2007. have shown that the qualitative features of these results are