Estimation of Robustness of Interdependent Networks against Failure of Nodes

Transcription

1 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. Abstract We consider a partially interdependent network and develop mathematical equations relating the fractional size of the connected component of the network, surviving the cascading failure, to the intra-layer degree distribution of the nodes. We show that these system equations can be mathematically analyzed and closed form expressions for the metrics of robustness can be obtained for the Erdos-Renyi (ER model of random graph generation. We have described the application of our analysis technique to networks with general degree distributions. In our analysis, we consider the two extremes of the attack model: randomized attack, where nodes are attacked at random without any knowledge of intra-layer degrees and perfect targeted attack, where nodes are attacked based on the strict descending order of their intra-layer degrees. Our results can enable researchers to gain a better understanding of the robustness of interdependent networks. Index Terms Interdependent networks, Critical fraction, network robustness. I. INTRODUCTION Over the past decade, inter-connectivity of systems has emerged as a prominent avenue for research in a wide variety of fields. Throughout the spectrum of cyber and physical systems, we can observe a steadfast transition to interdependent systems: smart power distribution grids which have inter-coupled power distribution and communication networks [] being the prime example in this case. The increased functionality of interdependent systems however comes at the price of enhanced sensitivity to node failures due to the phenomenon of cascading failure, where failure of a fraction of nodes initiates a recursive cascade of failures between the different layers of the network. A better understanding of the cascading failure phenomenon has been a prime focus of research work in this area. The traditional technique of analysis in this field utilizes the generating function of the intra-layer node degree distribution. Using this approach, researchers have obtained equations modeling the cascading failure for various scenarios like completely interdependent networks with one-to-one interdependence [], one-to-many interdependence [2] and partial interdependence [3], [4]. Some works have also considered the case of targeted attack [5]. Due to the inherent complexity of the generating function approach, closed form results relating the robustness to the structural parameters of the network were only obtained for a small subset of networks, particularly, completely interdependent networks with identically distributed layers. In this work, we have analyzed interdependent networks This work was supported in part by National Science Foundation under Grants CNS-626, ECCS and EARS from a percolation theory and branching process point of view, based on the classical work [6]. This method of analysis is more amenable to analysis and has been used in some recent works like [7] to obtain system equations modeling the cascading failure. Using this technique of analysis, we have been able to obtain relationships between the robustness of interdependent networks and its structural parameters for various network models and attack models which, to the best of our knowledge, do not exist in literature. The work presented in this paper is in two broad domains. Firstly, we have applied our technique of analysis on a general framework of partially interdependent network to mathematically obtain the robustness against randomized node failure for networks generated as ER graphs. Furthermore, we have been able to obtain closed form expressions for the critical fraction of nodes and phase transitions points, which are important properties [3], [7] governing the variation of robustness of networks for different strengths of attack. We have also discussed the application of the analysis techniques developed here to real world networks with known degree distribution. Secondly, we have considered the case of a targeted node failure model, where the attacker selects nodes in the order of their intralayer degrees and developed theoretical approximations for the corresponding robustness for any general network. The remainder of this paper is organized as follows. Section II presents the system model and system equations which have been analyzed for randomized attack in Section III and targeted attack in Section IV. Section V presents a comparison of the theoretically derived results with the simulations. Finally, we conclude and indicate possible future work in Section VI. II. SYSTEM MODEL We consider a system of partially interdependent network comprising two layers A and B with N nodes each. The layers consist of two types of nodes: autonomous nodes (X in Fig., which do not require support from nodes of the other layer; and interdependent nodes (Y in Fig., which are dependent on the other layer nodes for their survival. Let the fraction of interdependent nodes be denoted by q, where they are chosen randomly so that all nodes have the same probability (q of being interdependent. The two layers are constructed as random graphs with small average degrees λ A and λ B respectively (λ A, λ B N. This construction ensures the locally tree-like property [6] of the layers, which is necessary for the formulation of the system equations

2 Fig.. System model for partially interdependent network. modeling cascading failure. The interdependency between the two layers is taken to be one-to-one and bidirectional, which has been widely adopted in relevant literature as a first critical step towards understanding the operation and key phenomena of interdependent networks. Thus all interdependency links correspond to an interdependent pair of nodes, (a, b with a A and b B, which depend on each other for their survival. The interdependency links are assumed to be independent with respect to (w.r.t. the intra-layer degrees of the nodes, i.e. inter-links are constructed without the knowledge of the graph structure. In this model, we are interested in studying the effect of the failure of a fraction ( of nodes from both layers. For the first part of our analysis we will assume that the attack strategy is random in nature, i.e. the attacker (or the mother nature fails all nodes with the same probability. Subsequently in Section IV, we will consider a targeted attack scenario and investigate the corresponding robustness. Our main interest is to relate the robustness of the network to its structural parameters (λ, q. The fractional size of the mutually connected component (MCC of the network which survives the cascading failure is classically [] [5] taken to be the metric for robustness. It is known from works like [], [5] that for completely interdependent networks, the phase transition of the network robustness with varying attack strength ( is of first order, i.e. there exists a minimum = c which would lead to a non-zero MCC size in the steady state and for all < c the average size of the MCC is arbitrarily close to. In other words, the variation of robustness with attack strength is discontinuous at = c. Furthermore, partially interdependent networks [3], [4] undergo a phase transition from first order (discontinuous to second order (continuous as the fraction of interdependent nodes (q decreases below a certain threshold defined as the phase transition point (q c. In this work, we intend to obtain closed form expressions for these phenomena on networks with known degree distributions by formulating system equations for our particular network and attack models and mathematically analyzing these equations for networks generated as the celebrated ER graphs. ER graphs have been predominantly examined [], [5] as a first step in network science research to obtain insights, thanks to their amenability to analysis. Although such theoretical network structures are rarely encountered in the practical world, we strongly believe that their analysis is still important since the qualitative features of many of our results below are valid for general networks as will be shown in Section III-C. To formulate the system equations, we need to define node (ψ A and ψ B and edge (p A and p B 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]: 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 the above two conditions. On the basis of these conditions, we have obtained percolation theory [6], [7] based self-referencing system equations, the details of which can be obtained from the report [9]: ψ A = ( q k A f 2 (k A + q k A k B f 2 (k A f 2 (k B, ( p A = ( q k A f (k A + q k A k B f (k A f 2 (k B, (2 where f (k l = P [k l ] k l λ l [ ( p l kl ] and f 2 (k l = P [k l ][ ( p l k l ], and k l is the intra-layer degree of a node in layer l with mean degree λ l. Here f (or f 2 can be understood as the probability that a randomly chosen edge (or node is connected to a node of (is of degree k l and belongs to the MCC. Thus the summation of these individual contributions of edges and nodes gives the edge (p l and node percolation (ψ l probabilities. Note here that ψ is nothing but the fractional size of the MCC which is our metric quantifying the robustness. The system equations for layer B can be symmetrically obtained. Similar system equations exist in literature [7] and thus we have omitted the details of the derivation of these system equations. In this work, we have theoretically analyzed these system equations for networks generated as ER graphs to derive expressions describing the network robustness (ψ A and ψ B, the critical node fraction ( c, and percolation phase transition points (q c in terms of structural parameters (q, λ. III. ANALYSIS: RANDOM ATTACK We start our analysis considering random node failure, and consider the case when the network layers are generated according to the ER model. We analyze the system equations for layer A (analysis for layer B is symmetrical in parts by focusing on the individual terms. The details of the analysis is included in the report [9] which gives us: F {A} = ( + λ A P [k A ]( p A ka p A k A ( F {A} 2 = P [k A ]( p A k A k A, (3, (4 where F {A} = k A f (A and F {A} = k A f 2 (A. It can be seen that the term of interest here is the expression: F 3 ( k P [k]( pk, and its partial derivative with respect to p. The intra-layer degrees of the nodes in ER graphs is Poisson distributed, i.e. P [k] = e λ /λ k k!, where λ denotes the mean node degree. It has been shown in the report [9] that F 3 = e λp.

3 REMARK It can be observed that F 3 (λ, p/ p = λe λp, which upon substitution into (3,(4 gives F (λ, p = F 2 (λ, p. Thus from ( and (2, we get ψ = p for ER graphs, which is a very interesting result that proves ψ and p can be used interchangeably as a metric for network robustness. Using the simplifications of F 3 (λ, p, the system equations ( (2 can be written as: ψ A = ( q ( e λ Ap A + 2 q ( e λ Ap A ( e λ B p B = pa. (5 We present some interesting results below that help shed light on the resilience of interdependent networks against random node failure. A. Completely Interdependent Networks The system equations for this case (q = can be written as: ψ = 2 ( e λ Ap ( e λ Bp = p. (6 Thus for any strength of attack, we can obtain the robustness of the network (ψ by solving (6. It is known from literature [] that completely interdependent networks exhibit a firstorder phase transition, wherein ψ abruptly falls to when falls below a particular threshold c (critical fraction. THEOREM III. The robustness (ψ c = p c of an interdependent ER network with non-identically distributed layers at the critical point ( = c can be obtained by solving the following equations: f A log f A f A + f B log f B f B =, (7 f B = f r A, (8 where f A e λ Ap c and f B e λ Bp c, λ A and λ B are the mean degrees of the two layers, and r denotes the ratio r = λ B /λ A. The critical fraction of nodes ( c is then given by: c = [λ A f A ( f B + λ B f B ( f A ] /2. (9 Sketch of Proof. Note that the critical point ( c is the minimum value of for which the slopes of the functions g (p 2 ( e λap ( e λbp and g 2 (p p are equal. This idea has been used to derive the equations (7-(9, the details of which can be obtained from [9]. COROLLARY III.2 For identically distributed ER networks, c = /λ, and ψ c =.2864/λ. Proof. Here (7 simplifies to f log f = (f /2, which can be solved numerically to yield f = Thus ψ c = p c = log f/λ, and c = [2λf( f]. The result in the corollary is the same as that in literature []. Thus it establishes the equivalence of our method of analysis with existing ones. We have furthermore shown that the metrics of robustness (ψ vs of such non-identically distributed networks can be approximated by an equivalent identically distributed network whose mean intra-layer degree λ is the harmonic mean of λ A and λ B. This analysis is also included in the report [9], by which the network robustness (ψ and critical fraction of nodes ( c can be approximated by the following equations: ψ = p 2 ( e λ p 2, ( ( / c. ( Taking x e λap. the approximation error can be written as: f e (r = x r+ + 2x 2r r+ x x r x 4r r+. (2 It has been shown in report [9] that lim r f e (r =. Further graphical analysis gives us that f e (r becomes very close to for values of r >.5. Thus for practical purposes, we can consider the harmonic approximation to be an adequate measure for approximate network performance for interdependent networks generated as non-identically distributed ER graphs. λ B. Partially Interdependent Networks The analysis for partially interdependent networks is significantly more complicated due to the quadratic nature of the system equations. Thus we simplify the problem by focusing on the case of λ A = λ B, i.e. identically distributed layers. Under these conditions, the system equations can be written as: ψ = ( q( e λp + q 2 ( e λp 2 = p. (3 The phase transition properties of partially interdependent networks is much more interesting as compared to completely interdependent networks. This is motivated by the observation shown in literature [7], which indicates that phase transitions in partially interdependent networks is not exclusively first order in nature. In such systems, as the coupling between the two layers of the network (q is decreased, the phase transition order changes from first order (for higher values of q, where the robustness abruptly falls to after a certain threshold, to second order, where the robustness smoothly falls down to. Let this phase transition point be represented by q c. It should be noted here that for partially interdependent networks, the critical fraction of nodes ( c is only defined in the range of q for which the phase transition is first order in nature. To resolve this issue and to promote a unified analysis of such problems, we have defined the term critical strength of attack ( + as the minimum value of which results in a non-zero solution to the system equation (3. Let the non-zero solution to (3 at = + be represented by p +. Thus for q > q c, when the phase transition of the system is first order, + = c and p + = p c, whereas when the phase transition is second order (q < q c p gradually goes to with decreasing, i.e. p + at = +. THEOREM III.3 For a partially interdependent ER network with identically distributed layers, if q < q c, the critical strength of attack is given by: + = [λ( q], (4 where λ is the mean intra-layer degree, q is the fraction of interdependent nodes, and q c is the phase transition point. Sketch of Proof. Let us re-write the system equation under the critical strength of attack condition ( = +, p = p + using the idea that p + :

4 q Fig. 2. Comparison between the numerical solutions (markers and theoretically derived formulas (solid for + and q c. The three cases shown here: correspond to mean intra-layer degrees λ = [2, 6, 2]. The dotted vertical lines correspond to the theoretically obtained values of q c given by (6. lim p + [ ( q + e λp+ p + +q 2 + ( e λp+ 2 ] =. (5 p + Evaluating the above limit, we get the required result. THEOREM III.4 For a partially interdependent ER network with identically distributed layers, the phase transition point (q c is given by: q c = λ + 2λ +, (6 λ where λ is the mean intra-layer degree. Sketch of Proof. Note that the critical point solution to (3 below the phase transition point is p = since the network phase transition is of second order. For q > q c, the phase transition is of first order and thus (3 has a non-zero finite solution. We have used this idea to analytically derive the phase transition point for networks generated as ER graphs, the details of which can be obtained from [9]. Comparison of the numerical solutions to the system equations with the theoretically derived formulas for both + and q c is presented in Fig. 2. We can clearly observe from the figure that when the network phase transition is second order, i.e. q < q c, (4 accurately follows the numerical solution. The phase transition points (q c where the transition order changes is also very accurately presented in Fig. 2. However when the phase transition order of the network is first order, we cannot perform any simplification of the quadratic system equations since p + = p c is finite and positive. Thus for q > q c, we have no choice but to numerically solve the system equations (3 to theoretically estimate c. C. Applications: Real World Networks Although the analytical results presented in the previous section have been derived for networks constructed as ER graphs, the qualitative features of our results hold for random graphs with arbitrary degree distributions as well. Note that the essential step, which leads to the formulation of the mathematical equations relating robustness (ψ to the structural parameters (q, λ, was the simplification: F 3 (λ, p = k P [k]( pk = e λp. For networks with arbitrary degree distribution, such simplifications would not exist in most cases. However, if we know (or estimate the degree distribution of the nodes in the networks, we can numerically estimate the values of the function F 3 (λ, p and its derivative with respect to p. From the discussion in the previous section, it is evident that if the value and derivative of the function F 3 (λ, p is known to us, we can write the system equations ( (2 in terms of F 3. Thus for the theoretical estimation of the performance of real world networks whose degree distribution is known to us, we can numerically estimate the values of the function F 3 (λ, p and its derivatives at the required values of p and thereafter numerically solve the system equations modeling the cascading failure: ( (2. Note that the system equations are derived under the assumption that the network layers are locally tree-like. Random graphs have this property as long as the network is sparse [6]. However, real world networks would usually not follow such strict mathematical properties and thus the system equations derived in these cases are just theoretical approximations to the actual performance. The comparison of this theoretically predicted performance and simulation results will be presented in Section V. IV. ANALYSIS: TARGETED ATTACK The underlying assumption in the analysis presented in Section III is that the attacker randomly fails nodes. However, attacks on the nodes with high connectivity would undoubtedly be a better strategy from the perspective of disrupting the network. Previously, we have considered the case where the attacker has no knowledge about the node degrees and thus attacks nodes randomly. In this section, we analyze the other extreme situation: where the attacker has complete intra-layer degree information and attacks nodes in the strict descending order of the degrees. We present the analysis of completely interdependent networks with identically distributed layers. For a particular instance of network generation, we index the nodes in layers A and B in the increasing order of their intra-layer degrees. Let this index be represented by i: i N, where N is the number of nodes in each layer. Let k {i} A (or k{i} B be the intra-layer degree of the i th node in the layer A ( or B of the network. Let sequence {ρ} be a permutation of the number set {, 2,, N}, where N is the number of nodes in each layer, ρ i denotes the i th element of {ρ}, and the indices of any interdependent pair of nodes is represented by (i, ρ i. Thus ρ, representing an arbitrary ordering the nodes, can completely specify the one-to-one interdependence structure. As previously, the initial attack leads to the failure of a fraction ( of the nodes in the network. Due to the perfectly targeted attack model, all top ( N nodes in both layers fail due to the initial attack. It should be noted at this point that the metrics of network robustness ψ and p are defined in a mean sense over all instances of network generation with specified intra-layer degree distribution of the nodes. Thus for the purpose of obtaining system equations for ψ and p with respect to indexed nodes, we need to take an expectation over all instances for a specific degree distribution. Due to the identical distribution of the two layers of the network and the simultaneous attack on both layers, ψ and p would be the same for both layers. Thus we will drop the subscript denoting layer from the terms ψ l and p l for simplicity. Similar to the previous case, we define functions

5 (i and G{l} 2 (i to simplify the representation of the system equations and obtain: G {l} ψ = E [ N [ p = E N N i= N i= I g {A} 2 (i g {B} 2 (ρ i ], (7 ] I g {A} (i g {B} 2 (ρ i, (8 where g {l} (i = k l(i λ l [ ( p kl(i ] and g {l} 2 (i = [ ( p kl(i ], and k l (i is the intra-layer degree of the i th node in layer l having mean degree of λ l. Here E[ ] represents the expectation over all instances of network generation for a specific degree distribution of the nodes and I is an indicator function representing the targeted removal of the top ( N fraction of the indices i, i.e. I = if i (, N. THEOREM IV. The robustness of an interdependent network with identically distributed layers against targeted attack can be approximated by the two inequalities: ψ 2 [ ( p λ ][ ( p λ ], (9 p [ ( p λ ][ ( p λ ] N N i= Λ i λ, (2 where Λ i = E[k {i} ] denotes the order statistics of the intralayer degree (k {i} of the node with the i th lowest degree, λ is the intra-layer degree, and λ E[Λ i I ] where I is an indicator function representing the targeted removal of ( N nodes of highest degrees. Sketch of Proof. To prove this result, we use the concavity of the function f(x = ( p x for p [, ] and invoke the Jensen s Inequality on the system equations (7-(8.The details of this can be obtained from [9]. We have used (9,(2 to design a numerical method based algorithm to estimate the value of ψ and p for particular and λ. The order statistics [] for the degrees (Λ i can be estimated for random graph generators or real world networks if the degree distribution is known or can be estimated. For the ER model, the intra-layer degrees follow Poisson distribution whose order statistics can be readily obtained []. V. RESULTS AND DISCUSSIONS We have designed a test bench to simulate the mechanism of cascading failure in interdependent networks initiated by random or targeted removal of a certain fraction ( of nodes. The robustness is depicted by plotting the fractional size of the mutually connected component in the steady state (ψ, for various strengths of attacks (. The designed algorithm obtains the largest connected component in each layer at each stage of the recursive cascading failure and outputs the steady state fractional size of this component. We have considered networks where the constituent layers of size N = 5 are generated by the Erdos-Renyi model from the NetworkX Python library [] and the figures represent the average results of 2 independent runs ( network construction instances with 2 node removal instances each. The execution time ψ Fig. 3. Completely interdependent networks under randomized attack. The three sets of plots represent three cases with o : λ B =, : λ B = 5, and :λ B = 3; λ A = 3 for all three cases. ψ Fig. 4. Partially interdependent networks under randomized attack. The three sets of plots represent three cases with o : q =., : q =.5, and : q = ; λ = 4 for all three cases. for the simulation results, represented by the markers with dashed lines in Figs. 3-6, is on the order of hours, whereas the theoretical approximations, represented by the solid lines, can be obtained on the order of seconds. This demonstrates a key aspect of our work; fast approximation of robustness without expensive simulations. We present a brief discussion on the simulation figures next. Fig. 3 gives the comparison between the simulation results and theoretical estimation for completely interdependent networks with non-identically distributed layers. It can be easily verified from Fig. 3 that the analytical expressions very closely fit the simulation results for all three cases. This figure thus indicates that the robustness of interdependent networks can be very closely approximated by theoretical expressions instead of computationally intensive simulations. This idea is fundamental to this piece of work where the ultimate goal is to relate network robustness to the structural parameters of the network. Fig. 4 gives the same comparison as above for the case of partially interdependent networks. From Fig. 3 and Fig. 4, we can also compare the theoretical estimation and simulation results for the critical fraction of nodes ( c, which is defined as the maximum attack strength or equivalently the minimum value of which results in a non-zero MCC size in the steady state. For the case of completely interdependent networks, c is estimated by ( and gives the following numerical results: c = [.7294, 92, 47] for λ B = [, 5, 3]. We can clearly observe from the figure that the theoretical estimates are quite accurate. For the case of partially interdependent networks, c (or technically +

6 ψ Fig. 5. Targeted attack on completely interdependent ER networks with mean degrees of : λ = 4, and o : λ = 8. ψ Fig. 6. Real world networks with randomized attack. The three sets of plots represent three cases with o : q =., : q =.5, and : q =. as discussed previously can be estimated by (4 when the phase transition order is second order. For this case (λ = 4, the phase transition point is given by (6, which can be calculated to be q c =.5. Thus for the first two cases (q =.,.5, the theoretical values of + are.277 and.5 respectively which is also very accurate. The mathematical approximations for the case of targeted attack has been presented in Fig. 5. It can be observed that the gap between the simulated and theoretically predicted performance is larger than the case of randomized attack but still the approximations are close enough for us to gain a better qualitative understanding of the impact of structural parameters on the robustness (ψ and even obtain quantitative approximations for these metrics. For testing our method of analysis on real world networks, we have used the Gnutella peer-to-peer network dataset available from [2]. The interdependent network comprises two identical layers of this peer-topeer network and the interdependency has been assigned randomly. It can be clearly seen from Fig. 6 that the theoretical estimation of the performance follows the simulated performance reasonably accurately. VI. CONCLUSION AND FUTURE WORK Our aim in this work is to estimate the robustness of interdependent networks against failure of nodes and relate it to the structural parameters of network. Furthermore for networks generated as ER (Erdos-Renyi graphs, we have been able to mathematically derive closed form expressions for the critical fraction of node ( c and phase transition points (q c, the existence of which was known in literature [4]. Although such theoretical results may not accurately apply to real world networks due to the inherent simplicity of the ER model, we have shown that the qualitative features of these results are preserved even for the case of real world networks and thus the relationships obtained here are vital towards understanding the effect of structural parameters of network on its robustness against node failures. Another reason depicting the importance of such a theoretical framework for robustness estimation is that these can be used to predict the robustness of a network given its structural parameters without resorting to brute force simulations. The drawback of such simulations is that for many real world networks, they might be computationally very expensive. Furthermore simulations require the knowledge of the exact architecture of the constituent layers of the network, which can be unknown in many cases. For the case of targeted attack, we have developed numerical lower bounds for the performance metrics for any general network. One of the main aspects on which we have not focused is the impact of interdependency structures on the network, which we have considered in this work to be designed randomly without the knowledge of the intra-layer degree distribution. In future, we plan to generalize our current analysis and also explore similar estimation problems for more practical network models like Scale-Free graphs. We feel that our results would enable the academia to gain a better understanding of robustness of interdependent networks against node failures and its relationship with the structural parameters. 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