Assessing Network Vulnerability Under Probabilistic Region Failure Model

Transcription

1 2011 IEEE 12th Internationa Conference on High Performance Switching and Routing Assessing Networ Vunerabiity Under Probabiistic Region Faiure Mode Xiaoiang Wang, Xiaohong Jiang and Achie Pattavina State Key Laboratory of Nove Software Technoogy, Department of Computer Science and Technoogy, Nanjing University, China Emai: Future University Haodate, Hoaido, Japan Emai: Department of Eectronics and Information, Poitecnico di Miano, Miano, Itay Emai: Abstract The mission critica networ infrastructures are facing potentia arge region threats, both intentiona (ie EMP attac, bomb exposion and natura (ie earthquae, fooding. The avaiabe research on region faiure reated vunerabiity studies generay adopt a ind of simpe deterministic region faiure modes, which can not capture some important features of rea region faiure scenarios, where a networ component in the region ony fais with certain probabiity, and more importanty, such faiure probabiity tends to vary with both its dimension and its distance to faiure center. In this paper, we provide a more genera probabiistic region faiure mode to capture the ey features of a region faiure and appy it for the networ vunerabiity assessment. To faciitate such assessment, we adopt a grid partition-based scheme to estimate various statistica networ metrics under a random region faiure. A theoretica framewor is aso estabished to determine a suitabe grid partition such that a specified estimation error requirement is satisfied. The grid partition technique is aso usefu for identifying the vunerabe zones of a networ, which can guide networ designers to initiate proper networ protection against such faiures. The wor in this paper heps us more deepy understand the networ vunerabiity behavior under region faiure and faciitates the design and maintenance of future highy survivabe mission critica networs. I. INTRODUCTION Communication networ has been a tremendous success over ast decade, and it is now a very important part of our daiy ife. Peope are increasingy reying on arge-scae communication networs (e.g. Internet to provide reiabe and mission critica communication services for government, miitary, businesses, heathy care, etc. On the other hand, however, these mission critica networ infrastructures are now facing more and more potentia arge scae threats, both intentiona (ie bomb exposion, eectromagnetic puse (EMP attac and natura (ie earthquae and fooding 1], 2]. For instance, the Taiwan earthquae in Dec cutoff seven of eight cabes passing through the area, sowing down This wor is partiay supported by the Nationa Science Foundation of China under Grant Nos and , and JSPS grants (B X. Jiang is aso a visiting researcher of the State Key Laboratory for Nove Software Technoogy, Nanjing University, China. communications in part of Asia for months. Therefore, the networ vunerabiity studies under arge scae region faiure are essentia for the design and maintenance of future highy survivabe mission critica networs. The networ survivabiity and its reated routing, protection and restoration issues have been widey investigated (e.g., 3] 6]. These studies mainy focus on singe physica in faiure scenario as it is the main faiure mode that interrupts networ norma operation 7]. As the networ scae is increasing and networ robustness requirement is becoming more stringent, recenty researchers are paying attention to the mutipe faiure scenarios to ensure higher networ survivabiity. The wors in 8], 9] expored the connectivity recovery mechanism in presence of dua faiures, whie the wors in 10] 12] examined the in capacity provision issue under obivious routing to dea with mutipe independent faiures. It is notabe that these wors mainy focus on the networ s ogica connection whie negecting its rea geographica ayout. Recenty, some interesting research has been conducted to understand the impact of geographicay correated region faiures. A. Sen et a. 13] studied the impact of region faiure on networ physica connectivity, and reveaed that the max-fow is no ong equa to the min-cut in region faiure-oriented survivabe networ design. S. Neumayer et a. in 14], 15] examined the networ vunerabiity under region faiure of ine and circe shapes, proposed an efficient agorithm to identify the most vunerabe point in a networ, i.e., the ocation where the region faiure can induce the maximum capacity degradation. By using the computationa geometric toos, the authors further studied the statistica networ property under a random attac in the form of ine cut 16]. In genera, the region faiure modes adopted in avaiabe studies can be regarded as a ind of deterministic faiure modes, in the sense that any component faing within the faiure region wi aways be destroyed (i.e., be destroyed with probabiity 1. Such deterministic faiure modes significanty simpify the networ vunerabiity anaysis, but they faied to refect some important behaviors of physica attacs /11/$ IEEE 170

2 Fig. 1. Geographica ayout of U.S. bacbone networ. Fig. 2. Probabiistic region faiure mode, M = 3. (such as physica bomb exposion, EMP attac or natura disasters (such as earthquae, hurricane and fooding, where a networ component ony fais with certain probabiity, and more importanty, such faiure probabiity tends to vary with both its distance to faiure center and its dimension (ie the in ength. With these observations, we beieve a probabiistic region faiure mode addressing these properties wi be more suitabe for networ vunerabiity assessment under region faiures. In this paper, we consider such a probabiistic faiure mode and appy it for the networ vunerabiity assessment. Actuay, a probabiistic region faiure mode has just been proposed recenty by P. Agarwa et a. 17] for the reiabiity study of WDM optica bacbone networs, where a networ in wi fai with certain probabiity depending ony on its distance to the region faiure center. The main contributions of this paper are as foows: We provide a genera probabiistic region faiure mode, where a networ in may fai with certain probabiity due to a region faiure, and such probabiity depends on both the in ength and how the in is intersecting with faiure region. The mode covers the deterministic faiure modes in 13], 15] as specia cases. Based on the new faiure mode, we then adopt a grid partition scheme to efficienty estimate the statistica networ vunerabiity metrics. A theoretica framewor is aso estabished to determine a suitabe grid partition such that a specified estimation error requirement is satisfied. The grid partition technique can aso hep us to identify the vunerabe zones of a networ. We demonstrate through experimenta study that negecting probabiistic behavior of a region faiure may significanty over-estimate or under-estimate its impact on networ reiabiity. The rest of this paper is organized as foows. The new probabiistic faiure mode as we as concerned metrics are introduced in Section II. Section III deas with both the metric estimation based on the new faiure mode and the corresponding estimation anaysis. Finay, Section IV provides the numerica resuts to demonstrate the efficiency of the new faiure mode. II. MODELS AND PERFORMANCE METRICS We consider a physica ayer networ topoogy bounded in a two dimensiona Eucidean pane Z. To represent the networ geographica ayout, we proportionay shrin it as an undirected graph G(V,E with node set V and in set E. An exampe of such graph is shown in Fig. 1. A. Probabiistic Faiure Mode Based on the basic essons we earned from natura disasters (ie earthquae and attacs (ie bomb exposion, we can find some genera behaviors of a disaster and attac: 1 the damage region is usuay confined within a geographica area, beyond it the damage can be negected, and 2 the energy intensity at a ocation monotonousy attenuates as its distance from disaster center increases 18]. Based on these behaviors and aso the fact that the probabiity of causing damage to a networ component is proportiona to the energy intensity observed at the ocation of this component, we introduce here a genera probabiistic region faiure mode to mimic as much as possibe the behaviors of reaistic disasters (or attacs whie trying to eep the networ vunerabiity anaysis tractabe. Definition 1: Probabiistic Region Faiure (PRF is defined as foows: 1 The faiure region is divided into a set of M consecutive annuus by M concentric circes with radius m r, m = 1,...,M, as iustrated in Fig A in segment of arbitrary short ength δ fais with probabiity δ q m (0,1 if it fas within the m-th annuus. 3 The parameter q m is monotonousy decreasing, i.e., q m > q m+1, 1 m M 1. Based on the new PRF mode, the faiure probabiity P m for a in segment m that fas within the m-th annuus can be determined as 1 P m = 1 im n ( n 1 q m m = 1 e qmm (1 n 1 For simpicity, we use notation m (resp. to denote both a in segment (resp. a in and its ength. 171

3 In genera, a in spanning mutipe annuus of a faiure region can be regarded as mutipe in segments, each fas within one annuus, and the in wi fai if any of its in segment(s fais. Therefore, the faiure probabiity P of a in is given by P = 1 M m=1 (1 P m = 1 e M m=1 qmm (2 For the in iustrated in Fig. 2, its faiure probabiity P can be expressed as P = 1 e (q11+q22+q33 where 2 = 2a + 2b and 3 = 3a + 3b. B. Vunerabiity Metrics For the networ vunerabiity assessment, we adopt the foowing metrics: 1 (Remaining Lin Capacity The expected capacity of a remaining undisrupted ins after a PRF. 2 (Pairwise Capacity Reduction The expected traffic reduction between a pair of given nodes due to a PRF. 3 (Pairwise Connecting Probabiity The probabiity that a pair of given nodes with path protection is sti connected after a PRF. For a given networ, one straightforward approach to cacuating one above metric is to first partition the overa networ area into some disjoint and uniform region faiure ocation (RFL zones. Definition 2: (RFL Zone A RFL zone for a specified metric is a networ sub-area that any PRF with center faing within it wi aways induce the same vaue of to the networ. For a specified metric, suppose that we have aready divided the networ depoyment pane Z into a set of disjoint RFL zones {Z n }, where a PRF faing within Z n induces the vaue n of to the networ. Then the cacuation of the metric of overa networ can be expressed as = Z n Z n Z n (3 Here, Z n / Z is just the probabiity that the PRF s center fas within the RFL zone Z n. To appy (3 for the cacuation of a metric, we need to find out a the corresponding RFL zones. It is notabe, however, that with the new PRF mode the faiure probabiity of a in now varies with both its ength and how it intersects with the faiure region, which maes it hard to appy (3 for the evauation of. In the next section we adopt an efficient scheme for the estimation of. III. ESTIMATION OF VULNERABILITY METRICS In this section, we first introduce a grid partition scheme for the estimation of a metric, then provide a theoretica framewor to determine a suitabe grid partition such that a specified estimation error requirement is satisfied. Fig. 3. A grid-partitioned networ. A. Grid Partition-based Metric Estimation As shown in Fig. 3, we appy a grid to eveny partition the networ pane Z into N square grid ces {Z n,n = 1,...,N} with side ength d each, where N = Z /d 2. Based on this grid partition, one simpe way to estimate a metric is to regard each grid ce here as a RFL-zone, and tae the induced vaue n by a randomy seected point in each ce Z n as an estimation of this ce. In this way, we can get an estimation of the concerned metric based on (3. Agorithm 1: Estimation of a Metric Input: Networ topoogy G(V,E in Z and grid ce size d Output: Metric estimation 1: = 0 and N = Z /d 2 2: for each unvisited grid Z n in Z do 3: cacuate n by using a randomy seected point in Z n 4: = + n (1/N 5: end for 6: return Since each ce in the Agorithm 1 may not be a RFL zone, so an estimation error may be introduced between and. To define the estimation error between and, we adopt the so caed (1 ε-approximation. The estimation is caed an (1 ε-approximation to if the foowing inequaity hods ( min, (1 ε, 0 < ε < 1. (4 The foowing emma identifies a condition under which the estimation from the Agorithm 1 is an (1 ε-approximation to, which wi hep us to determine a suitabe grid partition (or equay a suitabe grid ce size d. Lemma 1: For a given grid partition {Z n,n = 1,...,N}, et (u n (resp. (v n denote the induced vaue of if a PRF happens at point u (resp. v in grid ce Z n. The Agorithm 1 based on this grid partition provides an (1 ε-approximation 172

4 to if the foowing condition hods: ( min (u n, (v n (v n (u n (1 ε, u,v Z n,n = 1,...,N. (5 Proof: For a grid ce Z n, et + n (resp. n denote the maximum (resp. the minimum vaue of that any point in Z n may induce. Based on the Agorithm 1 we can see that ( min, (1/N N n=1 n (1/N N N n=1 (1 ε + n N n=1 + n n=1 + n The ast inequaity is due to the condition (5. B. Remaining Lin Capacity = 1 ε. (6 For a networ G(V,E, we use c to denote the capacity of in and use P to denote faiure probabiity of this in due to a random PRF. Then the expected capacity C of a remaining undisrupted ins after a PRF is given by C = E(1 P c (7 In particuar, by setting c = 1 for a E, we can get the expected number of remaining undisrupted ins. The foowing theorem indicates that by a proper setting of grid ce size d, the Agorithm 1 can be used to get an (1 ε- approximation to the metric C. Theorem 1: The grid partition-based metric estimation Agorithm 1 induces an (1 ε-approximation to the metric C if the grid size d there satisfies the foowing condition: d rm (rm 2 (α/2 2 n(1 ε,α= 2 M m=1 q m Proof: For a grid ce, et C (u (resp. C (v denote the induced vaue of C if a PRF happens at point u (resp. v in the ce. The Lemme 1 indicates that the Agorithm 1 induces an (1 ε-approximation to the metric C if for any grid ce there the foowing condition hods: ( C (u C(v min, (1 ε (9 C (v C (u We use P (u (resp. P (v to denote the faiure probabiity of the in if a PRF happens at point u (resp. v in a ce, and further use P + (resp. P to denote the maximum (resp. the minimum vaue of such faiure probabiity in the ce, then the above condition (9 hods if for any grid ce the foowing condition hods: ( min 1 P (u 1 P (v (8, 1 P (v (1 ε, E. (10 1 P (u Fig. 4. Mutipath routing between a pair of end nodes. This is because if (10 is true, then we have ( C (u C(v min, (1 P + c C (v C (u (1 P c (1 ε(1 P c (1 P = 1 ε (11 c Without oss of generaity, we assume P (u P (v > 0 in this proof. Now, we want to determine a suitabe grid size d to ensure that 1 P(u (1 ε for any in (i.e., to ensure 1 P (v the condition (10. From (2 we now that 1 P (u 1 P (v = e M m=1 (qm(u m e = e M M m=1 (qm(v m m=1 (qm((u m (v m (12 where m (u (resp. m (v denotes the ength of in segment(s residing in the m-th annuus centered at the point u (resp. point v. From (12 we can see that the ratio 1 P(u decreases as 1 P (v the vaue of ( m (u m (v increases. Notice that the points u and v are in the same ce, so the distance between them is at most 2d. Thus, we have m (u m (v 2 (rm 2 (rm 2d 2,m = 1,...M. (13 Therefore, to ensure the condition (10, we just need to determine a suitabe ce size d such that 1 P (u ( e M ( 1 P (v m=1 qm 2 (rm 2 (rm 2d 2 1 ε (14 By setting α = n(1 ε/ M m=1 q m, the condition (8 foows. C. Pairwise Capacity Reduction The mutipath routing technique has been widey adopted to improve the transmission capacity and survivabiity 10], 11], 19]. As iustrated in Fig. 4, in mutipath routing the traffic between a node pair (S,D is routed through mutipe indisjoint paths. Thus, the maximum pairwise capacity between a pair of nodes is just the sum of the capacity of a these paths. For a given node pair (S,D, suppose that K in-disjoint paths τ = {τ 1,,τ K } are aocated between them. Let t and Q τ denote the capacity and connecting probabiity of the -th routing path, respectivey. Under a random PRF, the 173

5 expected pairwise capacity reduction R(τ of the node pair (S,D with routing path set τ = {τ 1,,τ K } is given by ] K K K K R(τ = (t Q τ / t = t (1 P / t (15 The (15 indicates ceary that the main probem in cacuating R(τ is to determine the connecting probabiity Q τ = (1 P for each path there. The foowing emma identifies a condition under which the Agorithm 1 induces an (1 ε-approximation to the connecting probabiity of any routing path of the node pair (S,D. Lemma 2: For a given node pair (S,D with routing path set τ = {τ 1,,τ K }, et ω denote the maximum number of ins of any singe path in τ that may simutaneousy intersect with the faiure region of a PRF, then the Agorithm 1 guarantees an (1 ε-approximation to the connecting probabiity of any routing path in τ if the grid of size d there satisfies the foowing condition: d rm (rm 2 (β/2 2 n(1 ε,β= 2 ω M m=1 q m Proof: For a grid ce of size d, et Q (u (16 (resp. Q (v denote the connecting probabiities of path if a PRF happens at point u (resp. v in the grid ce. The Lemma 1 indicates that the Agorithm 1 induces an (1 ε-approximation to Q τ if for any grid ce there the foowing condition aways hods: min ( Q (u Q (v, Q (v Q (u (1 ε (17 Without oss of generaity, we assume Q (v Q (u. Then we just need to ensure the foowing condition: (1 P (u (1 ε (1 P (v (18 By referring to the proof of Theorem 1 we now that if we choose a suitabe ce size d 1 according to (8 and use ε 1 in stead of ε there, the Agorithm 1 can induce an (1 ε 1 - approximation to the probabiity 1 P of any ( in, i.e., we can guarantee that (1 P (u (1 ε 1 τ (u (resp. τ (v 1 P (v. Let denote the set of ins intersecting with a PRF centered at point u (resp. v, then with the ce size d 1 the eft side of formua (18 wi satisfy (1 P (u = (1 P (u (1 ε 1 (1 P (v τ (u (1 ε 1 τ(u (1 ε 1 ω τ (u τ (u τ(v τ (u τ(v (1 P (v (1 P (v (19 For the right side of formua (18, it foows (1 ε (1 P (v = (1 ε (1 P (v τ (u τ(v (1 ε τ (v τ (u τ(v (1 P (v (20 Then we now that to guarantee (18 we just need to ensure that (1 ε 1 ω (1 P (v (1 ε (1 P (v τ (u τ(v (21 The above inequaity indicates that ε 1 shoud satisfy the foowing condition (1 ε 1 ω (1 ε i.e. ε 1 1 ω 1 ε (22 By choosing ε 1 according to the above condition and appy it in stead of ε in (8, the ce size requirement (16 foows. Remar 1: For a given node pair (S,D with routing path set τ, the maximum number of ins ω of any singe path in τ that may simutaneousy intersect with the faiure region of a PRF can be determined by appying the agorithm introduced in 15], 20]. Based on (15, Lemma 2 and a simiar proof as that of the Theorem 1, we are now abe to estabish the foowing theorem regarding the (1 ε-approximation of metric R(τ. Theorem 2: By setting the ce size d as that in Lemma 2, the (1 ε-approximation of R(τ can be achieved with the Agorithm 1. D. Pairwise Connecting Probabiity The path protection is a widey adopted technique for providing reiabe communication with ow protection cost, where two in-disjoint paths, a primary and a bacup, are usuay assigned to each connection. For a connection with primary and bacup path set τ = {τ 1,τ 2 }, its pairwise connecting probabiity A(τ can be expressed as A(τ = 1 (1 Q τ = 1 1 ] (1 P (23 where Q τ is the connecting probabiity of path. The foowing theorem indicates the condition for achieving the (1 ε-approximation of metric A(τ. Theorem 3: By setting the ce size d as that in Lemma 2, the (1 ε-approximation of A(τ can be achieved with the Agorithm 1. Proof: For a grid ce of size d, et A(τ (u (resp. A(τ (v denote the corresponding pairwise connecting probabiities of path set τ = {τ 1,τ 2 } if a PRF happens at point u (resp. v in the grid ce. Without oss of generaity, we assume A(τ (v A(τ (u, and et Q (u (resp. Q (v denote the 174

6 corresponding connecting probabiities of path. Then we just need to ensure the foowing condition: ] 1 (1 Q (u (1 ε 1 (1 Q (v (24 By referring to the Lemma 2, we now that if we choose a suitabe d 2 according to (16 and use ε 2 instead of ε there, the Agorithm 1 can induce an (1 ε 2 -approximation to Q τ of any path. Let Q denote the maximum possibe vaue of Q (u (and aso of Q (v, we can guarantee that Q (u (1 ε 2 Q. Therefore, the eft side of formua (24 wi satisfy 1 (1 Q (u 1 ( 1 (1 ε2 Q (25 Since Q Q (v, the right side of formua (24 satisfies ] ] (1 ε 1 (1 Q (v (1 ε 1 (1 Q (26 Then we now that to guarantee (24 we just need to ensure that ] ( 1 1 (1 ε2 Q (1 ε 1 (1 Q (27 By expanding the eft and right sides of (27, we can see that the above inequaity is satisfied when 1 1 ε ε 2 ε (28 To get a arge grid size d, we set ε 2 = ε and appy it instead of the ε in (16, then the requirement of ce size d foows. IV. NUMERICAL RESULTS In this section, we provide some numeric resuts to demonstrate the appication of the new PRF mode in networ vunerabiity anaysis. A. Experimenta Settings The U.S. bacbone networ shown in Fig. 1 is adopted in our anaysis, which consists of of 26 nodes and 41 bidirectiona edges. We quantify the average in ength as 111, the maximum in ength as 187, and the minimum in ength as 66. The capacity for each in is set as 1. The networ depoyment pane is set as a rectange with ength 760 and hight 450. Based on the formua (1 and data from reaistic faiure scenarios, we are abe to determine an appropriate vaue of parameter q m in the PRF mode. In our experiment, we regard a in segment of ength 10 as a basic in unit and use notation p m to denote its faiure probabiity if it fas within the m-th annuus of a PRF. Then, we have q m = n(1 p m /10. The parameters adopted in the experiment are isted in Tabe I. TABLE I PARAMETERS IN PRF MODEL p m q m p m , q m In our study, a the pairwise metrics are evauated for node pair (2, 25 in the U.S. bacbone networ, for which the Suurbae agorithm 21] is adopted to find two in disjoint paths for the node pair with the minimum tota ength. The experiment resuts based on the Agorithm 1 are generated with the setting of ε = 0.5. However, we notice that these resuts are simiar with the resuts achieved by setting ε = 0.2 and 0.1 respectivey. B. Vunerabiity Metrics vs.prf Mode We first show the efficiency of the PRF mode on evauating the networ vunerabiity. Fig. 5 iustrates how concerned networ metrics vary with the parameter r. It is notabe that the settings of ( = 1,p 2 = 1 and ( = 1,p 2 = 0 there correspond to the scenarios of deterministic region faiure mode. The resuts here indicate ceary that the deterministic mode may resut in a significant overestimation or underestimation of networ vunerabiity. We can aso see that as r increases (and thus faiure region becomes bigger, the estimation gap between the probabiistic mode and the corresponding deterministic ones tends to increase for a metrics. One attractive advantage of our networ grid partition framewor adopted in the Agorithm 1 is that it heps us in identifying the most vunerabe networ zones, where a region faiure can cause the most harmfu destruction to the networ. Fig. 6 iustrates the most vunerabe networ zones under the PRF mode with the setting of r = 20, = 0.7 and p 2 = 0.3. It is generay expected that to remove the maxima number of ins, the region faiure shoud be centered around the area with high in density. The resuts in Fig. 6(a indicate that such vunerabe area may not be exacty over the intersection points among ins. Moreover, due to the redundant routing or reiabe routing with protection, it is hard to competey disconnect any two nodes except the scenarios that the region faiure happens near either of them, as iustrated in Fig. 6(b and Fig. 6(c for the connection between nodes 2 and 25. V. CONCLUSION This paper studied how a random and geographicay correated region faiure affect networ performance, which is an essentia issue for designing highy survivabe networ infrastructure to support mission critica appications. A new and genera probabiistic region faiure mode is proposed for assessing the networ vunerabiity under arge scae region faiure, and the reated theoretica framewor has aso been deveoped to support the appication of this new mode. One of our our future wor is to deveop a more efficient approach based on samping theory to estimate the statistic networ metrics under more reaistic continuous PRF modes. 175

7 Remaining Lin Capacity =0 =0.1 =0.7 =1 r Pairwise Capacity Reduction =0 =0.1 =0.7 =1 r Pairwise Connecting Probabiity =0 =0.1 =0.7 =1 r (a Remaining in capacity vs. r (b Pairwise capacity reduction vs. r Fig. 5. Vunerabiity metrics vs. r. (c Pairwise connecting probabiity vs. r (a Vunerabe zones for remaining in capacity (b Vunerabe zones for pairwise capacity reduction between nodes 2 and 25 Fig. 6. Iustration of vunerabe networ zones for r = 40, = 0.7 and p 2 = 0.3. (c Vunerabe zones for pairwise connecting probabiity between nodes 2 and 25 REFERENCES 1] M. Liotine, Mission-Critica Networ Panning. Norwood, MA, USA: Artech House, Inc., ] A. Maropouou, G. Iannaccone, S. Bhattacharyya, C. Chuah, and C. Diot, Characterization of faiures in an IP bacbone, in Proceedings of INFOCOM 2004, March 2004, pp ] R. Cohen, K. Erez, D. Avraham, and S. Havin, Breadown of the internet under intentiona attac, Phys. Rev. Lett, no. 86, pp , ] J. Zhang and B. Muheriee, A review of faut management in WDM mesh networs: basic concepts and research chaenges, IEEE Networ, vo. 18, no. 2, pp , Mar-Apr ] Q. She, X. huang, and J. P. Jue, Maximum survivabiity using two disjoint paths under mutipe faiures in mesh networs, in IEEE Gobecom 2006, ] A. Tizghadam and A. Leon-Garcia, On congestion in mission critica networs, in In Worshop on Mission Critica Networs (MCN with IEEE INFOCOM, ] D. Turner, K. Levcheno, A. C. Snoeren, and S. Savage, Caifornia faut ines: understanding the causes and impact of networ faiures, SIGCOMM Comput. Commun. Rev., vo. 40, pp , August ] H. Choi, S. Subramaniam, and H. ah Choi, On doube-in faiure recovery in WDM optica networs, in in Proceedings of IEEE INFO- COM, 2002, pp ] S. Kini, S. Ramasubramanian, A. Kvabein, and A. F. Hansen, Fast recovery from dua in faiures in ip networs, in Proceedings INFO- COM 2009, ] R. Zhang-Shen and N. McKeown, Designing a faut-toerant networ with vaiant oad-baancing, in IEEE INFOCOM Mini-Conference, 2008, pp ] M. Kodiaam, T. V. Lashman, and S. Sengupta, Resiient routing of variabe traffic with performance guarantees, in in Proceeding of 17th IEEE Internationa Conference on Networ Protocos (ICNP 2009, October 2009, pp ] Y. Wang, H. Wang, A. Mahimar, R. Aimi, Y. Zhang, L. Qiu, and Y. R. Yang, R3: resiient routing reconfiguration, in Proceedings of the ACM SIGCOMM 2010 conference, 2010, pp ] A. Sen, S. Murthy, and S. Banerjee, Region-based connectivity - a new paradigm for design of faut-toerant networs, in IEEE Internationa Conference on High Performance Switching and Routing (HPSR, Paris, France, June 2009, pp ] S. Neumayer, G. Zussman, R. Cohen, and E. Modiano, Assessing the impact of geographicay correated networ faiures, in in Proc. IEEE MILCOM 08, Nov ], Assessing the vunerabiity of the fiber infrastructure to disasters, in IEEE INFOCOM 2009, Apri 2009, pp ] S. Neumayer and E. Modiano, Networ reiabiity with geographicay correated faiures, in IEEE INFOCOM 2010, March ] P. K. Agarwa, A. Efrat, S. K. Ganjugunte, D. Hay, S. Sanararaman, and G. Zussman, The resiience of WDM networs to probabiistic geographica faiures, Tech. Rep Onine]. Avaiabe: hdavid/aeghszreport.pdf 18] B. F. Howe, Jr., and T. R. Schutz, Attenuation of modified mercai intensity with distance from the epicenter, BULLETIN OF THE SEIS- MOLOGICAL SOCIETY OF AMERICA, vo. 65, pp , ] S. Vempaa, M. Motiwaa, M. Emore, and N. Feamster, Path spicing, in ACM SIGCOMM, ] X. Wang, X. Jiang, and A. Pattavina, Assessing networ vunerabiity under probabiistic region faiure mode, tech. Report. Onine]. Avaiabe: jiang/paper/regionfaiurereport.pdf 21] J. W. Suurbae and R. E. Tarjan, A quic method for finding shortest pairs of disjoint paths, Networs, vo. 14(2, pp ,