Robust Routing in Interdependent Networks

Transcription

1 Robut Routing in Interdependent Network Jianan Zhang and Eytan Modiano Laboratory for Information and Deciion Sytem, Maachuett Intitute of Technology Abtract We conider a model of two interdependent network, where every node in one network depend on one or more upply node in the other network and a node fail if it loe all of it upply node. We develop algorithm to compute the failure probability of a path, and obtain the mot reliable path between a pair of node in a network, under the condition that each upply node fail independently with a given probability. Our work generalize the claical hared rik group model, by conidering multiple rik aociated with a node and letting a node fail if all the rik occur. Moreover, we tudy the divere routing problem by conidering two path between a pair of node. We define two path to be d-failure reilient if at leat one path urvive after removing d or fewer upply node, which generalize the concept of dijoint path in a ingle network, and rik-dijoint path in a claical hared rik group model. We compute the probability that both path fail, and develop algorithm to compute the mot reliable pair of path. I. INTRODUCTION Many modern ytem are interdependent, uch a mart power grid, mart tranportation, and other cyber-phyical ytem [], [], [], [4], [5]. In interdependent network, one network depend on another to properly function. For example, in mart grid, power generator rely on meage from the control center to adjut to the power demand, while the control center relie on the electric power to operate. Due to the interdependence, failure in one network may cacade to another. It i important to undertand the robutne of interdependent network which are prone to cacading failure. Mot previou tudie on interdependent network have focued on the network connectivity baed on random graph model, in the aymptotic regime where the number of node approache infinity [6], [7]. The finite-ize arbitrary-topology graph model, which repreent real communication and phyical network, have been largely overlooked in the interdependent network literature. A few exception include [4], [5], which model interdependent power grid and communication network by graph with topologie pecified by the real network. Similarly, we abtract interdependent network by graph with pecified topologie, which can be tailored for a wide range of application. In thi paper, we tudy robut routing problem in interdependent network, by characterizing the effect of failure in one network on the other network. For an overview of the problem and challenge, it i helpful to conider a implified cenario where a demand network depend on a upply network, illutrated by Fig.. Every node in the demand network Thi work wa upported by DTRA grant HDTRA---00 and HDTRA i upported by one or more node in the upply network. Thu, node in the demand network and node in the upply network can be viewed a demand node and upply node, repectively. Given that a demand node fail if it loe all of it upply node, upply node failure may lead to correlated demand node failure, which make it difficult to route traffic through reliable path in the demand network. We develop technique to tackle the failure correlation. Thi implified one-way dependence exit in current ytem. For example, router and proceor in a communication network depend on the electric power. Moreover, a we will ee later, the analyi baed on thi implified cenario can be applied to interdependent network under certain aumption. demand network upply network Fig.. Every node in the demand network G i upported by two node in the upply network G. The robut routing problem have been extenively tudied under both independent failure and correlated failure cenario. If edge or node fail independently, the mot reliable path between a ource-detination pair can be viewed a a hortet path, where the length i a function of the failure probability. In the cae of correlated failure, it i difficult to find a path with any performance guarantee in general [8]. If correlation only exit among edge or node that fail imultaneouly, the network can be viewed uing a hared rik group model (Fig. ) [9], [0]. The hared rik group model capture correlated failure in an overlay network when underlay failure occur, and i commonly ued to tudy the cro-layer reliability, uch a logical link failure caued by fiber failure in optical network [], [], [], [4]. The mot reliable path contain the mallet number of rik if all rik are equally likely to occur, and can be obtained by integer programming []. Interdependent network have imilaritie with the claical hared rik group model, in that two demand node hare a rik if they have at leat one common upply node. However, t

2 4 Fig.. A hared rik node group model, where node labeled by the ame number hare the ame rik. the key difference i that a demand node doe not necearily fail if a rik occur (i.e., a upply node fail), ince a demand node may have multiple upply node, wherea a node fail if it aociated rik occur in the claical hared rik group model. The reliability of a path in interdependent network, in contrat to the claical hared rik group model, can no longer be characterized by the number of rik that the path contain. For example, if all the node in a path depend on a ingle upply node and thu the path ha a ingle rik, removing a ingle upply node would diconnect the path. In contrat, if every node in a path ha multiple upply node, the path would be more robut and can reit a larger number of upply node failure, although the path ha more rik. In addition to the mot reliable path, a backup path can be ued to further improve reliability, through divere routing. Intuitively, a pair of reliable path hould hare the minimum number of rik (or be rik-dijoint) in the hared rik group model [], []. However, in interdependent network, it i eay to contruct example where two path that hare many upply node can withtand a larger number of upply node failure than two path that hare a maller number of upply node (e.g., Fig. ). New metric, other than the number of rik hared by two path, need to be identified to characterize their reliability t Fig.. The two number in each node repreent it two upply node. The two t path in the left figure hare two upply node, and can be both diconnected after removing upply node {,}. The two t path in the right figure hare four upply node, but they cannot be both diconnected after removing any two upply node. Divere routing problem have been tudied under correlated link failure. The correlation between a pair of logical link i obtained either by meaurement [5] or by analyi of the underlay phyical topology [6]. Heuritic algorithm have been developed to find multiple reliable path, and their performance wa evaluated by imulation [5], [7], t [8]. In contrat, we explicitly bound the gap between the failure probability and the optimization objective, and develop algorithm that have provable performance. In thi paper, we develop an analytically tractable framework to tudy the following robut routing problem in interdependent network. Single-path routing: Compute the probability that a pecified path fail. Obtain the mot reliable path between a ource-detination pair. Divere routing: Compute the probability that two pecified path both fail. Obtain the pair of mot reliable path between a ource-detination pair. By generalizing the concept of dijoint path to interdependent network, we characterize the level of dijointne between two path to tudy divere routing. In contrat to the claical hared rik group model where a node fail if it rik occur, in interdependent network a node fail if a combination of rik occur. In view of thi, our method extend the hared rik group model. To the bet of our knowledge, thi paper i the firt to tudy the robut routing problem in interdependent network. The ret of the paper i organized a follow. In Section II, we tate our model for interdependent network and failure. In Section III, we prove the complexity, and develop approximation algorithm to compute the path failure probability. In Section IV, we develop algorithm to find the mot reliable path between a pair of node. In Section V, we tudy the divere routing problem in interdependent network, and find a pair of reliable path whoe failure probability i minimized. Section VI provide numerical reult. Finally, Section VII conclude the paper. II. MODEL We conider a demand network G and a upply network G, where every demand node in G depend on one or more upply node in G. We aume that every upply node provide ubtitutional upply to the demand node, and a demand node i functioning if it i directly connected to at leat one upply node. To tudy the impact of node failure in G on G, it i equivalent to tudy the following model. Conider a graphg(v,e,s V ), where nodev and edgee are identical to node and edge in G, and S V are the upply node et, each of which i a et of node in G that provide upply to a node in V. In thi model, each node v i V i a demand node, upported by a et of upply node S i S V, and v i fail if all the node in S i fail. (Note that node V may have different number of upply node.) Finally, let,t V be a ource-detination pair. Under the condition that upply node fail independently with given probabilitie, and following the convention that, t do not fail, we tudy the robut routing problem in G(V,E,S V ). Remark. The analyi for thi model can be directly applied to interdependent network, a long a the interdependence i bidirectional (i.e., if v G depend on u G, then u depend on v a well) and failure initially occur in one

3 network. It uffice to oberve that, given a et of failed node S G, a node v G fail if and only if it upply node are all in S. Notice that the failure of v doe not further lead to node failure in G, becaue all the node that v upport, which are exactly the upply node for v due to the bidirectional interdependence, have failed. III. COMPUTING THE RELIABILITY OF A PATH If every node ha a ingle upply node, the path failure probability i given by ( p) r, where each upply node fail independently with probability p and the path i upported byr upply node. In contrat, if every node ha more than one upply node, computing the path failure probability become #P -hard. The proof can be found in the Appendix of the technical report [9]. Theorem. Computing the failure probability of a path i #P -hard, if every node ha two or more upply node and each upply node fail independently with probability p. Although it i #P -hard to compute, the path failure probability can be well approximated. We apply the olution to the DNF probability problem and propoe an (ǫ, δ)-approximation algorithm baed on importance ampling, which approximate the path failure probability to within a multiplicative factor ±ǫ with probability at leat δ. The DNF probability problem compute the probability that a Dijunctive Normal Form (DNF) formula i true, when literal are et to be true independently with given probabilitie. A DNF formula i a dijunction of claue, each of which i a conjunction of literal, and take the following form: (x x n ) (x x n ) (x m xm n m ). Let v v m be a path in G(V,E,S V ). The key obervation i that computing the path failure probability can be formulated by a DNF probability problem, in which a claue C i repreent a node v i in the path and the literal x i j in claue C i repreent the upply node of v i. For completene, we tate Algorithm that approximate the path failure probability, by adapting the algorithm that approximate the DNF probability in [0]. The intuition behind thi importance ampling algorithm i a follow. Some event, although rare, are important in determining the path failure probability, epecially when the path failure probability i mall. The algorithm ample in a pace coniting of important event, each of which i a et of upply node failure U that lead to the path failure. In thi pace, the failure of U may appear multiple time, given that multiple choice ofv i in Step may lead to the ameu in Step. The algorithm then remove the duplicated U via ampling in Step 4. To prove the correctne of the algorithm, we take the following two tep. Firt, following a imilar analyi to [0], we prove that the path failure probability i given by E[I] k m j n (v k ) p(uk j ), where E[I] i the expectation of I in Step 4 of the algorithm. Second, by repeating the loop a ufficiently large number of time, E[I] can be approximated to within factor ± ǫ with probability at leat Algorithm Etimating the path failure probability baed on importance ampling. Initialization: ) Given a path{v,v,...,v m }, let{u i j j =,...,n (v i )} denote the et of upply node of v i, where n (v i ) i the number of upply node of v i. Main loop: ) Among {v,v,...,v m }, randomly chooe v i with probability p(u i j)/ p(u k j). j n (v i) k m j n (v k ) If every demand node ha an identical number of upply node, and the upply node failure probability p(u i j ) i identical, then node v i i choen with probability /m. ) If v i i choen, et all of it upply node {u i j j =,...,n (v i )} to be failed. The other upply node are randomly et to be failed with their repective failure probabilitie. Let U denote the et of failed upply node. 4) Tet whether v i i the firt failed node among {v,v,...,v m }, given that U fail (and no other upply node fail). If true, et I = ; otherwie, et I = 0. Repeat the loop for a = mln(/δ)/ǫ iteration. Reult: 5) Count the number of I = and denote the number by b. An (ǫ, δ)-approximation of the path failure probability i given by b/a k m j n (v k ) p(uk j ). δ. The detail of the proof can be found in the Appendix of the technical report [9]. The advantage of thi algorithm over a naïve Monte-Carlo algorithm (e.g., by repeatedly imulating the upply node failure event and counting the fraction of trial in which the path fail) i that the number of iteration in the naïve Monte- Carlo algorithm i large when the path failure probability i mall. In contrat, by ampling in a more important pace, the number of iteration i reduced. Note that the only quantity that need to be etimated in Algorithm by imulation i E[I], and that Pr(I = ) /m. We conclude thi ection by the following theorem, whoe proof i in the Appendix of the technical report [9]. Theorem. The path failure probability can be etimated to within a multiplicative factor ±ǫ with probability δ, in time O(m n ln(/δ)/ǫ ), where m i the path length and n i the maximum number of upply node for a demand node. Although the failure probability of a pecific path can be well approximated by the importance ampling algorithm, the algorithm hardly give an intuition for path propertie that characterize a reliable path. In the remainder of thi ection, we If F occur in b out of a trial, Pr(F) ( ± ǫ)b/a with probability δ, under the condition that b = Ω(ln(/δ)/ǫ ). The total number of trial a = Ω(ln(/δ)/ǫ )/Pr(F) i large when Pr(F) i mall.

4 develop indicator and bound on the path failure probability, which can be ued for finding the mot reliable path. A. Small and identical failure probability Conider a path v v m in G(V,E,S V ). Let F i denote the event that all the upply node of v i fail. Let F denote the event that the path fail. Clearly, the path fail if at leat one node v i loe all of it upply node (F = i m F i ). By the incluion-excluion principle, we have Pr(F) = Pr(F i ) Pr(F i F i ) i m i <i m + +( ) m Pr(F F F m ). () Directly computing the path failure probability i difficult, given that there are ( m j) ummation in the j-th term of the incluion-excluion formula. We firt reduce the number of event in the incluion-excluion formula, and then further implify the computation under the condition that the upply node failure probability i mall and identical. To reduce the number of event, ome redundant event can be ignored. For example, if F i occur only if F j occur, then the eventf i i redundant in determiningf with the knowledge of F j. To ee thi, note that ) if F j occur, then the path fail regardle of F i ; ) if F j doe not occur, then F i doe not occur a well. If the upply node of v j form a ubet of the upply node of v i, then F i i redundant. With an abue of language, we call a node v i redundant if F i (i.e., the tate of v i ) i redundant. With thi implification, we derive the following reult. Let n (v i ) denote the number of ditinct upply node of v i. Let n min = min i m n (v i ). After removing the redundant node equentially, let m be the number of remaining node that each have n min upply node. The path failure probability can be etimated by the following theorem. Theorem. If every upply node fail independently with probability p ǫ/m, then the path failure probability atifie ( ǫ) mp nmin Pr(F) (+ǫ) mp nmin. Proof. We firt reduce the number of failure event that appear in the incluion-excluion formula by removing the redundant node. Note that determining whether a node i redundant and removing the redundant node are done equentially. Thu, among the et of node that have the ame upply node, one node remain. Let D denote the node in the path excluding the redundant node. Firt, we conider the firt term in Eq. () that provide an upper bound on the path failure probability, known a the union bound. Let D D denote the et of node that each upply node, and let m = D. The remaining node D = D \ D each have n min + or more upply node. Thu, the firt term of Eq. () i at mot have n min for p ǫ/m. Pr(F) mp nmin mp nmin +(m m)pp nmin +ǫp nmin, Next, we conider the firt two term that provide a lower bound on the path failure probability (cf. Bonferroni inequalitie). For any pair of node v j,v k D, the union of their upply node et contain at leat max(n (v j ),n (v k )) + node, becaue neither upply node et include the other a a ubet. At leat n min + upply node have to be removed in order for a pair of node in D to fail. At leat n min + upply node need to be removed in order for a pair of node to fail if at leat one node belong to D. The abolute value of the econd term i at mot ( m ) p n min + +[ ( ) m ) ( m ]p n min +. A lower bound on Pr(F) i ( Pr(F) mp nmin m for p ǫ/m. mp nmin ppnmin ǫ mp nmin, ) + m p p nmin Thu, we have obtained the following two reliability indicator for a path. Thee combinatorial propertie are ueful in finding a reliable path, which will be tudied in the next ection. n min : the minimum number of ditinct upply node for a node in the path. m: the number of combination of n min upply node failure that lead to the failure of at leat one node in the path. B. Arbitrary failure probability In contrat with the cae where upply node failure probability i mall and identical, it i difficult to characterize the reliability of a path by it combinatorial propertie, with limited knowledge of node failure probabilitie. Therefore, we obtain bound on path failure probability that will be ueful in finding a reliable path. Firt, we develop an upper bound on the path failure probability. Let p(v i ) be the failure probability of node v i, under the condition that each of it upply node u i j fail independently with probability p(u i j ). The path failure probability, under the condition that the failure of V are poitively correlated, i no larger than the path failure probability by auming that the failure of V are independent. The proof can be found in the technical report [9]. Lemma. The failure probability of a path P where a upply node u i j fail independently with probability p(ui j ) i upper bounded by v ( p(v i P i)). Then, we develop a lower bound on the path failure probability. The intuition i a follow. After replacing a upply node that upport multiple demand node by multiple independent upply node with ufficiently mall failure probability, the path failure probability doe not increae. In the original graph G(V,E,S V ), conider a node v i V. Let U i denote the et of upply node of v i, let u i j Ui denote one upply node, let p(u i j ) denote the failure probability of ui j, and let n d (u i j ) denote the number of node that ui j upport. Let p(v i ) = u p(u i i j Ui j ) denote the failure probability of v i

5 if u i j fail independently with probability p(ui j ) = ( p(u i j ))/n d(u i j ). A lower bound on the path failure probability i a follow, whoe proof follow a imilar technique in [] and i in the technical report. Lemma. The failure probability of a path P where a upply node u i j fail independently with probability p(ui j ) i lower bounded by v ( p(v i P i)). Let n d denote the maximum number of demand node that a upply node upport, and let n denote the maximum number of upply node for a demand node. The following lemma bound the ratio between the upper and lower bound. It proof can be found in the Appendix of the technical report [9]. Lemma. For any path, the ratio of the upper bound on it failure probability obtained in Lemma to the lower bound obtained in Lemma i at mot (n d ) n. IV. FINDING THE MOST RELIABLE PATH In thi ection, we aim to compute the mot reliable path between a ource-detination pair,t V in G(V,E,S V ). We firt prove that it i NP-hard to approximately compute the mot reliable path. We then develop an algorithm to compute the mot reliable path when the upply node fail independently with an identically mall probability, and finally develop an approximation algorithm under arbitrary failure probabilitie. Hardne of approximation: Although the failure probability of any given path can be approximated to within factor ±ǫ for any ǫ > 0, it i NP-hard to obtain an t path whoe failure probability i le than + ǫ time the optimal for a mall ǫ. The proof can be found in the Appendix of the technical report [9]. Theorem 4. Computing an t path whoe failure probability i le than + ǫ time the failure probability of the mot reliable t path i NP-hard for ǫ < /m, where m i the maximum path length. A. Small and identical failure probability If every upply node fail independently with an identically mall probability, there are two reliability indicator: n min and m. Recall that n min i the minimum number of upply node for a node in the path, and that m i the number of combination of n min upply node failure that diconnect the path. With the two indicator, the path failure probability can be approximated to within a multiplicative factor ± ǫ by mp nmin, under the condition that p ǫ/m. Moreover, the indicator n min i more important (and ha a higher priority to be optimized) than m. We next develop algorithm to optimize the two indicator. Given a graph G(V,E,S V ) and a pair of node (,t), the problem of computing an t path with the maximum n min can be formulated a the maximum capacity path problem, where the capacity of a node equal the number of it ditinct upply node and the capacity of a path i the minimum node capacity along the path. The maximum capacity path can be obtained by a modified Dijktra algorithm, and can be obtained in linear time []. However, it i NP-hard to minimize m, even in the pecial cae where every demand node ha a ingle upply node. The reult follow from the NP-hardne of computing a path with the minimum color in a colored graph []. We develop an integer program to compute the path P with the minimum m, under the condition that n min (P) = min vi P n (v i ) i maximized. The following pre-proceing reduce the ize of the integer program. Firt, compute k = max P P n min (P), where P i the et of all the t path, uing the linear-time maximum capacity path algorithm. Then, remove all the node that have fewer than k ditinct upply node and their attached edge, and denote the remaining graph by G (V,E,S V ). The removed node and edge will not be ued by the optimal path. Let V V denote the node among which each ha exactly k ditinct upply node. We aim to find a path V P where the number of ditinct upply node et for V P V i minimized. Let S i denote the et of upply node of i V. Let S V denote the union of thee et. Let x ij denote the flow variable which take a poitive value if and only if edge (i,j) belong to the elected path. An t path i identified by contraint (). A node i i on the elected path if at leat one of x ij and x ji i poitive. Let h(s i ) denote whether removing upply node S i diconnect the elected path. If a node i i on the elected path and ha k upply node, then h(s i ) mut be one, guaranteed by contraint (4). All the other node either do not belong to the elected path or have more than k upply node, and their upply node failure are not conidered. The objective minimize m, which i the number of combination of k upply node failure that diconnect the path. min.t. S i S V {j (i,j) E } {j (i,j) E } h(s i ) () x ij {j (j,i) E } x ij + {j (j,i) E } x ij 0, (i,j) E, h(s i ) = {0,}, S i S V. B. Arbitrary failure probability, if i =, x ji =, if i = t, 0, otherwie. () x ji h(s i ), i V \,t,(4) If node Ṽ in a graph G(Ṽ,Ẽ) fail independently, the probability that a path urvive i the product of the urvival probabilitie of node along the path. The mot reliable path can be obtained by the claical hortet path algorithm, by replacing the length of travering a node ṽ i by ln( p(ṽ i )), where p(ṽ i ) i the failure probability of ṽ i. It i eay to ee that the length of a path P i ṽ ln( p(ṽ i P i)) = ln ṽ ( p(ṽ i P i)). The hortet path ha the mallet failure probability ṽ ( p(ṽ i P i)). Compared with the above imple model, the difficulty in

6 obtaining the mot reliable t path in interdependent network i the failure correlation of node V G(V,E,S V ). The failure probability of a path can no longer be characterized by v i P ( p(v i)). Moreover, let v i t be the mot reliable t path. The ub-path v i may not be the mot reliable path between and v i. Thu, the labelcorrection approach in dynamic programming (e.g., Dijktra algorithm) cannot be ued, even though the failure probability of a given path can be approximated. Given the bound obtained in the previou ection, we propoe Algorithm to compute a path whoe failure probability i within (n d ) n time the optimal failure probability. Recall that the bound on path urvival probability are the product of (original or new) node urvival probabilitie, which exactly match the path urvival probability in the cae of independent node failure. Algorithm An approximation algorithm to compute a reliable t path in G(V,E,S V ). ) For each v i V, compute p(v i ) a follow. Let u i j be a upply node of v i with failure probability p(u i j ). If u i j upport n d(u i j ) node, let p(ui j ) = ( p(u i j ))/n d(u i j ). Let p(v i ) be the failure probability of v i if u i j fail independently with probability p(ui j ). ) Compute the mot reliable t path auming that v i fail independently with probability p(v i ). The mot reliable path can be obtained by a tandard hortet path algorithm (e.g., Dijktra algorithm), by letting ln( p(v i )) be the length of travering node v i. Theorem 5. The failure probability of the path obtained by Algorithm i at mot (n d ) n time the failure probability of the mot reliable t path under arbitrary upply node failure probabilitie. Proof. Let the path obtained by Algorithm be P and let the path with the minimum failure probability be P. Let p(p ) and p(p ) denote their failure probabilitie. Moreover, let p(p ) and p(p ) denote their failure probabilitie by auming that each node v i fail independently with probability p(v i ). We have p(p ) n n d p(p ) n n d p(p ) n n d p(p ), where the firt inequality follow from Lemma and the lat inequality follow from Lemma. Remark. If n = and every upply node fail independently with an identically mall probability, our reult reduce to the following reult in the claical hared rik group model: The number of rik aociated with the hortet path i at mot n d time the number of rik aociated with the minimum-rik path []. V. RELIABILITY OF A PAIR OF PATHS To tudy divere routing in interdependent network, we conider the implet cae of two t path in thi ection. Given that computing the failure probability of a ingle path i #P hard if every node ha more than one upply node, it i alo #P hard to compute the failure probability of two path. To ee thi, note that if two path have the ame number of node and each node in the firt path ha identical upply node a it correponding node in the econd path, then the probability that both path fail equal the probability that a ingle path fail. Fortunately, we are till able to obtain ±ǫapproximation of the failure probability in polynomial time. A. Small and identical failure probability A central concept in divere routing i the dijoint path or rik dijoint path [], [], [0]. In the claical hared rik group model, if every rik occur independently with an identically mall probability p = o(/m ), the probability that two path fail i Θ(f(m)p ) if they are rik dijoint and Θ(f(m)p) if they hare one or more rik, where m i the maximum path length and f(m) i a function of m. Thu, rik-dijointne characterize the order of the reliability of two path. In interdependent network where every demand node ha multiple upply node, if node in P do not hare any upply node with node in P, then P and P are rik dijoint. However, rik-dijointne doe not uffice to characterize the reliability of two path, for the following two reaon. Firt, the failure probability of a demand node depend on the number of upply node for it, which i not related to rik-dijointne. Second, if P and P hare ome upply node, the failure probability depend further on the maximum number of upply node failure that the two path can withtand. To tudy the reliability of two path in interdependent network, we define d-failure reilient path a follow. Definition. Two path are d-failure reilient if removing any d upply node would not diconnect both path. Remark. In the claical graph model G( Ṽ,Ẽ), two dijoint path are -failure reilient while two overlapping path are 0-failure reilient. In the claical hared rik group model, two rik dijoint path are -failure reilient while two path that hare rik are 0-failure reilient. Two path can never be more than one failure reilient. Thu, the dijointne or rikdijointne uffice to characterize (the order of) the reliability of two path in thee model. ) Evaluation of failure probability: Conider two path P = v v v m t,p = v v v m t between a pair of node (,t). We tudy the event that at leat one node in P and at leat one node in P both fail. Let Fi k denote the event that all the upply node of vi k fail, and let F k denote the event that the k-th path fail, k {,}. Then F F = i m, j m (Fi F j ). For implicity of preentation, let F both = F F and F ij = Fi F j. Let S ij denote the union of upply node of vi and v j. Meanwhile, it i till imple to compute the failure probability of two path if every node ha a ingle upply node, by firt computing the probability that the firt path fail, and then computing the probability that the econd path fail while the firt path doe not fail (i.e., none of the upply node of the firt path fail), both in polynomial time, and umming the two probabilitie.

7 To decide whether two path are d-failure reilient, we conider the number of upply node failure that lead to the event F ij, and denote the number by d ij. Then d = min i m, j m d ij. Moreover, let m be the number of pair of node, one from each path, uch that each pair of node in total have d + ditinct upply node and any two pair do not have the ame et of d+ upply node. (I.e., m combination of d + upply node failure each diconnect both path.) The next theorem formalize the connection between the reliability of two path and d. The proof follow the ame manner a that for Theorem and can be found in the technical report [9]. Theorem 6. If every upply node fail independently with probability p ǫ/(m m ), then the probability that two d- failure reilient path with length m,m both fail atifie ( ǫ) mp d+ Pr(F both ) (+ǫ) mp d+. ) Finding the mot reliable pair of path: From Theorem 6, we know that the probability that two d-failure reilient path both fail i maller for larger value ofd. Moreover, for a fixed d, the failure probability i proportional to m, the number of combination of d + upply node failure that diconnect both path. We have obtained two reliability indicator for two path: d and m. Unfortunately, computing the pair of t path that have the maximum d and the minimum m are both NP-hard, even in the pecial cae where every demand node ha a ingle upply node. Thi pecial cae reduce to the claical hared rik group model. In thi pecial cae, d = if there exit two rik-dijoint path, and d = 0 otherwie. The NP-hardne of determining the exitence of two rik-dijoint path between an t pair ha been proved in []. Moreover, in thi pecial cae, for two path that hare common upply node, m i the number of overlapping rik between the two path (i.e., removing any of the m upply node diconnect both path). The NP-hardne of the leat coupled path problem, which compute a pair of path that hare the minimum number of rik in the claical hared rik group model, ha alo been proved in []. We develop an integer program to compute a pair of t path with the maximum d in G(V,E,S V ). Let variable x k ij denote whether edge (i,j) i part of the k-th path, and let variable b k i denote whether node i i part of the k-th path, k {,}. Same a before, let S i denote the upply node of node i. Contraint (6) guarantee that two path are nodedijoint. Notice that thee contraint can be dropped if there i no retriction on the phyical dijointne of two path. Contraint (7) guarantee that at leat d + upply node need to be removed in order for one node in each path to fail (i.e., b i = b j =, i,j V ), where M i a ufficiently large number, e.g., twice the maximum number of upply node for a demand node. max d (5).t. x k ij, if i =, x k ji =, if i = t, k {,}, {j (i,j) E} {j (j,i) E} 0, otherwie. x k ij + x k ji bk i, i V,k {,} {j (i,j) E} {j (j,i) E} b i +b i, i V \,t, (6) d+ S i S j +M( b i b j ), i,j V \,t,(7) x k ij {0,}, (i,j) E,k {,}, b k i {0,}, i V,k {,}. A lightly modified integer program uffice to minimize m under the condition that d i maximized. Let h(s i S j ) denote whether removing the union of upply node for i and j diconnect both path. Contraint (9) guarantee that if i and j belong to two different path, i.e., b i = b j =, then h(s i S j ) =. Otherwie, h(s i S j ) = 0 in the optimal olution. Let a poitive value w( S i S j ) denote it weight, which i a decreaing function of the cardinality S i S j. We aim to minimize the total weight of upply node failure that diconnect two path. In order to guarantee that d i maximized, w(l)/w(l + ) hould be ufficiently large for any integer l, e.g., V /. Since there are at mot V ( V )/ pair of node, larger d i alway preferable and ha a higher priority to be optimized over m. min.t. w( S i S j )h(s i S j ) (8) S i,s j S V x k ij, if i =, x k ji =, if i = t, k {,}, {j (j,i) E} 0, otherwie. x k ij + x k ji b k i, i V,k {,} {j (i,j) E} {j (i,j) E} {j (j,i) E} b i +b i, i V \,t, h(s i S j ) b i +b j, i,j V \,t, (9) {0,}, (i,j) E,k {,}, x k ij h(s i S j ) {0,}, i,j V. B. Arbitrary failure probability ) Evaluation of failure probability: We ue a imilar importance ampling approach to Algorithm and formulate the problem of computing the failure probability of two path a a DNF probability problem. A claue C ij repreent a pair of node vi and vj. Literal in C ij repreent the union of upply node of vi and vj. A literal i true if and only if the upply node that it repreent fail, and the probability that the literal i true i the ame a the upply node failure probability. The dijunction of claue i true if and only if at leat one claue i true, in which cae both path fail becaue at leat one node from each path fail. The ret of the computation follow the ame manner a Algorithm,

8 by replacing a node in Algorithm by a pair of node. An (ǫ,δ)-approximation of the failure probability Pr(F both ) can be obtained in O(m m n ln(/δ)/ǫ ) time. ) Finding the mot reliable pair of path: It i more difficult to find two path that have the mallet failure probability. Recall Theorem 6. The failure probability of two path i Θ(f(m,m )p d+ ) if they are d-failure reilient when the upply node failure probabilitypi mall, where f(m,m ) i a function of two path length. A a corollary of the fact that it i NP-hard to compute two path that have the maximum level of reilience d, it i alo NP-hard to compute two path whoe failure probability i within a factorαfrom the optimal, where α i any function of the network ize. Thu, we develop the following heuritic. After computing the failure probability p(v i ) of a nodev i in Step of Algorithm, let ln( p(v i )) be the length of travering node v i, and compute two node dijoint path with the minimum total length. The two path can be efficiently obtained uing a lightly modified hortet augmenting path algorithm []. The computation i outlined in Algorithm. The reaon for the graph tranformation in Step i to implify the computation of a reidual graph, to which the hortet augmenting path algorithm can be applied. Algorithm A heuritic to compute a pair of reliable t path in G(V,E,S V ). ) Tranform G(V,E,S V ) with node failure probabilitie to a directed graph G with edge failure probabilitie uing the tandard approach. (Split every node v into v in and v out. Add a directed edge from v in to v out, which ha length ln( p(v i )). Add a directed edge from v out to v in and a directed edge from v out to v in, both with zero length, if an edge exit between v and v in G(V,E,S V ).) ) Compute the hortet path P from out to t in in G. ) Compute the reidual graph a follow. Remove all the edge in P. Add a backward edge from v to v with a negated length if an edge from v to v i part of P. 4) Compute the hortet path P from out to t in in the reidual graph. 5) Combine P and P by cycle cancellation. The two path become node-dijoint and can be mapped to two path in G(V,E,S V ). VI. NUMERICAL RESULTS We tudy the robut routing problem in the XO backbone communication network with 60 node and 75 edge [4], by auming that the XO node are upported by 6 randomly generated upply node within the continental US. The XO network topology i depicted in Fig. 4, and the upply node are marked a triangle. The x-axi repreent the longitude and the y-axi repreent the latitude. We do not claim that the XO network need upply from thee randomly generated point, and we ue thi example only to provide a viualization of the robut routing problem uing available data. 50 Seattle Miami Fig. 4. Topology of the XO network and randomly generated triangle upply node. The mot reliable Seattle-Miami path i colored red under the condition that upply node failure probability i mall and identical and every XO node depend on two nearet upply node. Firt, we aume that every XO node depend on two nearet upply node and every upply node fail independently with probability 0. Since the upply node failure probability i mall and identical, we are able to obtain the mot reliable path and pair of path by optimizing the reliability indicator uing integer program. To identify the mot reliable path, ince n min (P) = for any path P, we only need to compute a path with the minimum m uing the integer program in Section IV. The mot reliable path i colored red in Fig. 4, for which m = 8. To evaluate the path failure probability, by a tronger analog of Theorem (Corollary in the technical report), etting ǫ = 4 0, Pr(F) [ ,8 0 4 ]. To compare, uing Algorithm, we obtain a a ( ± 0.0)-approximation of the path failure probability with probability Thee reult ugget that the two reliability indicator (n min, m) well characterize the path failure probability when the upply node failure probability i mall and identical. We compute the mot reliable pair of path connecting Seattle-Miami uing the integer program in Section V. The two path are plotted in Fig. 5, and they are -failure reilient (d = ). The failure probability of both path i approximately In contrat, the mot reliable pair of path connecting Seattle-Denver are -failure reilient and their failure probability i approximately Thu, the level of reilience well indicate the reliability of two path. Next, we aume that an XO node depend on N randomly choen upply node, where N i uniformly choen among,, and. Let the failure probability of each upply node be uniformly and independently choen from [0.005, 0.05]. We ue Algorithm to obtain a reliable path connecting Seattle- Miami. Averaged over 0 trial, the path failure probability i approximately.0 0, while the lower bound on the failure probability of the mot reliable path i The obtained path ha failure probability around four time the lower bound. Moreover, by uing the heuritic to find a pair of path, the path have average failure probability.97 0, which improve the reliability of a ingle path.

9 REFERENCES 50 Seattle Denver Miami Fig. 5. The mot reliable pair of path between Seattle-Miami are colored red, under the condition that upply node failure probability i mall and identical and every XO node depend on two nearet upply node. We compare the performance of the heuritic (Algorithm ) with the optimal pair of path. Since it i difficult to obtain the optimal pair of path under arbitrary failure probabilitie, we ue the integer program (8), under the condition that upply node fail independently with probability 0. If every XO node depend on two nearet upply node, the failure probabilitie of two optimal path and two path obtained by the heuritic are approximately and , repectively. If every XO node depend on three nearet upply node, the failure probability of two optimal path and two path obtained by the heuritic are approximately and , repectively. Thee experiment validate the performance of our heuritic algorithm. Finally, we report the running time of the algorithm, executed in a worktation that ha an Intel Xeon Proceor (E5-687W v) and 64GB RAM. The integer program that find the mot reliable path and pair of path (under mall and identical upply node failure probability) can both be olved within econd. The approximation algorithm to find a reliable path and the heuritic to find a pair of path (under arbitrary failure probabilitie) can both be olved within 0. econd. The evaluation of the failure probability of one path or a pair of path by Algorithm take everal minute, by etting ǫ = δ = 0.0. Thu, the algorithm (integer program and Algorithm and ) can be ued to find reliable route in realitic ize network. VII. CONCLUSION We tudied the robut routing problem in interdependent network. We developed approximation algorithm to compute the path failure probability, and identified reliability indicator for a path, baed on which we develop algorithm to find the mot reliable route in interdependent network. We alo tudied divere routing in interdependent network, and developed approximation algorithm to compute the probability that two path both fail and to find two reliable path. Our work extend the hared rik group model, and provide a new framework to tudy robut routing problem in interdependent network. [] O. Yagan, D. Qian, J. Zhang, and D. Cochran, Optimal allocation of interconnecting link in cyber-phyical ytem: Interdependence, cacading failure, and robutne, IEEE Tranaction on Parallel and Ditributed Sytem, vol., no. 9, pp , 0. [] V. Roato, L. Iacharoff, F. Tiriticco, S. Meloni, S. Porcellini, and R. Setola, Modelling interdependent infratructure uing interacting dynamical model, Int. J. Critical Infratructure, vol. 4, no. -, pp. 6 79, 008. [] C.-G. Gu, S.-R. Zou, X.-L. Xu, Y.-Q. Qu, Y.-M. Jiang, H.-K. Liu, T. Zhou et al., Onet of cooperation between layered network, Phyical Review E, vol. 84, no., p. 060, 0. [4] M. Parandehgheibi and E. Modiano, Robutne of interdependent network: The cae of communication network and the power grid, in Proc. IEEE GLOBECOM, Atlanta, 0, pp [5] M. Parandehgheibi, K. Turityn, and E. Modiano, Modeling the impact of communication lo on the power grid under emergency control, in Proc. IEEE SmartGridComm, Miami, 05, pp [6] S. V. Buldyrev, R. Parhani, G. Paul, H. E. Stanley, and S. Havlin, Catatrophic cacade of failure in interdependent network, Nature, vol. 464, no. 79, pp , 00. [7] J. Shao, S. V. Buldyrev, S. Havlin, and H. E. Stanley, Cacade of failure in coupled network ytem with multiple upport-dependence relation, Phyical Review E, vol. 8, no., p. 066, 0. [8] S. Yang, S. Trajanovki, and F. A. Kuiper, Shortet path in network with correlated link weight, in Proc. IEEE Computer Communication Workhop (INFOCOM WKSHPS), Hong Kong, 05, pp [9] D. Xu, Y. Xiong, C. Qiao, and G. Li, Failure protection in layered network with hared rik link group, IEEE network, vol. 8, no., pp. 6 4, 004. [0] D. Coudert, P. Datta, S. Pérenne, H. Rivano, and M.-E. Voge, Shared rik reource group complexity and approximability iue, Parallel Proceing Letter, vol. 7, no. 0, pp , 007. [] J. Q. Hu, Divere routing in optical meh network, IEEE Tranaction on Communication, vol. 5, no., pp , 00. [] S. Yuan, S. Varma, and J. P. Jue, Minimum-color path problem for reliability in meh network, in Proc. INFOCOM 005, Miami, 005, pp [] H.-W. Lee, E. Modiano, and K. Lee, Divere routing in network with probabilitic failure, IEEE/ACM Tranaction on Networking, vol. 8, no. 6, pp , 00. [4] M. F. Habib, M. Tornatore, F. Dikbiyik, and B. Mukherjee, Diater urvivability in optical communication network, Computer Communication, vol. 6, no. 6, pp , 0. [5] W. Cui, I. Stoica, and R. H. Katz, Backup path allocation baed on a correlated link failure probability model in overlay network, in Proc. IEEE Int. Conf. Network Protocol, Pari, 00, pp [6] Q. Zheng, G. Cao, T. F. La Porta, and A. Swami, Cro-layer approach for minimizing routing diruption in ip network, IEEE Tranaction on Parallel and Ditributed Sytem, vol. 5, no. 7, pp , 04. [7] C. Tang and P. K. McKinley, Improving multipath reliability in topology-aware overlay network, in Proc. IEEE International Conference on Ditributed Computing Sytem Workhop, Columbu, Ohio, 005, pp [8] T. Fei, S. Tao, L. Gao, and R. Guerin, How to elect a good alternate path in large peer-to-peer ytem? in Proc. INFOCOM 006, Barcelona, April 006, pp.. [9] J. Zhang and E. Modiano, Robut routing in interdependent network, Tech. Rep., 06. [Online]. Available: [0] R. M. Karp, M. Luby, and N. Madra, Monte-carlo approximation algorithm for enumeration problem, Journal of algorithm, vol. 0, no., pp , 989. [] W. Gatterbauer and D. Suciu, Obliviou bound on the probability of boolean function, ACM Tranaction on Databae Sytem (TODS), vol. 9, no., p. 5, 04. [] A. P. Punnen, A linear time algorithm for the maximum capacity path problem, European J. Oper. Re., vol. 5, no., pp , 99. [] J. Edmond and R. M. Karp, Theoretical improvement in algorithmic efficiency for network flow problem, Journal of the ACM (JACM), vol. 9, no., pp , 97. [4] XO Communication, Network map. [Online]. Available: