Assessing Attack Vulnerability in Networks with Uncertainty

Transcription

1 Assessig Attack Vulerability i Networks with Ucertaity Thag N. Dih Dept. of CS, Virgiia Commowealth Uiversity Richmod, VA, USA, tdih@vcu.edu My T. Thai CISE Dept., Uiversity of Florida Gaiesville, FL, USA, 36, mythai@cise.ufl.edu Abstract A cosiderable amout of research effort has focused o developig metrics ad approaches to assess etwork vulerability. However, most of them eglect the etwork ucertaity arise due to various reasos such as mobility ad dyamics of the etwork, or oise itroduced i data collectio process. To this ed, we itroduce a framework to assess vulerability of etworks with ucertaity, modelig such etworks as probabilistic graphs. We adopt expected pairwise coectivity (EPC) as a measure to quatify global coectivity ad use it to formulate vulerability assessmet as a stochastic optimizatio problem. The objective is to idetify a few umber of critical odes whose removal miimizes EPC i the residual etwork. While solutios for stochastic optimizatio problems are ofte limited to small etworks, we preset a practical solutio that works for larger etworks. The key advatages of our solutio iclude ) the applicatio of a weighted averagig techique that avoids cosiderig all, expoetially may, possible realizatios of probabilistic graphs ad ) a Fully Polyomial Time Radomized Approximatio Scheme (FPRAS) to efficietly estimate the EPC with ay desired accuracy. Extesive experimets demostrate sigificat improvemet o performace of our solutio over other heuristic approaches. I. INTRODUCTION Networked systems such as commuicatio etworks, electrical grids, ad trasportatio etworks are vulerable to atural disasters ad targeted attacks. Eve failures of few vital odes or liks ca severely compromise the etwork s ability to meet its quality-of-service (QoS), if ot cause total etwork breakdow []. Moreover, there is a icreasig cocer over such critical systems as targets for (cyber) terrorist attacks []. To develop proactive resposes ad mitigate the risk, it is importat to assess etwork vulerability, i.e., to idetify those crucial odes ad liks, beforehad. Despite of may studies o assessig etwork vulerability, little is kow about assessmet of etworks with ucertaity. This ucertaity ca arise due to various reasos from mobility ad dyamics of etworks to data collectio process. Particularly, liks i techological etworks, e. g., the Iteret, wireless sesor etworks ad mobile opportuistic etworks, are frequetly subject to disruptios. Typical abstractio of etworks as static graphs [3], [4], that fails to capture this ucertaity, may lead to serious misjudgemet o etwork vulerability. I this paper, we propose a framework to assess vulerability of etworks with ucertaity. We model the etwork as a probabilistic graph G ad formulate the vulerability assessmet as a stochastic optimizatio problem. The goal Pr G i Pr G i (a) A probabilistic graph (b) Realizatios Fig. : A probabilistic graph with existig probabilities o edges. I (a), ad hoc heuristics which target odes of highest cetrality, e.g., degree ad betweeess, remove ode 3, leavig the residual etwork itact. I (b), miimizig EPC leads to the removal of ode 4. It results i 5% chace of breakig the residual etwork (i G ad G 4) ad disrupts effectively 55% etwork coectivity. is to idetify a small set of odes that removal miimizes expected value of etwork performace. We associate each edge i G with a probability of existece, represetig the fractio of time that the lik is i a workig state. Additioally, we treat G as a geerative model for determiistic graphs. Each such determiistic graph is a possible realizatio/sample of G ad is also associated with a probability of beig geerated. Our basic measure for etwork performace is pairwise coectivity, defied as the umber of coected ode pairs i the etwork. This measure has bee recetly adopted to accout for the impact of attacks i determiistic graphs [3], [4], [5], [6]. It is favored for the strog discrimiatio i quatifyig the etwork coectivity level, eve for discoected etworks. Give a probabilistic graph G, we defie k-pcnd as the problem of fidig k odes that removal miimizes the expected pairwise coectivity (EPC), over all possible realizatios of G. The advatage of our assessmet framework over existig approaches is illustrated i Fig.. If oe ode is to remove from the graph accordig to either degree cetrality or betweeess cetrality, ode 3 will be removed. As show i Fig. a, the residual etwork remais coected i this case. However, Fig. b shows a more destructive attack o ode 4 through miimizig EPC. Sice most liks i the etwork have existig probabilities oe, except for two liks (, 3) ad (3, 5), we have four possible realizatios of the etwork, amed from G to G 4. Removig ode 4 will ot oly degrade EPC by

2 55% but also result i a 5% chace of breakig the residual etwork, as i the cases of G ad G 4. More sophisticated assessmet methods [4], [5] also fail o this simple example, idicatig the importace of cosiderig etwork ucertaity. Aother advatage of our approach is that we are able to costruct a efficiet Fully Polyomial-Time Radomized Approximatio Scheme (FPRAS) to compute EPC. Our FPRAS does ot cosider all, expoetially may, possible realizatios of G, yet it computes EPC with guarateed accuracy. As exact computatio of EPC is #P-complete [7], a FPRAS provides the best theoretical result. Such a result is ot kow for ay other measures for probabilistic graphs except all-termialreliability which admits the oly kow FPRAS i 995 [8]. Ulike the first FPRAS [8] which is oly practical for small etworks with equal edge probabilities, our FPRAS is scalable for large etworks with heterogeeous edge probabilities. Last but ot least, stochastic optimizatio problems are extremely difficult to solve. Commo techiques for stochastic programmig problems such as Beder s decompositio [9] ad Sample Average Approximatio [] ofte do ot scale beyod etworks with few dozes of odes. Thus it is critical that we desig a efficiet solutio for the problem. We summarize our cotributios as follows: We itroduce a framework to assess attack vulerability i etworks with ucertaity, formulatig it as stochastic optimizatio problems over probabilistic graphs. Besides EPC, the framework ca be itegrated with may other reliability ad performace measures such as size of largest compoets ad average maximum flow betwee ode pairs []. We formulate k-pcnd problem to assess etwork vulerability ad preset a practical solutio for the problem which utilizes a weighted averagig techique to avoid cosiderig all, expoetially may, possible realizatios of probabilistic graphs. We propose a FPRAS for computig EPC. The FPRAS is ot oly of theoretical iterest but also practical for large etworks. Extedig techiques i our FPRAS to other reliability problems, e.g., the two-termialreliability ad k-termial-reliability [8], is material for future work. We show sigificat performace improvemet of our algorithm over competitor heuristics via experimets. The experimets also reveal that the vulerability assessmet based o determiistic etwork aalysis is too optimistic i real scearios, as it greatly overestimates how resiliet etwork systems are. Orgaizatio. We summarize related work i Sectio II ad itroduce models, otatios, ad defiitio of the k-pcnd problem i Sectio III. Assumig the presece of a efficiet oracle to estimate EPC, we preset our algorithm for k-pcnd i Sectio IV. We preset our FPRAS to estimate EPC, the fial piece i our solutio, i Sectio V. We give simulatio results o real-world traces to show efficacy of our algorithm ad the isights i Sectio VI. II. RELATED WORK To our best kowledge, we are the first to study attack vulerability assessmet for ucertai etworks, optimizig directly a reliability measure desiged for such etworks. However, attack vulerability i determiistic etworks ad reliability (i the cotext of radom failures) for etworks with ucertaity have motivated may studies i commuicatio ad theoretical commuities. A. Vulerability Assessmet i Determiistic Networks. The fuctio ad performace of etworks rely o their resiliece, defied as the ability to cotiue fuctioig uder perturbatio. To measure robustess ad resiliet, prior works proposed to moitor differet measures icludig the diameter, size of the largest coected compoet, [], coectivity based o miimum ode/edge cut, algebraic coectivity, ad spectral radius [], [3], [5]. All these measures capture, i priciple, aspects of etwork coectivity. May metrics ad approaches have bee proposed to accout for etwork robustess ad vulerability [], [3], [6]. While each of these measures has its ow emphasis ad ratioality, they ofte come with several shortcomigs that prevet them from capturig desired characteristics of etwork coectivity ad resiliece. For example, measures based o shortest path are rather sesitive to small chages (e.g. removig edges or odes); algebraic coectivity ad diameter are ot meaigful for discoected graphs (all discoected graphs have the same values); umber of coected compoets ad compoet sizes, arguably, do ot fully reflect level of etwork coectivity. Pairwise coectivity, defied as the umber of ode pairs that remai coected, has bee proposed as a effective measure for etwork coectivity [3], [4], [5], [4], [6]. Arulselva et al. defied i [4] the Critical Node Detectio (CND) problem, the determiistic versio of k-pcnd. The problem seeks k odes that removal miimize the pairwise coectivity i the residual etwork. We proposed β-disruptor framework to assess attack vulerability i terms of pairwise coectivity [5], [4]. We preseted O ( log.5 ) bicriteria approximatio algorithms for assessig edge vulerability, ad a O (log log log ) bicriteria approximatio algorithm for the vertex versio of β-disruptor. Whe both odes ad liks i the etwork are subjected to attacks, we provide a O( log ) bicriteria approximatio algorithm [5] that immediately improve the results i [5], [4]. B. Reliability of Networks with Radom Failures. A sigificat amout of works has bee devoted for reliability of etworks whe their elemets are subjected to radom failures [6], [7], [7]. The well studied reliability assessmet framework is to calculate the probability that commuicatio ca be established amog a set of odes whe each ode ad/or lik ca fail idepedetly with some probability. The two-termial-reliability, with two special odes called source s ad destiatio t, cocers the probability that there exists a path betwee s ad t. More geeral cases ivolve k-termial reliability ad all-termial-reliability. The etwork reliability

3 problems were proved to be i #P-hard class, a super class of NP-hard problems [8]. Karger itroduced the first FPRAS for all-termial-reliability problem [8]. To our best kowledge, it is the oly kow FPRAS for etwork reliability problems. I [7], [9], Colbour itroduced etwork resiliece, defied as the average two-termial-reliability betwee all ode pairs. The exact computatio of etwork resiliece was proved to be #P-hard eve i plaar graphs. Ami et al. [] proposed pair-coected reliability, the expected umber of coected ode pairs. We ote that our measure EPC is the same as Pair-coected reliability ad both ca be obtaied by multiplyig etwork resiliece by ( ) V. We use the ame expected pairwise coectivity to be cosistet with amig of pairwise coectivity i [3], [4], [5]. Recetly, Neumayer et al. [] proposed a polyomial-time algorithm to compute etwork resiliece, which they referred to as average twotermial reliability (ATTR) [], [], for geometric etworks whe the disaster takes a special form of a straight lie. However, efficiet methods to compute etwork resiliece i geeral case is still a ope questio til this paper. We ote that the reliability literature, however, does ot cosider targeted attacks, which is the mai subject of this paper. III. MODEL AND DEFINITIONS I this sectio, we preset the probabilistic etwork model, ad the ecessary otatios to formulate the vulerability assessmet problem. A. Probabilistic Network Model We abstract a etwork with ucertaity as a probabilistic graph G = (V, E, P ) where vertices i V correspods to the set of odes; edges i E correspods to the set of liks i the etwork; ad P that maps each edge (u, v) E to a real umber i p uv [, ] that represets the probability that edge (u, v) exists. For each (u, v) / E, we have p uv =. A example of probabilistic graphs is Erdos-Reyi radom graphs [3] i which all edge probabilities are the same ad equal p. For clarity, we cosider oly udirected etworks ad assume idepedece amog edges. However our proposed solutio also applies i priciple to directed graphs or graphs with edge correlatios as log as expected values of edges ca be computed. A probabilistic graph G ca be see as a geerative model for determiistic graphs. A determiistic graph G = (V, E s ) is geerated from G by selectig each edge (u, v) E, idepedetly, with probability p uv. We refer to G as a realizatio or a sample of G ad write G G. The probability that G is geerated from G is Pr[G] = p e ( p e ). s \E s Let m = E, there are W = m possible realizatios of G. We umber those realizatios as G = (V, E ), G = (V, E ),..., G W = (V, E W ), where E, E,..., E W are all possible subsets of E. B. Expected pairwise coectivity Our mai measure for the etwork reliability is expected pairwise coectivity (EPC), which is the expected umber of coected pairs i the etwork. Formally, deote by P(G) the umber of coected pairs or pairwise coectivity of a determiistic graph G. The expected pairwise coectivity (EPC) of G is defied as EPC(G) = E [P(G)] = G G Pr[G]P(G). EPC has a tight coectio to two-termial reliability as stated i the followig lemma. Lemma : [7] Let REL u,v (G) deote the two-termialreliability betwee ode u ad v i G, i.e. the probability that v is reachable from u. We have EPC(G) = REL u,v (G). () u,v V ;u v Thus EPC ca be computed as the total of two-termialreliability betwee all ode pairs. However, this approach is problematic due to the facts that exact computatio for twotermial-reliability is NP-hard, ad that eve we apply existig heuristics to approximate two-termial-reliability, computig EPC requires a large umber, ( ) V, of calls to such heuristics. Istead, EPC ca be computed efficietly as show i Sectio V. This is a importat advatage of EPC over other reliability measures ad the reaso for the adoptio of EPC. C. Vulerability Assessmet i Probabilistic Networks We formulate vulerability assessmet as the followig stochastic optimizatio problem. Probabilistic Critical Nodes Detectio (k-pcnd). Give a probabilistic etwork G = (V, E, p) ad a iteger k, fid a k odes subset S V that removal miimizes EPC i the residual etwork. Whe all edge probabilities are oe, we obtai the CND problem i [4]. Sice the CND problem is NP-hard, k-pcnd, geeralizig CND, is also NP-hard. IV. VULNERABILITY ASSESSMENT IN PROBABILISTIC NETWORKS I this sectio, we ivestigate the Probabilistic Critical Nodes Problem (k-pcnd). We formulate the problem as a two-stage stochastic program i Subsectio IV-A; ad devise efficiet approaches to overcome the difficulty of havig a expoetial umber of costraits i the mathematical formulatio i Subsectio IV-B. Our solutio assume the presece of a efficiet oracle to compute EPC, which we preset later i Sectio V. A. Two-stage Stochastic Liear Program Stochastic programmig has bee a commo approach for optimizatio uder ucertaity whe the probability distributio govers the data is give. A comprehesive itroductio to stochastic programmig ca be foud i referece [5]. Give a probabilistic graph G = (V, E, p) ad a iteger < k <, we first use iteger variables s i to represet whether or ot ode i is removed, i.e, s i = if ode i is

4 removed, ad, otherwise. Here = V is the umber of odes ad we assume odes are umbered from to. We impose o s the costrait i= s i k to guaratee o more tha k odes are removed. Variables s are kow as first stage variables. The values of s are to be decided before the actual realizatio of the ucertai parameters i G. We associate with each ode pair (u, v) a radom Berouli variable ξ uv satisfyig Pr[ξ uv = ] = p uv ad Pr[ξ uv = ] = p uv. For each realizatio of G, the values of ξ uv are revealed to be either or ad we ca compute the pairwise coectivity i the residual graph after removig k odes idicated by s. To do so, we defie iteger variables x ij to be the discoectivity betwee a ode pair i ad j i the residual etwork, i.e., x ij = if i ad j are still coected ad, otherwise. Pairwise coectivity i the residual graph ca be computed usig a secod stage iteger programmig, deoted by P (s, x, ξ) as follows. P (s, x, ξ) = mi i<j( x ij ) () s. t. x ij s i + s j + ξ ij, (i, j) E, (3) x ij + x jk x ik, (i, j) E, k =.. (4) s i {, }, x ij {, } (5) This secod stage programmig formulatio is essetially the same with the formulatio for the CND problem i [4] (except for the cardiality costrait o s). Ideed, we adopt the improved formulatio of CND i [4]. This formulatio reduces the umber of costraits from θ( 3 ) to O(m) ad shorte the solvig time substatially. The two-stage stochastic liear formulatio for the k-pcnd problem is as follows. mi E [P (s, x, ξ)] (6) s {,} s. t. s i k (7) i= where P (s, x, ξ) is give i ()-(5) (8) The objective is to miimize the expected coectivity i the residual etwork E [P (s, x, ξ)], where P (s, x, ξ) is the optimal value of the secod-stage problem. This stochastic programmig problem is, however, ot yet ready to be solved with liear algebra solver. Discretizatio. To solve a two-stage stochastic problem, oe ofte eed to discretize the problem ito a sigle (very large) liear programmig problem. That is we eed to cosider all possible realizatios G l G ad their probability masses Pr[G l ]. Deote by {ξ l } ij the adjacecy matrix of the realizatio G l = (V, E l ), i.e., ξij l =, if (i, j) El, ad, otherwise. Sice the objective ivolves oly the expected cost of the secod stage variables x ij, the two-stage stochastic program ca be discretized ito a mixed iteger programmig, deoted by MIP F as follows. mi W ( x l ij) (9) Pr[G l ] l= i<j s. t. s i k () i= x l ij s i + s j + ξ l ij, (i, j) E, l =..W () x l ij + x l jk x l ik, (i, j) E, k =.., l =..W () x l ij = x l ji, i, j =.., l =..W (3) s {, }, x l [, ], l =..W (4) The major challege i solvig this discretized form is that there is a expoetial umber of variables ad costraits. Thus, solvig MIP F is itractable eve for very small istaces of G. To overcome this difficulty, we preset i ext subsectio a compact relaxatio of MIP F. Solvig this polyomial size relaxatio leads to high quality solutios for k-pcnd problem, as we will show i the experimetal sectio. B. Algorithm We preset our solutio, amed, for the stochastic optimizatio problem i Algorithm. First, the algorithm costructs a liear relaxatio of the expoetial size formula MIP F ad select k vertices via a iterative roudig procedure. The result is a subset D of cardiality k. The algorithm follows by a local search procedure that improves D via swappig vertices. A vertex u D ad a vertex v / D are swapped places if doig so reduces the EPC. The key of the local search is to compute EPC quickly ad accurately. This is doe with the CSP algorithm (preseted later i Algorithm ). The local search stops whe o more swaps ca reduce the EPC. The relaxatio of MIP F is costructed by applyig a weighted-averagig of all costraits i MIP F. Costraits ivolvig the realizatio G l are give weights Pr[G l ]. Thus costraits () are reduced to a sigle costrait x ij s i + s j + Pr[G l ]ξij, l G l G which ca be further simplified ito x ij s i +s j + p ij. The other costraits ca be averaged i the same way, givig us the followig relaxatio of MIP F. mi ( x ij ) (5) s. t. i<j s i k (6) i= x ij s i + s j + p ij, (i, j) E (7) x ij + x jk x ik, (i, j) E, k =.. (8) s i {, }, x ij [, ], i, j =.. (9) We shall refer to this relaxatio of MIP F as MIP R. Note that the o-itegrality of x ij is essetial for the above relaxatio, deoted by MIP R. If we restrict x ij to {, } the costrait x ij s i +s j ξ ij is equivalet to x ij s i +s j +.

5 The the costrait holds trivially for ay values of x ad s. Thus the iformatio ecoded i the edge probabilities is ot itegerated i the formulatio. Algorithm. Roudig the Expected Graph Algorithm () ) Obtai a LP relaxatio of MIP R with the relaxed costraits s [, ]. ) Iitialize the set of selected odes D =. 3) Repeat k times the followig steps Solve the LP relaxatio Select u = arg max i V \D s i. Add u to D ad set s u 4) Repeat 5) For each pair (u, v) D (V \ D) 6) Estimate the EPC after removig D {u} + {v} usig CSP 7) Update D = D {u} + {v}, if the ew EPC is lower 8) Util o possible update 9) Output D. Lower-boud. Oe of the ice feature of MIP R is that its optimal objective provides a lower-boud o the optimal objective of MIP F. This provides a useful tool to assess the quality of proposed algorithms, especially whe fidig the optimal solutios of MIP F is likely itractable. We prove the objective lower-boud i the followig lemma. Lemma : The optimal objective value of the MIP R is a lower-boud o the optimal objective value of MIP F. Proof: To show that the the objective of the MIP R is a lower-boud o that of the MIP F, we costruct a feasible solutio ( s, x) of MIP R that gives a objective equal to the optimal objective of MIP F. Let ( ŝ, ˆx,..., ˆx ) W ( be a optimal solutio of the MIP F. Costruct a solutio s = ŝ, x = W l= Pr[Gl ]ˆx ). l The objective value of MIP R give by that solutio is x ij ) = i<j( W ( Pr[G l ]ˆx l ij) i<j l= = W Pr[G l ]( ˆx l ij) i<j which is exactly the optimal objective of MIP F. The last equality holds because the probabilities Pr[G l ] add up to oe. The rest is to show that ( s, x) is a feasible solutio of MIP R. Clearly, s satisfy (6) ad the itegral costraits. Also sice x is a covex combiatio of ˆx l, l =..W with the masses Pr[G l ], x satisfy the costraits (8), (7),& (9) as they ca be iferred from the same covex combiatio of the costraits from () to (4). We ote that due to the high similarity i programmig formulatios of critical elemets detectio problems, our algorithm ca be easily modified to solve extesios of other vulerability assessmet problems to etworks with ucertaity. Examples iclude the Critical Edge Detectio [6], β-vertex disruptor, ad β-edge disruptor [5], [6]. l= V. COMPUTING EPC This sectio focuses o efficiet methods to compute EPC of probabilistic graph, the fial but importat piece of the algorithm. Sice it is itractable to compute the exact value of EPC [7], we preset efficiet methods to approximate EPC with ay desired accuracy. A. Compoet Samplig Procedure to Approximate EPC We develop a Mote Carlo method to approximate the EPC withi a arbitrary small error with a high probability. We also reveal why the aive Mote Carlo method caot guaratee a polyomial time complexity. Give a pair of ɛ, δ >, our Mote Carlo method returs a estimatio of EPC(G) accurate to withi a relative error of ɛ with a probability at least δ. Mathematically, our proposed method is a (ɛ, δ)-approximatio of EPC, which is defied as follows. Defiitio ( (ɛ, δ)-approximatio): A fuctio ˆF (G) is a (ɛ, δ)-approximatio for the expected pairwise coectivity EPC(G) if [ Pr ( ɛ)epc(g) ˆF ] (G) ( + ɛ)epc(g) > δ. A (ɛ, δ)-approximatio is called a fully polyomial radomized approximatio scheme (FPRAS) if its ruig time is bouded by a polyomial i terms of /ɛ, log(/δ), ad the iput size. I geeral, a FPRAS is the best theoretical result oe ca hope for a #P-hard computatioal problem. We preset our Compoet Samplig Procedure (CSP) with two importat advatages over the aive Mote Carlo method. First, it has a polyomial time complexity ad is, thus, a FPRAS for the EPC(G) problem. Secod, it has a smaller average time complexity, ad is up to times faster tha aive Mote Carlo methods. CSP is summarized i Algorithm. The algorithm computes the sum of edge probabilities P E = p e. If P E is sufficietly small (at most ɛ ), the algorithm returs P E as a ubiased estimator of EPC(G). Otherwise, it performs a importace samplig method to estimate EPC(G) i steps 4 to 6. I the importace samplig method, we select a ode u V uiformly ad perform a Bread-First Search procedure from u, util reachig all odes i the coected compoet that cotais u. The algorithm the computes the average of the size of the compoet that cotais u less oe, ad multiply the result by to obtai a ubiased estimator E. Oe advatage of CSP over direct Mote-Carlo approaches is that it avoids geeratig too may graph samples whe the EPC is predicted to be small. This is the key to guaratee that the algorithm is polyomial-time. Further, the algorithm does ot geerate the whole sample graph at oce, but oly reveal the availability of edges alog the Bread-First Search procedure. This characteristic substatially reduces CSP s average ruig time, as aalyzed later i Theorem 3.

6 Algorithm. (ɛ, δ) Compoet Samplig Procedure to Approximate EPC(G) ) Let P E = pe ) if P E < ɛ the 3) retur E = P E. 4) C. 5) for i = to N(ɛ, δ) do Select a ode u V uiformly. Start a Breath-First Search from u. For each ecoutered edge (v, w), flip a coi of bias p vw to determie its availability. Let S i be the umber of visited odes, icludig ode u. C = C + (S i ). 6) Retur E = C N B. Correctess as a ubiased estimator of EPC(G). We determie whether or ot the value EPC is too small based o the value of P E = p e. This is based o a observatio that EPC is sadwiched betwee two fuctios of P E, as show i the followig propositio. Propositio : Let G = (V, E, p) be a probabilistic graph ad P E = p e, the followig iequality holds ( P E EPC(G) + P ) m E. () m The bouds i Propositio are asymptotic tight i the sese that there are arbitrary large graphs i which the bouds are oly differet from the actual values of EPC(G) by a factor of two. For example, cosider G as a star graph of size that cosists of oe ceter vertex ad leaves. All edges are assiged the same probability /( ). Oe ca verify that the lower-boud, EPC(G), ad the upper boud are, 3 ( ), ad ( + ) < e, respectively. Further, we state several iequalities eeded for (ɛ, δ)- approximatio proof i the followig propositio. Propositio : Let G = (V, E, p) be a probabilistic graph ad q i be the probability that G has exactly i edges, for i =.. E. If P E = p e < /, the followig iequalities hold P E q exp( P E ), () P E q P E q q, () P E m q k PE. (3) k= For the sake of completeess, we preset the proofs of Propositio ad i the Appedix. We ow derive N(ɛ, δ), the umber of ecessary samples to be draw usig the followig Geeralized Zero-Oe Estimator Theorem itroduced by Dagum et al. [7]. Theorem : (Geeralized Zero-Oe Estimator [7]) Let X, X,..., X N be idepedet idetically distributed radom variables takig values i [, ], with mea µ >. If < ɛ < ad N 4(e ) l(/σ)/(ɛ µ), where e.78 is Euler s umber, the [ Pr ( ɛ)µ N ] N X i ( + ɛ)µ > δ. i= The required umber of samples to obtai a (ɛ, δ) approximatio is N(ɛ, δ) = 4(e ) l ( ) σ ɛ EPC(G), as proved i the followig lemma. Lemma 3: If N(ɛ, δ) 4(e ) l ( ) σ ɛ EPC(G), the E, the output of CSP, is a (ɛ, δ)-approximatio for EPC(G). Proof: We cosider two cases of P E. Case P E < ɛ : CSP returs P E (step ). We show that P E is ideed a (ɛ, δ)-approximatio for EPC(G) by provig the followig iequalities. P E EPC(G) ( + ɛ)p E. From Lemma, we already have P E EPC(G). Thus we oly eed to show EPC(G) ( + ɛ)p E. Deote by q k, k =..m, the probability that G has exactly k edges. Sice i ay (determiistic) graph with m edges ad vertices, the pairwise coectivity caot exceeds ( mi{m,} ), we obtai the followig iequality. EPC(G) ( ) q + ( ) q + ( ) m q k (4) Apply iequalities (), ad (3) p E EPC(G) q + ( p E ) P E. Usig iequality (), we arrive P E k= EPC(G) exp( P E ) + ( P E ) P E. Sice P E ɛ < /, we apply the iequality exp( x) x + x, x (, ) to yield EPC(G) ( + P E + ( ) PE ) PE ( + ɛ)p E. (5) This completes the proof of P E beig (ɛ, δ)-approximatio of EPC(G) whe P E < ɛ. Case P E ɛ : The importace samplig is carried out i steps 4 to 6 i Algorithm. Withi the loop i Step 5, we ca compute S i with the followig equivalet procedure: ) Draw a sample graph G i ; ad ) Select a ode u i G i uiformly ad compute S i as the size of coected compoet that cotais u. Assume that there are t coected compoets with sizes s, s,..., s t i G i. We have E[S i G = G i ] = k i= s i(s i ) k i= s i = P(Gi ).

7 EPC Time(secods) Hece (a) Erdos-Reyi etwork EPC (b) Barabasi-Albert Network EPC (c) Small-world Network EPC (d) US backboe etwork Fig. : Comparig performace of the algorithms o differet etwork topologies ad edge probabilities Time(secods) (a) Erdos-Reyi etwork (b) Barabasi-Albert Network (c) Small-world Network (d) US backboe etwork Fig. 3: Comparig ruig time of the algorithms (y-axis) with differet edge probabilities (x-axis). E[S i ] = EPC(G)/. (6) By applyig Theorem to i.i.d. radom variables Y i = (S i )/( ) with mea µ = EPC(G)/ ( ), it follows that E is a (ɛ, δ)-approximatio of EPC(G). C. Time Complexity Aalysis Lemma 4: CSP has a time complexity O(m 4 ɛ 3 ). Proof: If P E < ɛ, the algorithm takes a O(m) time, as we oly eed to compute P E. Otherwise, the algorithm performs N(ɛ, δ) times the BFS algorithm i step 5. Sice the BFS algorithm takes a time at most O(m + ), the total time take by Algorithm is upper bouded by θ(m + )4(exp ) l δ ( ) ɛ EPC(G) By Propositio, EPC(G) P E ɛ. Thus the worstcase time complexity is at most O(m 4 ɛ 3 ). Lemmas 3 ad 4 immediately lead to our mai result for approximatig EPC, stated i the followig theorem. Theorem (Mai theorem): CSP is a FPRAS for the EPC computatio problem that outputs a (ɛ, δ)-approximatio of EPC i a O(m 4 ɛ 3 ) time. CSP ot oly has a polyomial ruig time, it s also faster tha aive Mote Carlo methods. I geeral, the time eeded for the BFS procedure i CSP is ofte less tha the time to geerate a sample graph. The reaso is that the BFS procedure oly eeds to be aware about the surroudig of the selected vertex u, while geeratig a graph sample might ivolve all edges ad vertices i the graph. To formally prove this observatio, we give the expected ruig time of CSP i the followig theorem. ( Theorem 3: CSP has a expected time complexity O ɛ mi{ + m EPC(G), m EPC(G) ). } The proof of this theorem ca be foud i the Appedix. Time(secods) Time(secods) VI. EXPERIMENTS We demostrate through our experimets the efficiecy of our proposed algorithm ad the eed of ew assessmet methods for etworks with ucertaity. A. Experimet Setup Dataset. We aalyze the performace our proposed algorithm through experimets o differet etwork models ad a real commuicatio etwork, as described below. Erdos-Reyi: A radom graph of vertices ad edges followig the Erdos-Reyi model [3]. Barabasi-Albert: A radom graph of vertices ad edges. The graph follows power-law model usig preferetial attachmet mechaism [8]. Watts Strogatz: A radom graph that is geerated from the small-world model [9] with the dimesio of the lattice ad the rewirig probability.3 [9]. US Backboe etwork: The US backboe cablig etwork of XO compay [3] with 78 odes ad 9 liks. Compared Methods. We compare the performace of, Algorithm, with the followig methods : Sample Average Approximatio method [], a commo techique to solve stochastic optimizatio problem. The solutio is furthered optimize usig the same local search procedure i. The umber of samples to optimize is T = 3., a greedy algorithm that removes the odes with the highest betweeess cetrality values., aother heuristic that removes the edge with the highest values. The dampig factor is.85. The umber of samples draw i the local search procedure i both ad are. Durig our experimets, we observe that the local search procedure are quite isesitive to the umber of samplig times. The fial EPC i each etwork is, however, measured by settig the umber of sample times to, to guaratee high accurate estimatio of EPC.

8 Eviromet. All algorithms are implemeted i C++ ad compiled with GCC 4.4 compiler o a 64 bit Widow machie with a i7 3.4Ghz processor ad 6 GB memory. The mathematical optimizatio package to solve liear programmig formulatio is GUROBI 5.5. B. Experimetal results. The solutio quality, i.e., the expected pairwise coectivity (EPC) i the residual etworks are show i Figure. The lower EPC value, the better the algorithm performs. Thus both ad are much better tha the adhoc heuristics based o cetrality. The results of ad are highly similar, except for the largest test cases, where shows a slight advatage over. The ruig time of the algorithms are show i Figure 3 i log-scale. There is o doubt that heuristics based o cetrality takes oly a fractio of secod to complete ad is much faster tha ad. rus much slower tha, up to times slower. This expected behavior is due to the larger size of the liear program that has to cope with. Overall, turs out to be the best choice i terms of both quality ad ruig time. It rus much faster tha, ad also provide much better solutio quality tha the aive cetrality-based heuristics. VII. CONCLUSION Assessig vulerability i etworks with ucertaity is a challegig problem. While the NP-hardess of exact computatio for etwork reliability measures is a sigificat obstacle, such obstacle ca be overcome with efficiet computatioal methods, e.g., the FPRAS to compute EPC. I future, we aim to ivestigate efficiet methods to compute other etwork reliability measures as well as desig more efficiet solutios for the vulerability assesmet i forms of optimizatio problems. VIII. ACKNOWLEDGEMENT This work is partially supported by the NSF CAREER Award ad by the DTRA grat HDTRA A. Proof of Propositio APPENDIX Proof: We prove the lower ad upper bouds separately. Lower boud: By Lemma, we have EPC(G) = REL uv (G) u,v V ;u v REL uv (G) (u,v) E (u,v) E p uv Upper boud: First, we show that EPC(G) (+p e). The we ca apply the iequality of arithmetic ad geometric meas for positive umbers ( + p e ) e E to obtai EPC(G) ( ( ) m + p e ) + p e. m We prove EPC(G) ( + p e) by iductio o µ E the umber of udetermied edges (those with probabilities strictly less tha oe). Basis: If µ E =, we have a determiistic graph with m = E edges. Sice, the size of the largest compoet caot exceed m +, the pairwise coectivity is at most /(m + ) < /m(m + ) < m m. Thus, the iequality holds for µ E =. Iductio step: Assume that the iequality holds for µ E = t, we show that the iequality also holds whe µ E = t+. Assume that µ E = t+, select a arbitrary udetermied edge (u, v) E ad perform a brachig procedure o (u, v) we have EPC(G) = p uv EPC(G + ) + ( p uv )EPC(G ), where G + is obtaied from G by settig the (u, v) s probability to oe ad G is obtaied from G by removig (u, v). Sice, both G + ad G have exactly µ E udetermied edges, we ca apply the iductio hypothesis to obtai EPC(G) p uv ( + ) ( + p e ) + ( p uv ) e (u,v) e (u,v) ( + p e ) = ( + p uv ) + p e ) = e (u,v)( ( + p e ). Thus, the iequality holds for all µ E. B. Proof of Propositio Iequalities o q : O oe had q = ( p e ) p e = P E. (7) O the other had, we have q ( p e m )m (AM-GM iequality) = ( ( P E m )m/p E ) P E < exp( PE ). (8) The last step holds due to the iequality ( x) /x exp( x), x (, ),. Iequalities o q : Iterate through all edges i E, we have q = p e p e ( p e ) = q. p e e e Sice p e P E <, it follows that p e p e p e. p e P E Hece P E q P E q q. P E Iequalities o m k= q k: Apply () ad the (), we obtai m q k = q q q ( + P E ) k= ( P E )( + P E ) = P E.

9 C. Proof of Theorem 3 Proof: I step 5 of CSP, each vertex u is chose uiformly with a probability /. The BFS procedure starts from u ad gradually reveals the availability of edges whe eeded. For ay vertex v visited by the BFS procedure (icludig the startig vertex u), it takes O(d v ) times to check the availability of the icidet edges. Thus the expected umber of edges that are icidet at v ad visited by the BFS procedure at u will be REL u,v (G)d v. Ad the expected umber of visited edges by the BFS procedure at u will be v V REL u,v(g)d v. Sice u is chose uiformly, the expected umber of visited edges by the BFS procedure is REL u,v d v = ( du u V v V u V v u REL u,v ) + m From (6), the expected umber vertices visited by the BFS procedure is EPC(G)/ +. Thus the expected time complexity of the BFS procedure is O ( ) m du REL u,v + + EPC(G) u V v u Apply the iequality d u, the expected time complexity of the BFS procedure ca be simplified to ( O EPC(G) + m ). 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