Using Partial Probes to Infer Network States

Transcription

1 Using Prtil Proes to Infer Network Sttes Pvn Rngudu, Bijy Adhikri, B. Adity Prksh, Anil Vulliknti Deprtment of Computer Siene, Virgini Teh NDSSL, Bioomplexity Institute, Virgini Teh {ijy, ABSTRACT In mny pplitions, suh s the Internet nd infrstruture networks, nodes fil or get ongested dynmilly. We study the prolem of inferring ll the filed nodes, when only smple of the filures is known, nd there exist orreltions etween node filures/ongestion in networks. We formlize this s the Grph- StteInf prolem, using the the Minimum Desription Length (MDL priniple. We propose greedy lgorithm for minimizing the MDL ost, nd show tht it gives n dditive pproximtion, reltive to the optiml. We evlute our methods on syntheti nd rel dtsets, whih inludes one from WAZE whih gives trffi inident reports for the ity of Boston. We find tht our method gives promising results in reovering the missing filures. 1 INTRODUCTION Most network pplitions ssume the network is stti, nd is known hed of time. This is not true in prtie, nd networks re inferred y indiret mesurements, e.g., s in the se of the Internet router/as level grphs, whih re onstruted using treroutes, e.g., [5], or iologil networks, whih re inferred y experimentl orreltions, e.g., [16]. Further, network elements n fil dynmilly, or their stte my hnge with time. For instne, links in the Internet router network or the trnsporttion network n get ongested or fil. Reonstruting the network topology dynmilly nd inferring network sttes in suh settings re fundmentl prolems. Suh prolems hve een studied s prt of the re of network tomogrphy, espeilly in ommunition networks, e.g., [9, 11, 19]. Suh networks re not pulily essile, nd indiret proes re the only mens of otining informtion; exmples of proes inlude queries of the tivity sttes of seleted nodes nd end-to-end mesurements of delys etween seleted pirs of nodes. These eome very hllenging prolems, nd ll prior work in this diretion in network tomogrphy hs een foused on simple models of independent link filures nd delys, e.g., with exponentilly distriuted proilities [11]. In mny settings, suh s disster events in infrstruture networks, however, filures might e sptilly orrelted, s in [1, 3, 15]. For instne, in the model onsidered in [1], the proility tht node j fils deys with the distne from soure s. This motivtes the prolem GrphStteInf of inferring the network sttes under suh sptil orreltions, whih is the fous of our pper. A losely relted topi is the inferene of the soure of n infetion nd other missing infetions in the se of epidemi spred on networks these re typilly modeled s SI/SIR proesses, where the infetion spreds from one node to its neighors with some proility [12, 13, 17, 18]; see [8, 10] for introdution to epidemi models. An pproh sed on the Minimum Desription Length (MDL priniple [7, 14] ws introdued for missing infetion prolems in [12, 13, 18] for the SI proess. However, there re key differenes in our setting, nd the methods developed for missing node infetion in epidemi proesses do not seem to diretly work for the GrphStteInf prolem. Our ontriutions re summrized elow. (1 We develop novel formultion for the GrphStteInf prolem using the Minimum Desription Length (MDL priniple, whih tkes orrelted filures into ount. We present Greedy, n lgorithm for inferring the missing filed nodes, given smple of the filures. We prove tht the MDL ost of the solution omputed y Greedy is within n dditive pproximtion of the minimum ost MDL solution. Typilly, pprohes using MDL re sed on heuristis nd getting ounds is non trivil s MDL ost funtions re not onvex. To the est of our knowledge, our lgorithm is the first to otin rigorous ounds on the ojetive vlue mong MDL sed pprohes for network inferene. (2 We evlute our results on different kinds of syntheti nd rel dtsets, nmely, one week s worth of trffi sttus nd inident reports from WAZE for the ity of Boston, nd eletri disturne events in the power grid (desried in Setion 4.1. We study the preision, rell nd F1-sore for Greedy, ompred with seline. We oserve tht our lgorithm is quite effetive in inferring unknown/missing filures in the network, nd hs lower MDL ost thn the seline. The rest of the pper is orgnized in the usul wy, with the formultion first, then methods, nd experiments. We then finlly give the relted work nd onlusions. 2 OUR PROBLEM FORMULATION We re given n undireted grph G(V, E representing n infrstruture network. We ssume there is n initil filure t node, referred to s the seed node, whih uses other nodes to fil. Further, suset Q I of the tul filed nodes re ssumed to e known. The ojetive in the GrphStteInf prolem is to infer ll the missing filures. 2.1 Filure Model Next, we will disuss the filure model we use to desrie the filures in the given network G. This model is motivted y the geogrphilly orrelted filure model introdued y Agrwl et l [1], to pture filures in infrstruture networks due to lrge sle dissters. In suh events, there is n initil lolized filure, whih uses other nodes to fil with some proility tht deys with the distne from the soure.

2 ,, Pvn Rngudu, Bijy Adhikri, B. Adity Prksh, Anil Vulliknti Following [1], we ssume there is n initil single seed node s nd ll the filures I in G re used due to the influene of tht seed node. We ssume disrete proility distriution funtion p s : V [0, 1] tht gives the proility of eh node v V eing seed nd onditionl filure proility distriution funtion F : V V [0, 1] tht gives the filure proility of node v V given seed node s. Note tht the p s (v is the proility of v eing the only seed, i.e, v V p s (v = 1. These proility distriutions re preomputed from historilly oserved filures. We ssume tht the onditionl filure proilities given y F re independent i.e., for-ll v 1,v 2 V nd v 1 v 2, 2.2 Proes F(v 1 v 2 s = F(v 1 sf(v 2 s (1 Bsed on our model given ove, we ssume tht some seed filed using multiple orrelted filures ross the network G. The finl set of true filures is represented y I V. Further, we re lso given set of filed nodes represented y Q I, whih we will refer to s proes in rest of this pper. We ssume tht the given set of input proes Q re smpled uniformly t rndom from the true filure set I with proility γ. 2.3 MDL We formulte our prolem using the Minimum Desription Length (MDL priniple [6]. We will use two-prt MDL, or the senderreeiver frmework. Our gol here is to trnsmit the given set of proes Q from sender to reeiver y ssuming tht oth of them know the lyout of the network G. We do this y identifying the model tht est desries the given dt in terms of forml ojetive or ost funtion. This ost funtion onsists of two prts: (1 Model ost tht signifies the omplexity of the seleted model tht explins the filures in the network; nd (2 Dt ost tht represents the ost of oserving the given proe dt Q given the model. More formlly, given set of models M, MDL identifies the est model M s the model tht minimizes L(M + L(D M, in whih L(M is the model-ost (length in its to desrie model M, nd L(D M is the dt-ost (the length in its to desrie the dt using M. Note tht the dt we need to desrie in our sitution is the proes set Q (nd not the true filures set I. Next we desrie the model spe nd the model nd dt ost, whih we will optimize. 2.4 Model Spe nd Cost Model Spe: The most nturl model for our prolem would hve een M = (s, I (the soure s V nd the full filure set I V, s it diretly mimis the genertive proess of the filure model. However, this model hs severl disdvntges. Firstly, note tht this model spe is intuitively frgile : smll hnges in I or the soure s n hve vstly different osts. Hene due to dt sprsity, we expet it would e very hrd to lern the true soure whih generted the filures indeed, in our experiments, we find tht it ws not roustly lerning the true soure. As result, we lso found tht the solutions with minimum MDL ost were finding very few missing filed nodes (i.e. I Q, leding to very low rell. How to design etter model spe for our prolem? We oserve tht this model intuitively tries to explin more thn wht is needed. Note tht while our originl gol ws to mp the missing filures only, this pproh tries to explin the soure s well s the set of filures. Hene we dopt different pproh, where we try to mrginlize over the seeds, nd fous only on the filures. This mkes our model spe more roust s well. This motivtes our proposed model, whih onsists of three omponents, nmely, M = ( Q, I, I. In other words, we send the size of proes Q, the size of true filure set I, nd then identify the set itself. After sending the model, we will then identify the tul proes set Q s the dt. Model ost: The MDL model ost, L( Q, I, I hs three omponents L( Q, I, I = L( Q + L ( I ( Q + L I Q, I. ( We derive these elow. We hve L( Q = log Pr( Q, y using the Shnnon-Fno ode to enode Q. Similrly we hve: L ( I ( Q = log Pr ( I Q ( ( Pr Q I Pr ( I (2 = log Pr ( Q From the smpling ssumption for Q, we n get: Pr ( Q ( I I = γ Q (1 γ I \Q (3 Q Also oserve tht: L ( I Q, I = log ( Pr ( I Q, I = log ( Pr ( I ( Pr (I (4 I = log Pr ( I Comining ll of the ove, the omplete model ost is: L( Q, I, I =L( Q + L ( I Q + L ( I Q, I 2.5 Dt Cost ( ( ( Pr Q I Pr ( I = log Pr ( Q log Pr ( Q ( Pr (I log Pr ( I ( = log Pr ( Q ( ( Pr (I I log Pr ( I log Pr ( I ( ( I = log γ Q I (1 γ \Q Q ( log Pr (I sp(s s V ( I = log Q log(γ ( I Q log(1 γ Q ( ( log p s (s F (v s 1 F (v s s V v I v I (5 Now, we need to desrie the given input proes Q in terms of the model. Given model M = ( Q, I, I, desriing Q is the sme s speifying the djustments tht needs to e pplied to the filure set I in the model to reh Q, whih n e done y desriing the following sets: (1 Unoserved filures i.e., Q + = I \ Q (2 Oservtion errors i.e., Q = Q \ I

3 Using Prtil Proes to Infer Network Sttes,, In this pper, we ssume tht there re no oservtion errors, i.e., Q = (s Q I. Aording to the smpling ssumption we hve, Q is smpled uniformly t rndom from I with proility γ. This implies tht Q + = I \ Q is smpled from I with uniform proility (1 γ. Hene we n ompute the proility of seeing set Q + when smpled from I s follows Pr(Q + I = γ Q (1 γ Q+ Now, using this proility distriution of oserving the set Q + given the filure set I we n ompute the optiml numer of its required to trnsmit Q + enoded in terms of model M s follows (gin using the Shnnon-Fno ode: L(Q + I = log (γ Q (1 γ Q+ (7 = Q log(γ ( I Q log(1 γ 2.6 Our Forml Prolem Putting it ll together, we n stte our forml prolem: Given n undireted grph G(V, E, where node filures tken ple in the network s per the model desried in Setion-2.1, nd set of oserved filures Q V, whih re smpled independently from the true filure set I with uniform proility γ, find the omplete set of filures I V y minimizing the MDL ost funtion L ( Q, I, I, Q given y L ( Q, I, I, Q = L( Q +L ( I Q +L ( I ( Q, I + L Q Q, I, I ( I ( ( = log log p s (s F (v s 1 F (v s Q s V v I v I 2 Q log(γ 2( I Q log(1 γ (8 where p s (s is the seed proility of s nd F(v s is the filure proility of node v given seed node s. 3 PROPOSED METHODS Clerly the serh spe for the prolem is lrge, nd there exists no trivil struture for fst serh. We now desrie two pprohes for finding solutions with low MDL ost. The first, LolSerh, inrementlly dds node tht gives the most redution in MDL ost, till no further improvements our. The seond, Greedy, guesses the size k of the optiml solution, nd greedily piks the k nodes tht would minimize the ost. We show tht the ost of the solution produed y Greedy is within n dditive ftor of the optimum. 3.1 Algorithm LolSerh This is n intuitive lol serh lgorithm whih is populrly used in mny MDL optimztions. We initilize Î to Q, nd just repetedly dd node tht redues the MDL ost. This is desried formlly in Algorithm Algorithm Greedy In this setion we will disuss n effiient lgorithm for finding filure set I whih provides n dditive pproximtion gurntee on the MDL ost of the solution. (6 Algorithm 1 Algorithm LolSerh Input: Instne (V, Q,p, P,γ Output: Solution Î tht minimizes L( Q, Î, Î, Q 1: Î Q 2: while v V \Î : L( Q, Î, Î, Q L( Q, Î +1, Î {v}, Q > 0 do 3: u rg mx L( Q, Î, Î, Q L( Q, Î + 1, I {v}, Q v V \Î 4: Î Î {u} 5: end while 6: Return Î First, let A = log ( I Q 2 Q log(γ. We rewrite the MDL ost funtion in the following mnner: L ( Q, I, I, Q ( ( =A log p s (s F(v s 1 F(v s v I v I 2( I Q log(1 γ ( ( =A log p s (s 1 F(v s v V F(v s 2( I Q log (1 γ (1 F(v s v I ( ( =A log p s (s 1 F(v s (1 γ 2 Q v V F(v s(1 γ 2 I (1 F(v s v I ( =A log д(s f (s,v, v I (9 where д(s = (1 γ 2 Q p s (s ( v V 1 F(v s nd f (s,v = F (v s(1 γ 2 1 F (v s. Therefore, the prolem redues to finding set Î suh tht { ( I Î = rg min log I Q ( log д(s + 2 Q λ 1 v I } f (s, v. (10 The min ide of this lgorithm is to use the quntity f (s,v = F (v s(1 γ 2 1 F (v s defined ove s the weight for eh pir (s,v. For eh seed node s V, we guess the size of the solution I s, if the soure were to e s, nd pik the set of I s nodes whih minimizes the MDL ost. This is desried formlly in Algorithm 2, nd nlyzed in formlly Theorem 3.1. Theorem 3.1. Let I e the set minimizing the MDL ost, nd let I denote the solution omputed y Algorithm Greedy. Then, L( Q, I, I, Q L( Q, I, I, Q + log(n, where n is the numer of seed nodes. Proof. the definitions of д(s, f (s, v nd A ove. Then, ( L( Q, k, I, Q = log ϕ(s, I + A,

4 ,, Pvn Rngudu, Bijy Adhikri, B. Adity Prksh, Anil Vulliknti Algorithm 2 Algorithm Greedy Input: Instne (V, Q,p, P,γ Output: Solution Î tht minimizes L( Q, Î, Î, Q 1: for eh s V do 2: for eh k [ Q, V ] do 3: I s (k Top k Q nodes in V \ Q with highest weight f (s,v 4: I s (k I s (k Q 5: end for 6: end for 7: S {I s (k : s V &k [ Q, V ]} 8: Î rg min L( Q, I, I, Q I S 9: Return Î where ϕ(s, I = д(s v I f (s,v. Note tht ϕ(s, I is mximized for the set I s (k defined in Algorithm Greedy, sine this onsists of the set of top k Q nodes in V \ Q, with respet to the quntity f (s,v, long with ll nodes in Q. Therefore, we hve ϕ(s, I s (k ϕ(s, I Adding over ll possile seed nodes, we hve ϕ(s, I s ( I ϕ(s, I whih implies for some seed ŝ, we hve ϕ(ŝ, Iŝ (k 1 ϕ(s, I n ( ( 1 log ϕ(ŝ, Iŝ (k log ϕ(s, I n ( = log ϕ(s, I + log(n This, in turn, implies ( L( Q, k, Iŝ (k, Q = log ϕ(s, Iŝ (k + A ( log ϕ(ŝ, Iŝ (k + A ( log ϕ(s, I + log(n + A L( Q, k, I, Q + log(n, where the first inequlity follows euse ϕ(s, I 0 s, I, so tht ϕ(s, Iŝ (k ϕ(ŝ, Iŝ (k. Sine, the Algorithm 2 serhes over ll possile solution sizes k, the theorem follows. Lemm 3.2. Algorithm Greedy runs in O( V 3 time. Proof. The quntities д(s nd f (s,v defined erlier in (9 n e omputed for ll s,v in O( V 3 time. The lgorithm involves two for loops. The inner loop in lines 2-5 runs in O( V time, sine it finds solution I s (k for eh k. The set S hs size O( V 2. Step 8 of the lgorithm involves omputing L( Q, I s (k, I s (k, Q for eh set I s (k. Done nively, it tkes O( V 2 time to ompute this. However, y keeping the intermedite solutions, we n ompute L( Q, I s (k, I s (k, Q for given s nd for ll k inrementlly in O( V 2 time, leding to the time ound in the lemm. 4 EXPERIMENTS We evlute performne of our lgorithms on vrious syntheti nd rel networks, whih re disussed next. 4.1 Dtsets Syntheti Grid Dtset. We reted simple grid where eh ell is onsidered s node in rod network, leding to 3600 nodes. We ssumed n uniform seed proility distriution ross ll nodes. We omputed onditionl filure proilities (PlinCF etween pir of nodes (s, v sed on Geogrphilly Correlted Filure (GCF Model [1] i.e., if s is the seed node then F(v s = 1 d(s,v (11 where d(s,v is distne funtion d : V V [0, 1]. In our se, d(, is the Mnhttn distne etween the nodes normlized y the mximum distne. We will refer to this set of onditionl filure proilities s GCF. Rel Dtsets. We reted three dtsets from rel world node filure logs in trnsporttion nd power-grid networks. We use filure logs in rod networks from WAZE lerts dt, whih is pulily ville on the City of Boston s wesite 1. These lerts hve een reported y users vi the rowd soured pplition WAZE etween Mondy 23 rd Ferury, 2015 nd Sundy 1 st Mrh, The lerts in the dtset re sptilly distriuted ross Boston, Cmridge, nd Brook-line regions of Msshusetts. The lerts inlude different types filures suh s trffi jm, extreme wether, idents, nd rod losures. Additionlly, the ltitude nd longitude of the ffeted lotions, nd strt nd end time of the lert is lso given. From these lerts, we reted two dtsets sed on trffi jm (JAM nd extreme wether (WEATHER. Similrly, we use list of Eletri disturne events from Energy.gov 2 this list inludes reported events of eletri emergenies nd disturne in power supply from 2002 to Eh event log ontins informtion regrding dte nd time of the eginning nd restortion of the event, geogrphil res ffeted y the event, numer of ustomer ffeted, nd so on. We reted POWER-GRID dtset from the log of eletri emergenies nd disturnes. Dtset retion. As disussed in Setion 2.1, we need to define the seed proility nd pir-wise onditionl filure proility distriutions over ll nodes in the network. This is done in the following mnner. For WAZE lert dt, we hve prtitioned the omplete geogrphil region oupied y these filures y using grid s shown in Figure 1, where eh ell is squre nd ts s node in our virtul rod network. For the POWER-GRID dt, eh lotion referred in the dtset ts s node. Seed Proility. Let n v denote the ell/nodev, s disussed ove, nd let N = v n v denote the totl numer of filures ross ll nodes. We define the seed proilities s p s (v = n v (12 N Conditionl Filure Proilities. We onstrut Binry Filure Stte Time Series (BinTS for spn of 7 dys, y using the temporl informtion tht is ville from WAZE lerts dt. This time series

5 Using Prtil Proes to Infer Network Sttes,, filure likelihood nd onditionl filure proilities re desried in Setions 4.1 nd Figure 1: Prtitions of Boston region oupied y WAZE lerts using grid gives inry (0 or 1 vlue for eh time step whih represents the filure stte of the respetive node i.e., BinT Sv (t = 1 implies tht there is t-lest one filure in v t time t. Using BinTS we were le to ompute the pir-wise onditionl filure proilities for our dtsets in the following mnner. For two nodes v 1 nd v 2, we define the Plin Conditionl Filure Proility (PlinCF of v 1 given v 2 s the rtio etween numer of time steps in whih oth v 1 nd v 2 re filed (i.e. with vlue 1 in BinTS to the numer of time steps in whih only v 2 is filed. Pl inc F (v 1 v 2 = {t t, BinT Sv1 (t = 1&BinT Sv2 (t = 1} (13 {t t, BinT Sv2 (t = 1} We study two more syntheti onditionl filure proilities, nmed URndCF nd NRndCF, for eh dtset; these re defined in the following mnner: URndCF is n ritrry smple from uniform distriution nd NRndCF is n ritrry smple from norml distriution ( N (5, 1 over the vlues [0, 1]. We follow the sme proedure to generte onditionl proility filures for the POWER-GRID dtset. Dtset desriptions. We onstrut three different dtsets following the ove steps. JAM: This is dtset tht we reted from WAZE lerts dt using dt of filure type JAM. The resultnt dtset onsists of rod network with 2650 nodes long with seed proility distriution nd onditionl filure proilities omputed s disussed in Setions 4.1 nd 4.1. Figure 2 presents sptil nd frequeny distriutions of the seed nd onditionl filure proilities of this dtset. WEATHER: Similr to the JAM dtset this dtset is reted y using WEATHERHAZARD filure dt from WAZE lerts dt. The resultnt dtset onsists of rod network with 1520 nodes long with seed proility distriution nd onditionl filure proilities omputed s disussed in Setions 4.1 nd 4.1. POWER-GRID: As mentioned erlier, we reted this dtset from the log of eletril emergenies nd disturnes. We filtered out the events in the log whih were not relted to loss of eletri servie. For this dtset, whih onsists of 24 nodes, we omputed Performne evlution In this setion we disuss the performne of our lgorithms ginst vrious dtsets tht re desried in Setion 4.1 ross vrious vlues of γ [, ]. We exmine the preision, rell nd F1-sore for Greedy, ompred with LolSerh. We oserve tht our MDL sed pproh does indeed llow us to infer unknown/missing filures in the network using the proes. The speifi MDL formultion we onsider in Setion 2.6, whih inludes I in the model seems to perform muh etter thn other nturl MDL formultions. Comprison of Greedy with LolSerh. Figure 3 presents omprison of the trends in performne of oth lgorithms on the JAM dtset with PlinCF proilities ross γ [, ] nd MDL ost of their respetive solutions. For oth lgorithms, the performne vries with the smpling rte, γ. We find tht the solution omputed y Greedy hs lower MDL ost, ompred to the seline. One interesting oservtion from Figure 3 is tht the rell for Greedy deys with γ till nd then inreses. Greedy hs higher F1-sore ompred to LolSerh, for most vlues of γ. In the rest of our evlution, we only onsider Greedy. Performne of Greedy for different dtsets. Figures 4, 5, 6 nd 7 present the performne of Algorithm Greedy for ll the dtsets from Setion 4.1, nd for the three different wys of defining onditionl proilities. Aross ll these results, on verge we re le to find 80% of the filed nodes with n verge preision of 79% ross vrious vlues of γ [, ]. In other words, we re suessful in inferring resonle frtion of unknown/missing filures in the network from prtil set of oservtions with resonle preision. 5 RELATED WORK Some of the different res relted to our work inlude network nd stte inferene in ommunition networks, reonstruting networks from sdes nd inferring missing infetions in the se of epidemis. We riefly disuss these elow. The re of network tomogrphy involves inferring link sttes, suh s delys nd filures, in the Internet nd other ommunition networks; see, e.g., [2, 4, 9, 11, 19]. Proes suh s end-to-end delys re the only mesurements tht re ville in suh networks. At n strt level, the prolem here involves solving for the link dely vetor x, given the mesured delys ross the proes. This eomes very hllenging prolem nd it is typilly ssumed tht link hrteristis suh s delys re modeled s independent rndom vriles with known distriutions, ut potentilly unknown prmeters. Xi et l. [19] solve this ssuming link delys re exponentilly distriuted. Ni et l. [11] study different kinds of proing models, inluding multist proes whih n give estimtes on tree, nd develop methods for inferring the topology in dynmi networks. There hs lso een work on designing proes to infer prt of the network struture, s in [2]. Another relted topi is the inferene of the soure of n infetion nd other missing infetions, in the se of epidemi spred on networks. Epidemis re modeled s stohsti proesses, e.g.,

6 Pvn Rngudu, Bijy Adhikri, B. Adity Prksh, Anil Vulliknti,, : Sptil distriution of seed proilities. : Sptil distriution of PlinCF for rndom seed node. (L(I, Q/L(I*, Q / / / / / / Figure 2: JAM dtset reted from WAZE lerts dt of Boston, Cmridge, Brook-line regions Quik Lol Greedy / / / / / / / / / Figure 3: Performne of the LolSerh ( nd Greedy ( on the JAM dtset with PlinCF proilities. The MDL osts of the solutions is shown in (. Figure 4: Performne of Greedy on the Syntheti Grid Dtset with GCF (, URndCF (, nd NRndCF ( onditionl proilities. SI/SIR, in whih the infetion spreds from n infeted node to its suseptile neighors. Usully, only prtil informtion out the infetions is known, nd some of the prolems tht hve een studied inlude identifying the soure of n infetion nd finding other missing nodes [12, 13, 17, 18]. The Minimum Desription Length (MDL priniple [7, 14] hs een suessfully used in [12, 13, 18] for these prolems, wheres [17] develop n MLE method. A different lss of filure models from the SI/SIR type of epidemis hs een studied extensively, motivted y settings suh s disster events, e.g., [1, 3, 15]. These studies ssume n initil filure, nd susequent filures whose proility is orrelted with the soure. For instne, in [1], the proility p(j i tht node j fils, given tht i is the soure is funtion of the distne from i to j, with the proilities deying with the distne. Our work is motivted y these models.

7 Using Prtil Proes to Infer Network Sttes,, / / / / / / / / / Figure 5: Performne of Greedy on the JAM Dtset with PlinCF (, URndCF (, nd NRndCF ( onditionl proilities. / / / / / / / / / Figure 6: Performne of Greedy on the WEATHER Dtset with PlinCF (, URndCF (, nd NRndCF ( onditionl proilities. / / / / / / / / / Figure 7: Performne of Greedy on the Power Grid Dtset with PlinCF (, URndCF (, nd NRndCF ( onditionl proilities. 6 CONCLUSIONS Our results show tht n MDL sed pproh is quite useful in the prolem of inferring missing filures in settings with orrelted filures. This motivtes its use in other inferene prolems with prtil informtion. We hve onsidered the simplest notion of proe here informtion out speifi nodes whih hve filed. Extending our work to other kinds of proes (like onnetivity queries is n interesting nd nturl prolem. Inferring the stte of the network using suh proes, nd supporting dditionl queries re interesting prolems. Further we hve ssumed there re no oservtionl errors designing roust lgorithms in fe of errors is lso interesting future work. Aknowledgements This pper is sed on work prtilly supported y the NSF (IIS , the NEH (HG , ORNL (Order nd from the Mrylnd Prourement Offie (H C-0127, Feook fulty gift, DTRA CNIMS Contrt HDTRA1-11-D , NSF BIG DATA Grnt IIS nd NSF DIBBS Grnt ACI REFERENCES [1] Pnkj K. Agrwl, Alon Efrt, Shshidhr K. Gnjugunte, Dvid Hy, Swminthn Snkrrmn, nd Gil Zussmn The Resiliene of WDM Networks to Proilisti Geogrphil Filures. IEEE/ACM Trns. Netw. 21, 5 (Ot. 2013, DOI: [2] Animshree Anndkumr, Avintn Hssidim, nd Jonthn Kelner Topology Disovery of Sprse Rndom Grphs with Few Prtiipnts. SIGMETRICS Perform. Evl. Rev. 39, 1 (June 2011, DOI:

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