COMPRESSED NETWORK MONITORING. Mark Coates, Yvan Pointurier and Michael Rabbat

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1 COMPRESSED NETWORK MONITORING Mark Coates, Yvan Ponturer and Mchael Rabbat Department of Electrcal and Computer Engneerng McGll Unversty 348 Unversty Street, Montreal, Quebec H3A 2A7, Canada ABSTRACT Ths paper descrbes a procedure for estmatng a full set of network path metrcs, such as loss or delay, from a lmted number of measurements. The approach explots the strong spatal and temporal correlaton observed n path-level metrc data, whch arses due to shared lnks and statonary components of the observed phenomena. We desgn dffuson wavelets based on the routng matrx to generate a bass n whch the sgnals are compressble. Ths allows us to explot powerful non-lnear estmaton algorthms that strve for sparse solutons. We demonstrate our results usng measurements of end-to-end delay n the Ablene network. Our results show that we can recover network mean end-to-end delay wth 95% accuracy whle montorng only 4% of the routes. Index Terms network montorng, dffuson wavelets, compressed sensng 1. INTRODUCTION Drect montorng of a network ether at the path level or the lnk level does not scale n any practcal settng. For the past decade, researchers have been actvely nvestgatng technques for nferrng network characterstcs from ncomplete or ndrect measurements [1, 2]. Ths paper descrbes a scheme for estmatng performance metrcs such as delay or loss rates on many end-to-end paths n a network usng measurements taken on only a few of these paths. Smlar to related prevous work [3, 4], we explot the dea that, typcally, each lnk s used by many paths. Thus, by only measurng a subset of paths, t s possble to nfer the performance on other unmeasured paths usng knowledge of the routng topology. Our approach combnes the use of wavelets, for effcent sgnal representaton, wth recent developments n nonlnear estmaton, for recoverng a sparse sgnal from few measurements. We use dffuson wavelets [5] to compress the pathlevel performance sgnal nto a set of sparse coeffcents (only a few coeffcents have sgnfcant energy). The wavelets are desgned to specfcally account for antcpated correlatons Ths work was supported by the Natural Scences and Engneerng Research Councl (NSERC) of Canada and ndustral and government partners, through the Agle All-Photonc Networks (AAPN) Research Network. between dfferent path performance values, based on knowledge of the network topology and routng polcy. Then, we use l 1 optmzaton technques to dentfy the most lkely set of sparse wavelet coeffcents gven measurements on a few paths, and these estmated wavelet coeffcents are used to predct performance values on unobserved paths. In comparson to the prevous state-of-the-art network krgng estmator [4], we fnd that our method makes more accurate estmates wth few measurements on real data sets. 2. PROBLEM FORMULATION Suppose we wsh to montor performance metrcs on n p paths n a network wth n l lnks. Call x (k) R n l the vector of lnklevel performance metrcs and y (k) R np the vector of pathlevel performance metrcs at a partcular tme nstant, k. We assume we are gven the routng matrx, G (k) {, 1} np n l, where G (k),j = 1 f and only f lnk j belongs to path. We also assume that x and y are lnearly related va the expresson y (k) = G (k) x (k), (1).e., the performance metrc of a path s equal to the sum of the performance metrcs of the lnks comprsng that path. Examples of relevant performance metrcs satsfyng ths assumpton nclude mean delay, delay varance, and the (log) rate of successful transmsson. In addton to the routng matrx, G, we observe n s metrcs y s, on a subset s of paths (n s = s ). Our goal s to nfer the metrcs for the remanng n p n s paths. Observatons of end-to-end metrcs are made accordng to a selecton matrx A (k) {, 1} ns np whch effectvely selects the routes that are observed (sampled); each row of A (k) contans exactly one 1 and each column contans at most one 1. The presence of a 1 n the p th column of A (k) ndcates we observe the performance metrc on path p. The observed samples (e.g., end-to-end delays) y s (k) R ns are therefore y (k) s = A (k) y (k). (2) We perform estmaton of metrcs over a block of measurement ntervals, allowng us to explot both spatal and

2 temporal correlaton of the path-level metrcs. We denote by y = [y (1)T,..., y (τ)t ] T the vector of path-level metrcs over τ tmesteps. Smlarly, the correspondng lnk metrcs are x = [x (1)T,..., x (τ)t ] T, and we ntroduce block-dagonal matrces G R npτ nlτ and A R nsτ npτ, wth dagonal submatrces G (1),..., G (τ) and A (1),...,A (τ), respectvely. Wth ths new notaton we can wrte y = Gx, and y s = AGx. In general, routng n a network changes very slowly over tme, and unless reroutng occurs, we have G (k) = G for all k. We focus on ths specal case here, although ncorporatng tme-varyng routng and load balancng s a straghtforward extenson. 3. DIFFUSION WAVELETS Wavelet transforms are a staple of modern compresson and sgnal processng methods due to ther ablty to effcently represent pece-wse smooth sgnals (sgnals whch are smooth everywhere, except for a few dscontnutes). Tradtonally, dscrete wavelet transforms provde a mult-scale decomposton of functons defned on a regularly sampled nterval or grd. A mother wavelet s dlated by powers of two and translated to obtan orthonormal wavelet bases. However, n the context of network montorng, we seek to effcently represent a functon (performance metrcs) defned on a network topology whch does not, n general, have a regular structure, so standard wavelets cannot be drectly appled. Crovella and Kolaczyk [6] descrbe one method of constructng wavelets on a graph for decomposng traffc on an arbtrary topology based on dlatng and scalng a mother wavelet, smlar to the tradtonal approach. The prmary shortcomng of ths approach s that t does not lead to an orthogonal bass, lmtng ts use as a mechansm for generatng a compressble representaton of a network functon. More recently, Cofman and Maggon [5] have ntroduced dffuson wavelets, generalzng the concept of wavelets to functons supported on a graph through the use of dffuson operators. The constructon of a dffuson wavelet bass s based on a dffuson operator, D, defned on the support of the underlyng graph. For a graph wth n nodes, D s an n n matrx where D,j > f and only f there s a lnk between nodes and j. The magntude of D,j models the strength of the correlaton or smlarty between the functon values at nodes and j. Much lke tradtonal wavelets, dffuson wavelets recursvely splt the space over whch the sgnal s observed nto smaller, orthogonal subspaces. Consder a functon f R n defned on a network of n nodes, where f corresponds to the value at node ; the functon s ntally defned on the space V = R n. At scale j = 1,...,L, for some pre-specfed depth L, the dffuson wavelet constructon recursvely splts the space V j nto a scalng subspace, V j+1, and a wavelet subspace, W j+1, by analyzng egenvectors of the jth dyadc power of the dffuson operator, D 2j. The matrx D 2j s, ntutvely, related to averagng or smoothng over neghborhoods of radus 2j hops n the orgnal graph, and the study of egenvectors of ths matrx s analogous to Fourer spectral analyss on a regular space. The ensung orthonormal wavelet bass, adapted to the representaton of the data (functon values) over the graph, s obtaned by concatenatng bases for V L and the wavelet subspaces, {W j } L j=1. We refer the reader to [5] for the precse detals of the constructon. Let B j R n, j = 1,...,n, denote the fnal collecton of orthonormal wavelet bass vectors. A functon on the graph can be represented as a vector y R n, where y s the value at the th node, and the wavelet decomposton of y s gven by y = n j=1 β jb j, where β j = y T B j s the jth wavelet coeffcent. Stackng the coeffcents, β j, nto a vector, β, and concatenatng the bass vectors, {B j } nto an n n matrx, B, we can wrte y = Bβ. In the followng secton we propose a dffuson operator D desgned such that the correspondng wavelet representaton of a path performance vector, y, s hghly compressed;.e., most of the energy n y can be captured n a few β j. To be more precse, let us rearrange the wavelet coeffcents n order of decreasng magntude so that β (1) β (2) β (n), and defne the best m-term approxmaton of y n B to be ŷ (m) = m j=1 β (j)b (j). We say that y s compressble n B when the approxmaton error y ŷ (m) decays rapdly as a functon of m, meanng that y s effcently represented usng only a few bass vectors, B (1),...,B (m). In ths case, we only really need to estmate values of the few large coeffcents n order to obtan a hgh qualty estmate of end-to-end performance on many paths. Moreover, n ths settng we can make use of recent breakthroughs n the area of non-lnear estmaton of compressble functons to quantfy the number of paths that need to be measured to obtan estmates of performance at a specfed level of accuracy. 4. COMPRESSIBLE REPRESENTATIONS In order to construct a compressble representaton, we develop a dffuson wavelet bass where the dffuson operator s related to the antcpated correlaton between lnk metrcs. We frst defne the graph of nterest G = (V, E). We are measurng (weghted sums of) a performance metrc functon defned on the physcal lnks of the network. Accordngly, the vertex set V for our dffuson wavelet bass has one vertex for }, over the estmaton nterval k = 1,...,τ. The edges n the graph model correlaton and ther assocated weghts defne the dffuson operator. We form the edge set E by nsertng two types of edges nto the graph. We add a each lnk at each tmestep: V (k) = {v (k) spatal-correlaton edge between the vertces v (k) and v (k) j f there s at least one route n the network that uses both lnks

3 and j durng tme nterval k. We assgn a weght w(v (k), v (k) j ) to ths edge to model the spatal correlaton. Denote by R the set of paths that use lnk. We set w(v (k), v (k) j ) = R R j R R j, (3) thereby assgnng greater weghts to edges that jon network lnks whch have more routes n common. We also add a temporal-correlaton edge between vertces v (k) and v (k+1) for k = 1,...,τ 1, reflectng the dea that performance of a lnk at one tmestep s a good predctor for the same lnk at the next tmestep. We assgn a weght w(v (k), v (k+1) ) = c to these edges, where c > s a constant that s adjusted accordng to the (antcpated) relatve strength of temporal correlaton. In ths paper, we set c =.5, because we antcpate strong correlaton. The dffuson wavelet procedure n [5] requres a dffuson operator D to generate a wavelet representaton over G. To obtan a dffuson operator D from the constructon descrbed above, we apply Snkhorn balancng [7] to the matrx of weghts, [w], to form a doubly stochastc matrx. 5. NON-LINEAR ESTIMATION Now, suppose we have made observatons y s of the end-toend performance for a subset of the paths we are nterested n, and we wsh to estmate y. We have y s = Ay, where A s the selecton matrx, ndcatng whch paths we observe drectly. Our estmaton procedure s based on the belef that the lnk metrcs are hghly compressble when represented usng the dffuson wavelet bass B. Call β the lnk metrcs coeffcents n the dffuson wavelet bass B, such that x = Bβ. We can express y n terms of ts wavelet coeffcents as y = GBβ, where most of the energy n y s captured n a few entres of β. Combnng ths expresson wth the expresson for y s above leads to y s = AGBβ. In ths formulaton, we have a strong pror belef that β has only a few large entres, wth most beng very small n magntude or even zero. Ths motvates the adopton of estmaton strateges that produce sparse estmates. A straghtforward approach to obtanng a sparse estmate of β s to solve an l optmzaton of the form: β = arg mn β subject to y s = AGBβ, β where β counts the number of non-zero entres of β. It s well known that ths problem s NP-hard, requrng one to enumerate all possble subsets of non-zero coeffcents. It has recently been shown that the soluton to a smpler l 1 optmzaton problem, β = arg mn β β 1 subject to y s = AGBβ, (4) Fg. 1. Ablene backbone: 11 nodes, 3 (undrectonal) lnks. s equvalent to the l problem f certan condtons on A, GB, and β are satsfed [8 1]. Here, β 1 = n =1 β. Because the l 1 optmzaton (4) s convex, t s computatonally tractable, and a soluton can be obtaned usng lnear programmng. 6. PATH SELECTION So far, we have dscussed why and how t s possble to estmate accurately end-to-end metrcs from a lmted number of observatons. However, we have not dscussed how to choose the path selecton matrc A, that s, select the n s routes whch should be observed. Ths problem s challengng and the approprate approach depends on the measurement constrants or costs. Here, we consder that the cost (or constrant) s the number of measurements made per tmestep. It thus does not matter here whch paths we measure. For example, f we are allowed one observaton per tmestep, then we can measure the same route every tmestep or we can measure a dfferent route every tmestep. We adapt the path selecton technque presented by Chua et al. n [4] to nclude our correlaton model. The path selecton procedure n [4] strves to mnmze the mean square of the predcton error of a lnear end-to-end delay estmator. The exact mnmzaton procedure s NP-complete (t amounts to the problem of subset selecton) and hence heurstcs are needed. Chua et al. propose a heurstc that conssts of fndng the rows of the routng matrx G that approxmate the span of the frst n s left sngular vectors of GC, where C s a nonsngular matrx that satsfes Σ = GC and Σ s the covarance of x. Note that the estmaton methodology n [4] s restrcted to metrcs of the form y = Gx, whch leads to the ncorporaton of a lnk-level covarance matrx n ths path selecton procedure. In the case where ths covarance matrx s not known, reasonable results can be often by obtaned by settng Σ = I. An algorthm (see Alg. 1) that mplements ths heurstc can be found n [11]. The ntuton behnd ths heurstc s that most of the energy of the path metrc sgnal should resde n the space

4 Coeffcent magntude In orgnal bass In dffuson wavelet bass Coeffcent ndex (k) Fg. 2. Lnk delays for the complete network over 8 tmesteps, sorted by magntude, n the orgnal bass and n the dffuson wavelet bass. spanned by the n s left sngular vectors of GC. Identfyng a set of paths that approxmately span ths space s thus a desrable goal. Here, we use as the covarance Σ = D τ (recall D s the dffuson operator and τ the number of tmesteps used to account for tme-correlaton), and thus selecton s performed by Alg. 1 where the nput matrx M s equal to GC. Algorthm 1 path selecton Input: Matrx M Number of paths to select n s Output: Selecton matrx A 1: Perform SVD on M : M = USV T where S contans the sngular values of G, n descendng order. 2: Perform QR decomposton wth column pvotng on the frst n s columns of U: QR = U T (1,...,n s) P T. 3: Return A = P (1,...,ns), the matrx formed from the frst n s columns of P. 7. NUMERICAL RESULTS To llustrate the estmaton technque presented n ths paper, we use expermental delay data collected on the Ablene backbone network depcted n Fg. 1. The network conssts of 11 nodes and 3 undrectonal lnks. Mean end-to-end delay measurements are collected between every par of nodes over 48 fve-mnute ntervals. There are thus 121 path metrcs to be estmated at each tme step. We frst verfy the compressblty of the data. Fg. 2 shows the delays for all lnks and the absolute values of the dffuson wavelet coeffcents, over τ = 8 tmesteps, sorted n descendng order. The decay of the delays expressed n the orgnal bass s very slow and exhbts a heavy tal, whereas n the dffuson wavelet bass, the decay s much faster. Fgure 3 compares the performance of the proposed technque (nonlnear estmaton n the dffuson wavelet bass) wth network krgng, the current state-of-the-art estmaton technque for network end-to-end metrcs [4]; network krgng uses an egenbass formed usng the routng matrx and Relatve error Mean end-to-end delay error Nonlnear est. n dffuson wavelet bass, τ = 8 Nonlnear est. n dffuson wavelet bass, τ = 1 Lnear est. l 2 error Number of samples per tmestep Fg. 3. Top panel: Relatve mean end-to-end delay error (top) and relatve l 2-norm error (bottom) as functons of the average number of measurements per tmestep. lnear (MMSE) estmaton. We estmate the average end-toend delay over the entre network, y = 1 np n pτ =1 y, and the ndvdual delays of all paths. Estmaton s performed for τ = 1 and τ = 8 for a total of 48 tmesteps and 5% confdence ntervals are depcted. We assess performance based on the relatve mean end-to-end delay error y y est / y and the relatve l 2 -norm error y y est l2 / y l2. We provde results for the sngle tmestep case (τ = 1), where only spatal correlaton between the routes s accounted for, and the mult tmestep case (τ = 8) where both spatal and temporal correlatons are accounted for. In each case, we bult a dffuson wavelet bass wth the maxmum depth allowed by the numercal approxmatons used n the constructon procedure: L = 1 for τ = 1 and L = 11 for τ = 8. In general, accountng for tme correlaton decreases the estmaton error, but the mprovement s margnal and not unversal, Snce there s substantal temporal correlaton n the data, ths suggests that there s room for mprovement n how our methodology ncorporates temporal correlaton nformaton. When less than 1 samples per tmestep are collected, non-lnear estmaton n a dffuson wavelet bass usng temporal correlaton nformaton exhbts much lower estmaton error than the lnear estmator. Our results suggest that by makng only 5 measurements per tmestep (4% of the network paths) we can hope to recover the mean network endto-end delay wth an error of less than 5%. Fg. 4 shows the recovered end-to-end delay over tme for two example paths. We see that lnear estmaton exhbts substantal bas. In contrast, the nonlnear estmator exhbts much less bas but more varablty, an artfact nduced by the utlzaton of dffuson wavelet bases.

5 End to end path delay (ms) Sample path 1 Orgnal data Nonlnear estmaton n wavelet bass Lnear estmaton Sample path Tme (tmestep) Fg. 4. Comparson between non-lnear estmaton and lnear estmaton of path delays for two example paths. 8. DISCUSSIONS AND FUTURE WORK In ths paper we focused on developng a compressble representaton for the case where we know that performance on a path s the sum of performances on lnks, but we do not necessarly need to make ths assumpton. When ths s not true, we can stll apply the same methodology. As long as we have some pror nformaton about how performance on dfferent paths s correlated, we can desgn a dffuson wavelet bass to compress the path performance sgnal. We have appled our framework to a specfc example (end-to-end delay estmaton n a network whose topology s known) and showed that the observaton of only a few sample end-to-end delays was suffcent to recover most of the nformaton regardng the unobserved end-to-end delays. In partcular, we showed that our technque outperformed network krgng, whch uses lnear estmaton. Our technque s very general and can be appled to many dfferent setups. For nstance, see [12] for more on applyng the methodology descrbed n ths paper to estmate qualty of servce metrcs n all-optcal networks. There are three key components to the network montorng framework descrbed n ths artcle: 1) a compressng transformaton to sparsfy the target sgnal, 2) a nonlnear estmaton scheme whch favors sparse solutons, and 3) a path selecton algorthm for determnng the optmal montorng strategy. Our ongong work nvolves refnng each of these components. More specfcally, we report here that dffuson wavelets provde a compressng transformaton, but the resultng estmates have oscllatory artfacts due to the nature of the dffuson wavelet bass. We are currently nvestgatng the use of alternatve sparsfyng transformatons such as graph wavelets [6] and a novel mult-scale decomposton on graphs nspred by Haar wavelets. Smlarly, we are expermentng wth alternatve nonlnear estmators and approaches to path selecton. Last, our framework was valdated usng data from the Ablene backbone network. Investgatng theoretcal performance bounds for our framework s a nterestng and challengng extenson to ths work. Acknowledgment The authors would lke to thank Dr. Davd Chua (Department of Mathematcs and Statstcs, Boston Unversty) who provded the Ablene data set used n [4]. 9. REFERENCES [1] Y. Vard, Network tomography: Estmatng sourcedestnaton traffc ntenstes from lnk data, Journal of the Amercan Statstcal Assocaton, vol. 91, no. 433, pp , March [2] M. Coates, A. Hero, R. Nowak, and B. Yu, Internet tomography, IEEE Sgnal Processng Mag., pp , May 22. [3] Y. Chen, D. Bndel, H. Song, and R. Katz, An algebrac approach to practcal and scalable overlay network montorng, n Proc. ACM SIGCOMM, Portland, Oregon, August 24. [4] D. Chua, E. Kolaczyk, and M. Crovella, Effcent montorng of end-to-end network propertes, n Proc. Infocom, vol. 3, Mam, FL, USA, Mar. 25, pp [5] R. Cofman and M. Maggon, Dffuson wavelets, Appled and Computatonal Harmonc Analyss, vol. 21, no. 1, pp , July 26. [6] M. Crovella and E. Kolaczyk, Graph wavelets for spatal traffc analyss, n Proc. IEEE Infocom, San Francsco, USA, March 23. [7] R. Snkhorn, A relatonshp between arbtrary postve matrces and double stochastc matrces, Ann. Mathematcal Statstcs, vol. 35, no. 2, pp , June [8] E. Candès, J. Romberg, and T. Tao, Robust uncertanty prncples: Exact sgnal reconstructon from hghly ncomplete frequency nformaton, IEEE Trans. Inform. Theory, vol. 52, no. 2, pp , Feb. 26. [9] D. Donoho, Compressed sensng, IEEE Trans. Inform. Theory, vol. 52, no. 4, pp , Aprl 26. [1] J. Haupt and R. Nowak, Sgnal reconstructon from nosy random projectons, IEEE Trans. Inform. Theory, vol. 52, no. 9, pp , September 26. [11] G. Golub and C. V. Loan, Matrx Computatons. Baltmore: The Johns Hopkns Unversty Press, [12] M. Coates, Y. Ponturer, and M. Rabbat, Compressed network montorng, n IMC 7, n submsson.