# Compressed network monitoring for IP and all-optical networks

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3 wavelets, diffusion wavelets recursively split the space over which the signal is observed into smaller, orthogonal subspaces. Consider a function f R n defined on a network of n nodes, where f i corresponds to the value at node i; the function is initially defined on the space V = R n. At scale j =,...,L, for some pre-specified depth L, the diffusion wavelet construction recursively splits the space V j into a scaling subspace, V j+, and a wavelet subspace, W j+, by analyzing eigenvectors of the jth dyadic power of the diffusion operator, D j. The matrix D j is, intuitively, related to averaging or smoothing over neighborhoods of radius j hops in the original graph, and the study of eigenvectors of this matrix is analogous to Fourier spectral analysis on a regular space. The ensuing orthonormal wavelet basis, adapted to the representation of the data (function values) over the graph, is obtained by concatenating bases for V L and the wavelet subspaces, {W j } L j=. We refer the reader to [7] for the precise details of the construction. Let B j R n, j =,...,n, denote the final collection of orthonormal wavelet basis vectors. A function on the graph can be represented as a vector y R n, where y i is the value at the ith node, and the wavelet decomposition of y is given by y = n j= β jb j, where β j = y T B j is the jth wavelet coefficient. Stacking the coefficients, β j, into a vector, β, and concatenating the basis vectors, {B j } into an n n matrix, B, we can write y = Bβ. In the following sections we propose diffusion operators D designed such that the corresponding wavelet representation of a path performance vector, y, is highly compressed; i.e., most of the energy in y can be captured in a few β j. To be more precise, let us rearrange the wavelet coefficients in order of decreasing magnitude so that β () β () β (n), and define the best m-term approximation of y in B to be ŷ (m) = m j= β (j)b (j). We say that y is compressible in B when the approximation error y ŷ (m) decays rapidly as a function of m, meaning that y is efficiently represented using only a few basis vectors, B (),..., B (m). In this case, we only really need to estimate values of the few large coefficients in order to obtain a high quality estimate of end-to-end performance on many paths. Moreover, in this setting we can make use of recent breakthroughs in the area of nonlinear estimation of compressible functions to quantify the number of paths that need to be measured to obtain estimates of performance at a specified level of accuracy..3 Estimation of Compressible Signals Now, suppose we have made observations y s of the end-to-end performance for a subset of the paths we are interested in, and we wish to estimate y. We have y s = Ay, where A is the selection matrix, indicating which paths we observe directly. In the following sections we describe diffusion wavelet bases B which efficiently compress the vector of end-to-end path metrics. We can express y in terms of its wavelet coefficients as y = Bβ, where most of the energy in y is captured in a few entries of β. Combining this expression with the expression for y s above leads to y s = ABβ. This begs the question: can we accurately recover the vector of coefficients, β, from measurements y s? In particular, we would like to take advantage of the fact that β only has a few large entries, and most are very small in magnitude or even zero. A straightforward approach to obtaining a sparse estimate of β is to solve an l optimization of the form: β = arg min β subject to y s = ABβ, β where β counts the number of non-zero entries of β. It is well known that this problem is NP-hard, requiring one to enumerate all possible subsets of non-zero coefficients. It has recently been shown that the solution to a simpler l optimization problem, β = arg min β β subject to y s = ABβ, () is equivalent to the l problem if certain conditions on A, B, and β are satisfied [,, ]. Here, β = n i= β i. Because the l optimization () is convex, it is computationally tractable, and a solution can be obtained using linear programming. 3. COMPRESSIBLE REPRESENTATIONS In order to construct a compressible representation, we develop a diffusion wavelet basis where the diffusion operator is related to the anticipated correlation between path metrics. We first define the graph of interest G = (V, E). We are measuring a performance metric function defined on the physical paths of the network. Accordingly, the vertex set V for our diffusion wavelet basis has one vertex for each path at each timestep, V = {v i } np i=, over the estimation interval k =,...,τ. 3. Single Timestep: Spatial Diffusion First consider the case where τ =, that is, n s routes are observed during one timestep k and we want to recover the metrics of the non-observed routes immediately. Let G = (V, E ) be the undirected graph over which we apply the diffusion wavelet framework. Notice that, in this work, the terms vertex and edge refer exclusively to the graph G defined in this section, while the terms node, link, and path refer to the physical nodes (e.g., routers), links and paths of the network; therefore, we can refer to edges be- 3

7 Queuing delay/coefficients magnitude Raw delay values Diffusion wavelet coefficients Reference power law decay (power:.7) (a) Single timestep graph. Raw delay values Diffusion wavelet coefficients Reference power law decay (power:.) Coefficient index Figure 3: Link delays for the complete network over (top panel) and 8 timesteps (bottom panel), sorted by magnitude, in the original basis (raw delay values) and in the diffusion wavelet basis (wavelet coefficients magnitudes). For comparative purposes, we also show the power-law decay functions k αk p (for constant α, for p =.7 in the -timestep case, and p =. in the 8-timestep case). (b) Multiple timesteps graph (The labeled axis is the time axis.) Figure : Representation of an example wavelet basis vector on G. Each vertex depicted here corresponds to a link of the network, and the thickness of the edge between vertices i and j increases relative to the weight w ij. Each vertex is scaled according to the magnitude of the wavelet basis vector at the vertex. Vertex layout is determined by application of Isomap [9]. follows: we penalize in () the coefficients associated to deeper scales of the diffusion wavelet basis by them assigning weights ω i, such that () becomes: β = arg min β β subject to y s = ABΩβ, (5) where Ω is the diagonal matrix such that Ω i,i = ω i. Recall each β i is a coefficient in the diffusion wavelet basis. Following the discussion above, the weights should increase with the depth of the scale associated to β i. Here, we chose a geometric increase in the weights: denoting by k the scale associated to a diffusion wavelet coefficient β i, then ω i = α k where α is a parameter that is fixed to in the remainder of this section. We show how our estimation techniques performs in Figure, which plots performance as the number of paths-per-timestep varies from to 3, with the blocksize set to τ = and τ = 8 timesteps. For the τ = 8 case, end-to-end delays are estimated by blocks. Our dataset includes measurements for all paths in the network, so we can verify the accuracy of our estimation procedure against ground truth. We assess performance in terms of the relative end-to-end mean delay error, y y est / y, and the relative l error, y y est / y, where y = i y i. Performance is averaged over timesteps. First consider the single timestep case, τ = in Fig.. We verify that the rank of G is equal to the number of links n l and thus the observation of n l paths ensures exact recovery of all end-to-end link delays with any estimation technique we are presenting. Our technique outperforms linear estimation (network kriging), with the performance improvement being most substantial when there are few measurements per timestep. However, to fully harness the power of the nonlinear estimator, we need to consider data (and its diffusion, via the diffusion operator) over several timesteps. Now consider the block estimation case, τ = 8, in Fig. ; when less than samples per timestep are collected, nonlinear estimation in a wavelet basis exhibits much lower estimation error than the linear estimator in terms of average end-to-end delay; l error is also lowered when nonlinear estimation in a wavelet basis is used. 7

8 Relative l error Relative mean error Linear estimation Nonlinear estimation in diffusion wavelet basis, τ= Nonlinear estimation in diffusion wavelet basis, τ= Number of samples per timestep Figure : Relative l end-to-end delay error (top) and relative mean error (bottom) as functions of the average number of measurements per timestep (for τ = and τ = 8), for our nonlinear estimation framework and the linear estimator [5]. In terms of mean end-to-end delay (Fig., bottom panel), our results suggest that by making only 3 measurements per timestep we can hope to recover the mean network end-to-end delay with an error of less than %. The error stabilizes for larger number of samples per timestep and decreases to as the number of samples per timestep approaches. In the last figure (see Fig. 5), we provide a more detailed insight into the nature of our end-to-end delay recovery techniques. We show the recovered end-to-end delay (original data, estimation via nonlinear estimation in the wavelet basis and path selection accounting for time-correlation, and linear estimation) over time for different paths. We used τ = 8 and samples per timestep in the estimation procedure. In Fig 5 (bottom panel), for example, we see that linear estimation severely underestimates the end-to-end delay for the chosen path. In general, the linear estimation exhibits substantial bias. In contrast, the nonlinear estimator exhibits much less bias but more variability. It is possible to estimate the bias if we are provided measurements of all link-level queueing delays (or can make sufficient estimates to form unbiased estimates.) However, such observations are not always available and we focus here on the case where estimating the bias is not possible. In the following section, we study an application to our technique where physical constraints on the observations prohibit the utilization of full-ranked observations to precompute the bias The presented nonlinear estimation technique is out- Queueing delay (ms) Original data Linear estimation Nonlinear estimation Time (timestep) Figure 5: Comparison between nonlinear estimation and linear estimation of path delays for two example paths. performs the standard linear estimation technique. In terms of computation time, on standard hardware, most of the time is spent computing the basis B (seconds to minutes depending on τ for the Abilene topology). This is a one-time cost since B only depends on the network topology. The nonlinear estimation part is typically an order of magnitude slower than linear estimation, however it only takes tens to hundreds of milliseconds, depending on τ, to estimate a block of end-to-end delays for all paths (τ end-to-end delays), making the technique deployable for real-time monitoring in networks with tens of nodes. 6. ALL-OPTICAL NETWORK MONITORING In this section, we apply the compressive network monitoring framework derived in this paper to the particular case of bit-error rate (BER) monitoring in alloptical networks. We address the problem of monitoring circuit-switched all-optical networks with no wavelength conversion subject to a variety of physical impairments. More specifically, we tackle the specific case where signal statistics (which are used to determine signals bit-error rates) can only be measured at certain locations. This is a key issue in all-optical networks since the equipment needed to take measurements at one location is extremely costly. The problem is then two-fold: given fixed BER monitors and hence the BER of observed lightpaths, what is the best estimate of the BERs of unobserved lightpaths? Then, how should BER monitors be placed to facilitate the estimation problem? 8

13 Fraction of estimated or observed lightpaths Estimated lightpaths (Nonlinear estimator) Estimated lightpaths (Linear estimator) Estimable lightpaths Number of monitors Figure : Fraction of estimable and estimated lightpaths. The BERs of some lightpaths are not estimable because none of their links is observed through any other lightpath; among estimable lightpaths, some are not estimated at all because the estimates returned by the estimator were physically meaningless (e.g., negative BER). paths are estimable. This proportion rises to 9% if 5 monitors are used. The nonlinear estimator estimates the BER of all of the estimable lightpaths, whereas the linear estimator consistently leaves the BER of a small proportion (5 %) of estimable lightpaths unestimated unless a very high number (35 and more) monitors are installed in the network. In optical networks, only the order of magnitude of the BER is relevant and hence we work solely with log(ber) to evaluate the performance of the estimators. We compare in Fig. the performance of the linear and the nonlinear estimators in the diffusion wavelet framework for the relative l error (y y est ) / y (top panel) and the relative mean error y y est / y (bottom panel), where y is the vector containing the log of the BER for each lightpath, at each time instant. We also give 5% confidence intervals. The performance improvement achieved by the nonlinear estimation technique is largest when few monitors are available. In particular, when 5 monitors or less are placed in the network (out of a maximum of monitors), corresponding to a maximum of 9% of estimated lightpaths, the nonlinear estimation technique exhibits a significant advantage over the linear estimator in terms of l norm. In terms of mean BER, the nonlinear estimator is able to predict the true mean BER over the network even with a very small number of monitors (less than % error in mean on log(ber) with 5 monitors), while the linear estimator requires 5 monitors to achieve the same Relative l error Relative mean error.3.. Nonlinear estimation Linear estimation Nonlinear estimation bound Number of monitors Number of monitors Figure : Relative l (top panel) and mean error (bottom panel) for the two estimators, for log(ber). Also depicted is the lower bound on the nonlinear estimation described in the text. performance. As was the case for end-to-end delays, the nonlinear estimator has a very low bias. When the number of estimators increase the gap between the nonlinear and linear estimation techniques closes and linear estimation actually performs slightly better than the nonlinear estimation. We emphasize that practical situations are really those where the number of monitors is small, which is when our nonlinear framework applies best and performs best. Moreover, the nonlinear estimation technique applies to more general situations than the linear estimation framework. Indeed, for the linear estimation framework to apply, we need to identify a linear relationship between link-level (x) and lightpath-level metrics (y). For the case of lightpath BER estimation, we were able to define an approximately linear relationship for transformed metrics. This artificial construct is unnecessary in our nonlinear estimation framework, since correlation between lightpaths is naturally modeled through the diffusion operator. Finally, we give in Fig. lower bounds on the performance of the nonlinear estimator. These lower bounds are substantially lower than what is achieved by our nonlinear estimator, which is expected given we picked coefficients directly in the diffusion wavelet basis to construct the bound. 7. CONCLUSION We have presented a framework for monitoring path metrics based on incomplete end-to-end measurements. The core of the framework is the development of a basis 3

14 in which the path metric signal is compressible, which allows us to use powerful nonlinear estimators from the theory of compressed sensing. Diffusion wavelets provide an appealing mechanism for developing the basis, because the specification of a diffusion operator allows us to create very general models for the correlations between metrics on different paths. Case studies involving the estimation of mean end-to-end delays and the monitoring of lightpath BERs in all-optical networks indicate the promise of our framework. Currently we are investigating the development of alternate bases which can better capture spatial localization of signal changes. We are also developing theoretical bounds on the number of paths that need to be measured to achieve a specified accuracy. 8. REFERENCES [] G. Agrawal. Fiber-Optic Communications Systems. John Wiley & Sons, Inc., third edition,. [] E. Candès, J. Romberg, and T. Tao. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory, 5():89 59, Feb. 6. [3] Y. Chen, D. Bindel, H. Song, and R. Katz. An algebraic approach to practical and scalable overlay network monitoring. In Proc. ACM SIGCOMM, Portland, USA, Aug.. [] I. Chlamtac, A. Ganz, and G. Karmi. Lightpath communications: a novel approach to high bandwidth optical WANs. IEEE Trans. Commun., (7):7 8, July 99. [5] D. Chua, E. Kolaczyk, and M. Crovella. Efficient monitoring of end-to-end network properties. In Proc. Infocom, Miami, USA, Mar. 5. [6] M. Coates, A. Hero, R. Nowak, and B. Yu. Internet tomography. IEEE Signal Processing Mag., May. [7] R. Coifman and M. Maggioni. Diffusion wavelets. Applied and Computational Harmonic Analysis, ():53 9, July 6. [8] M. Crovella and E. Kolaczyk. Graph wavelets for spatial traffic analysis. In Proc. IEEE Infocom, San Francisco, USA, Mar. 3. [9] T. Deng, S. Subramaniam, and J. Xu. Crosstalk-aware wavelength assignment in dynamic wavelength-routed optical networks. In Proc. Broadnets, Oct.. [] P. D. Dobbelaere, K. Falta, L. Fan, S. Gloekner, and S. Patra. Digital MEMS for optical switching. IEEE Commun. Mag., pages 88 95, Mar.. [] D. Donoho. Compressed sensing. IEEE Trans. Inform. Theory, 5():89 36, Apr. 6. [] E. Goldstein and L. Eskildsen. Scaling limitations in transparent optical networks due to low-level crosstalk. IEEE Photon. Technol. Lett., 7():93 9, Jan [3] G. Golub and C. V. Loan. Matrix Computations. The Johns Hopkins University Press, Baltimore, 996. [] J. Haupt and R. Nowak. Signal reconstruction from noisy random projections. IEEE Trans. Inform. Theory, 5(9):36 8, Sept. 6. [5] Y. Pointurier, M. Brandt-Pearce, T. Deng, and S. Subramaniam. Fair QoS-aware adaptive Routing and Wavelength Assignment in all-optical networks. In Proc. IEEE ICC, June 6. [6] B. Ramamurthy, D. Datta, H. Feng, J. Heritage, and B. Mukherjee. Impact of transmission impairments on the teletraffic performance of wavelength-routed optical networks. J. Lightwave Technol., 7():73 73, Oct [7] R. Ramaswami and K. Sivarajan. Optical Networks: A Practical Perspective. Morgan Kaufmann Publishers, second edition,. [8] R. Sinkhorn. A relationship between arbitrary positive matrices and double stochastic matrices. Ann. Mathematical Statistics, 35(): , June 96. [9] J. Tenenbaum, V. de Silva, and J. Langford. A global geometric framework for nonlinear dimensionality reduction. Science, 55(9):39 33, Dec.. [] Y. Vardi. Network tomography: Estimating source-destination traffic intensities from link data. J. American Statistical Assosciation, 9(33): , Mar [] A. Willner, M. Cardakli, O. Adamczyk, Y.-W. Song, and D. Gurkan. Key building blocks for all-optical networks. IEICE Trans. Commun., E83-B:66 77, Oct.. [] B. Xu and M. Brandt-Pearce. Comparison of FWM- and XPM-induced crosstalk using the Volterra Series Transfer Function method. J. Lightwave Technol., (): 53, Jan. 3. [3] B. Xu and M. Brandt-Pearce. Analysis of noise amplification by a CW pump signal due to fiber nonlinearity. IEEE Photon. Technol. Lett., 6():6 6, Apr..

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