Ongoing work on Multiresolution Analysis of Traffic Matrices

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Ongoing work onMultiresolution Analysis of
Traffic Matrices

• David Rincón
• Matthew Roughan

EuroNFTraf 2009 workshop Paris, 7-8 december
2009
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Outline

• Introduction
• Seeking a sparse model for TMs
• Multi-Resolution Analysis on graphs with
  Diffusion Wavelets
• MRA of TMs preliminary results
• Conclusions and open issues

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Example Abilene traffic matrix
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Traffic matrices

• Basic input for network planning, dimensioning,
  traffic engineering, etc
• Direct inspection
• Netflow-like solutions
• Difficult to obtain experiment setting, router
  performance
• Indirect estimation Inference
• SNMP link traffic volumes (5 minutes)
• yAx
• Underconstrained problem need of models
• Gravity model Traffic exchanged between two
  nodes is proportional to the total traffic
  entering/exiting the nodes
• Prior regularization

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Traffic matrices

• Open problems
• Need of good TM models
• Synthesis of TMs for planning / design of
  networks
• Traffic prediction anomaly detection
• Traffic engineering algorithms
• Traffic and topology are intertwined
• Hierarchical scales in the global Internet apply
  also to traffic
• How to reduce the dimensionality catch of the
  inference problem?

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Context topology

• Spatial hierarchy

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Our goal can we find a general model for TMs?

• Criterion the TM model should be sparse
• Sparsity energy concentrates in few coefficients
  (M ltlt N2)
• Tradeoff between predictive power and model
  fidelity
• Easier to attach physical meaning
• Could help with the underconstrained inference
  problem
• Multiresolution analysis (MRA)
• Classical MRA wavelet transforms observe the
  data at different time / space resolutions
• Wavelets (approximately) decorrelate input
  signals
• Energy concentrates in few coefficients
• Threshold the transform coefficients ? sparse
  representation (denoising, compression)
• Successfully applied in time series (1D) and
  images (2D)

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Multi-Resolution Analysis

• Intuition to observe at different scales
• Approximations coarse representations of the
  original data

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Wavelet transform example

• 2D wavelet decomposition of the image for j2
  levels
• Vertical/horizontal high/low frequency subbands

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MRA on graphs?

• A TM is not an image
• Image uniform sampling or R2
• The TM is defined on a graph (manifold)
• Example swiss roll
• Available MRA techniques
• Graph wavelets (Crovella Kolaczyck, 2003)
• Sampled 2D wavelets
• Non-orthogonal, lack of fast algorithm
• Diffusion Wavelets (Maggioni Coifman, 2006)

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Diffusion Wavelets (Coifman, Maggioni 2006)

• Diffusion operator
• A diffusion operator T learns the underlying
  geometry
• Tk represents the probability of a transition in
  k time steps
• Example (Coifman, Lafon 2006)
• 3 clusters, 300 random Gaussian-dist points, with

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How to perform MRA on TMs?
Eigenspectrum of T (normalized)
Operator T (10x10 matrix)
W1
V1
W2
V2
?
Eigenvalues (low to high frequency)
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Diffusion Wavelets and our goals

• Unidimensional functions of the vertices F(v1)
  can be projected onto the multi-resolution spaces
  defined by the DW.
• Network topology can be studied by defining a
  random- walk-like diffusion operator and
  representing the coarsened versions of the graph.
• But Traffic Matrices are 2D functions of the
  origin and destination vertices, and can also be
  functions of time TM(V1,V2,t)

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2D Diffusion wavelets

• Extension of DW to 2D functions defined on a
  graph
• F(v1,v2)
• Construction of separable 2D bases by projecting
  twice into both directions
• Tensor product
• Similar to 2D DWT
• Orthonormal, invertible, energy conserving
  transform

Operator T
VW1
WW1
WV1
VV1
VW2
WW2
WV2
VV2
VW3
WW3
WV3
VV3
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2D Diffusion wavelets

• Extension of DW to 2D functions defined on a
  graph

Operator T
VW1
WW1
WV1
VV1
VW2
WW2
WV2
VV2
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MRA of Traffic Matrices

• More than 20000 TMs from operational networks
• Abilene (2004), granularity 5 mins
• GÉANT (2005), granularity 15 mins
• Acknowledgments Yin Zhang (UTexas), S. Uhlig
  (Delft),
• Adjacency operator
• A unweighted adjacency matrix
• Symmetrised version of the random walk same
  eigenvalues
• Precision e 10-7

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2D Diffusion wavelets Abilene example
V0
12
V4
6
W1
V1
W5
V5
W2
V2
W6
V6
W3
V3
W7
V7
W4
V4
W8
V8
eigenvalues at each subspace
Wj WVj VWj WWj
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2D Diffusion wavelets Abilene example
STTL
SNVA
DNVR
LOSA
KSCY
HSTN
IPLS
ATLA
CHIN
NYCM
WASH
ATLA-M5
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2D Diffusion wavelets Abilene example
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2D Diffusion wavelets Abilene example
DW coefficients Abilene 14th July 2004 (24 hours)
Time (5 min intervals)
Coefficient index (high to low freq)
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2D Diffusion wavelets Abilene example

• How concentrated is the energy of the TM?
• Wavelet coefficients for the Abilene TM
• 12 x 12 144 coefficients

Coefficients high to low frequency
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Compressibility of TMs
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Rank signature
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Rank signature anomaly detection?
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Other operators

• Gravity operator
• G normalized gravity model (rank 1) from fan-out
  and fan-in probabilities
• Needs symmetrisation (undirected graph)
• Actual operator Max-eig-normalized T
  (non-stochastic)

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Gravity operator
Gravity operator
Topology operator
Coefficients high to low frequency
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Gravity operator
1.4 coeff ? 80
4.9 coeff ? 90
11 coeff ? 95
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Conclusions and open issues

• Representation of TMs in the DW domain
• TMs in the DW domain seem to be sparse
  (compressible)
• Consistency along time
• Ongoing work
• Develop a sparse model for TMs
• Exploit DWs dimensionality reduction in the
  inference problem
• Exploring weighted / routing-related diffusion
  operators
• Introducing time correlations in the diffusion
  operator
• Diffusion wavelet packets best basis algorithms
  for compression
• DW analysis of network topologies

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Time-based operators
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Flow/traffic operators
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• Thank you !
• Questions?

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MRA of TMs Why?

• Applications of MRA in Signal Processing
• Denoising
• Keep the low-frequency components, discard the
  high-frequency details
• Compression
• Keep the best coefficients for highest perceptual
  quality
• Potential applications for TMs
• Denoising
• Compression express a TM with few
  coefficients
• Lower-dimension model of the TM, easier to
  predict/analyze
• Could this help with the inference problem?

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Flow/traffic operators
Traffic operator
Flow operator
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Géant
23 nodes (2005)
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The tools Graph wavelets

• Graph wavelets for spatial traffic analysis
  (Crovella Kolaczyk 03)
• Exploit spatial correlation of traffic data
• Sampled 2D wavelets

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The tools Graph wavelets

• Graph wavelets for spatial traffic analysis
  (Crovella Kolaczyk 03)
• Link analysis
• Definition of scale j j-hop neighbours

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The tools Graph wavelets

• Graph wavelets for spatial traffic analysis
  (Crovella Kolaczyk 03)
• Anomaly detection in Abilene

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Multi-Resolution Analysis

• Scaling functions averaging, low-frequency
  functions
• Wavelet functions differencing, high-frequency
  functions

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Multi-Resolution Analysis (2D)

• Separable bases horizontal x vertical
• Example 2D scaling function

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Diffusion wavelets

• Eigenvalues of the diffusion operator
• Every operator can be defined in terms of its
  eigenspectrum
• Eigenvalues ?i, eigenvectors vi
• 0 ?i 1
• Eigenvalues of Tk ?ik
• Amount of important eigenvalues vectors
  decreases with k
• Those under certain precision ? related to
  high-frequency detail
• Those over ? are related to low-frequency
  approximations