D. Rincón, M. Roughan, W. Willinger – Towards a Meaningful MRA of Traffic Matrices1/36
Towards a Meaningful MRA for Traffic Matrices
D. Rincón, M. Roughan, W. Willinger
IMC 2008
D. Rincón, M. Roughan, W. Willinger – Towards a Meaningful MRA of Traffic Matrices2/36
Outline
Seeking a sparse model for TMs
Multi-Resolution Analysis on graphs with Diffusion
Wavelets
MRA of TMs: preliminary results
Open issues
D. Rincón, M. Roughan, W. Willinger – Towards a Meaningful MRA of Traffic Matrices3/36
Context: Abilene 12 nodes (2004)
Abilene topology (2004)
STTL
SNVA
DNVR
LOSA
KSCY
HSTN
IPLS
ATLA
CHINNYCM
WASH
ATLA-M5
D. Rincón, M. Roughan, W. Willinger – Towards a Meaningful MRA of Traffic Matrices4/36
Example: Abilene traffic matrix
STTL
SNVA
DNVR
LOSA
KSCY
HSTN
IPLS
ATLA
CHIN
NYCM
WASH
ATLA-M5
ST
TL
SN
VA
DN
VR
LOS
A
KS
CY
HS
TN
IPLS
AT
LA
CH
IN
NY
CM
WA
SH
AT
LA-M
5
80 Mbps
0 Mbps
20 Mbps
40 Mbps
60 Mbps
Traffic matrix from Abilene (March 2nd 2004, 12:00-12:05)
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Traffic matrices
Open problems 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
Time evolution of TMs How to reduce the dimensionality catch of the inference
problem?
Our goals Can we find a general model for TMs? Can we develop Multi-Resolution machinery for jointly
analyzing topology and traffic, in spatial and time scales?
D. Rincón, M. Roughan, W. Willinger – Towards a Meaningful MRA of Traffic Matrices6/36
Can we find a general model for TMs?
Our criterion: the TM model should be sparse Sparsity: energy concentrates in few coefficients (M << 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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How to perform MRA on TMs?
Traffic matrices are 2D functions defined on a graph
2D Discrete Wavelet Transform of TM as images• Uniform sampling in R2
• TMs are NOT images! – the intrinsic geometry is lost
Graph wavelets (Crovella & Kolaczyk, 2003)• Spatial analysis of differences between link loads -anomaly
detection
• Drawbacks of the graph wavelets approach
– Non-orthogonal transform - overcomplete representation
– Lack of fast computation algorithm
D. Rincón, M. Roughan, W. Willinger – Towards a Meaningful MRA of Traffic Matrices8/36
How to perform MRA on TMs?
Diffusion Wavelets (Coifman & Maggioni. 2004)
MRA on manifolds and graphs Diffusion operator “learns” the
underlying geometry as powers increase – random walk steps
Amount of “important” eigenvalues -vectors decreases with powers of T
Those under certain precision are related to high-frequency details, while those over are related to low-frequency approximations
Operator T
W1 V1
W2 V2
W3 V3
D. Rincón, M. Roughan, W. Willinger – Towards a Meaningful MRA of Traffic Matrices9/36
How to perform MRA on TMs?
Diffusion Wavelets (Coifman & Maggioni. 2004) Operator T
W1 V1
W2 V2
Eigenvalues (low to high frequency)
Cv2 5 CW2 3 CW1 2
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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 the right 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
VW1WW1 WV1 VV1
VW2WW2 WV2 VV2
VW3WW3 WV3 VV3
D. Rincón, M. Roughan, W. Willinger – Towards a Meaningful MRA of Traffic Matrices12/36
2D Diffusion wavelets
Extension of DW to 2D functions defined on a graph Operator T
VW1WW1 WV1 VV1
VW2WW2 WV2 VV2
D. Rincón, M. Roughan, W. Willinger – Towards a Meaningful MRA of Traffic Matrices13/36
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),
Diffusion operator: A: unweighted adjacency matrix “Symmetrised” version of the random walk – same
eigenvalues Double stochastic (!) Precision ε = 10-7
2
1
2
1
ADD
D. Rincón, M. Roughan, W. Willinger – Towards a Meaningful MRA of Traffic Matrices14/36
2D Diffusion wavelets – Abilene example
W1 V1
W2 V2
W3 V3
W4 V4
V0
W5 V5
W6 V6
W7 V7
W8 V8
V4
# eigenvalues at each subspace Wj = WVj + VWj + WWj
12
120
120
102
64
6
5 1
32
21
11
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2D Diffusion wavelets – Abilene example
Original TM - 12 eigenvaluesApprox3 - 10 eigenvaluesApprox4 - 6 eigenvaluesApprox
5 - 5 eigenvaluesApprox
6 - 3 eigenvaluesApprox
7 - 2 eigenvaluesApprox8 - 1 eigenvalue
STTL
SNVA
DNVR
LOSAKSCYHSTN
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)
Coefficient index (high to low freq)Time (5 min intervals)
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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, low- to high-frequency
0 20 40 60 80 100 120 1400
2
4
6
8x 10
10
Coefficients - low to high frequency
Sq
ua
red
co
effi
cie
nts
(e
ne
rgy)
Coefficients – high to low frequency
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Compressibility of TMs
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Stability of DW coefficients
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Rank signature
Coefficient index
Tim
e (5
min
inte
rval
s)
Coefficient rank – Abilene March 2004
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Rank signature – anomaly detection?
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Conclusions and open issues
Representation of TMs in the DW domain TMs seem to be sparse in the DW domain Consistency across time and different networks
Ongoing work Develop a sparse model for TMs How the sparse representation relates to previous models (e.g.
Gravity) ? Exploit DW’s dimensionality reduction in the inference problem Exploring weighted / routing-related diffusion operators Exploring bandwidth-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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Thank you !
Questions?
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Extra slides
D. Rincón, M. Roughan, W. Willinger – Towards a Meaningful MRA of Traffic Matrices26/36
Géant
23 nodes (2005)
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Context: topology
Spatial hierarchy
AS1
AS2
AS3
PoPs
Access Networks
Network/AS
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Multi-Resolution Analysis
Intuition: “to observe at different scales”
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Multi-Resolution Analysis
Approximations: coarse representations of the original data
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Multi-Resolution Analysis
Mathematical formalism
Set of nested scaling subspaces (low-frequency approximations)
generated by the scaling functions
The orthogonal complement of Vi inside Vi+1 are called detail (high-frequency) or wavelet subspaces Wi,
generated by wavelet functions
0121 ...... VVVVV nn
)(xi
)(xiW1 V1
W2V2
W3 V3
V0
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-0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
-0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
Multi-Resolution Analysis
Scaling functions: averaging, low-frequency functions
Wavelet functions: differencing, high-frequency functions
0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
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Multi-Resolution Analysis (2D)
Separable bases: horizontal x vertical Example: 2D scaling function
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Wavelet transform example
2D wavelet decomposition of the image for j=2 levels Vertical/horizontal high/low frequency subbands
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Our approach
Can we develop Multi-Resolution machinery for analyzing topology and traffic, in spatial and time scales?
Classical 1D or 2D wavelet transforms are not an option
We need a new graph-based wavelet transform! Graph wavelets for spatial traffic analysis (Crovella &
Kolaczyk 03) Diffusion wavelets (M. Maggioni et al, 06)
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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
j=1
j=3
j=5
traffic
Graph wavelets for spatial traffic analysis (Crovella & Kolaczyk 03) Anomaly detection in Abilene