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Internet Measurement Conference 2003
27-29 Of October, 2003
Miami, Florida, USA
http://www.icir.org/vern/imc-2003/
Date for student travel grant applications: Sept 5th
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An Information-Theoretic Approach to Traffic Matrix
Estimation
Yin Zhang, Matthew Roughan, Carsten Lund – AT&T ResearchDavid Donoho – Stanford
Shannon LabShannon Lab
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Want to know demands from source to destination
Problem
Have link traffic measurements
A
B
C
...
...
...,, CABA xx
TM
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Example App: reliability analysis
Under a link failure, routes changewant to find an traffic invariant
A
B
C
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Approach
Principle *“Don’t try to estimate something
if you don’t have any information about it”
Maximum Entropy Entropy is a measure of uncertainty
More information = less entropy To include measurements, maximize entropy subject to
the constraints imposed by the data Impose the fewest assumptions on the results
Instantiation: Maximize “relative entropy” Minimum Mutual Information
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Mathematical Formalism
Only measure traffic at links
1
3
2router
link 1
link 2
link 3
Traffic y1
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Mathematical Formalism
1
3
2router
route 2
route 1
route 3
Problem: Estimate traffic matrix (x’s) from the link measurements (y’s)
Traffic y1
Traffic matrix element x1
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Mathematical Formalism
1
3
2router
route 2
route 1
route 3
311 xxy
Problem: Estimate traffic matrix (x’s) from the link measurements (y’s)
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Mathematical Formalism
1
3
2router
route 2
route 1
route 3
311 xxy
Problem: Estimate traffic matrix (x’s) from the link measurements (y’s)
3
2
1
3
2
1
110
011
101
x
x
x
y
y
y
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Mathematical Formalism
1
3
2router
route 2
route 1
route 3
311 xxy
For non-trivial networkUNDERCONSTRAINED
y = Ax
Routing matrix
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Regularization
Want a solution that satisfies constraints: y = Ax Many more unknowns than measurement: O(N2) vs O(N) Underconstrained system Many solutions satisfy the equations Must somehow choose the “best” solution
Such (ill-posed linear inverse) problems occur in Medical imaging: e.g CAT scans Seismology Astronomy
Statistical intuition => Regularization Penalty function J(x) solution:
xJAxyx
22minarg
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How does this relate to other methods?
Previous methods are just particular cases of J(x)
Tomogravity (Zhang, Roughan, Greenberg and Duffield) J(x) is a weighted quadratic distance from a gravity model
A very natural alternative Start from a penalty function that satisfies the
“maximum entropy” principle Minimum Mutual Information
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Minimum Mutual Information (MMI)
Mutual Information I(S,D) Information gained about Source from Destination I(S,D) = -relative entropy with respect to independent S
and DI(S,D) = 0S and D are independentp(D|S) = p(D)gravity model
Natural application of principle * Assume independence in the absence of other information Aggregates have similar behavior to network overall
When we get additional information (e.g. y = Ax) Maximize entropy Minimize I(S,D) (subject to
constraints) J(x) = I(S,D)
equivalent
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MMI in practice
In general there aren’t enough constraints Constraints give a subspace of possible solutions
y = Ax
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MMI in practice
Independence gives us a starting point
y = Ax
independent solution
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MMI in practice
Find a solution which Satisfies the constraint Is closest to the independent solution
solution
Distance measure is the Kullback-Lieber divergence
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Is that it?
Not quite that simple Need to do some networking specific things e.g. conditional independence to model hot-potato
routing
Can be solved using standard optimization toolkits Taking advantage of sparseness of routing matrix A
Back to tomogravity Conditional independence = generalized gravity model Quadratic distance function is a first order
approximation to the Kullback-Leibler divergence Tomogravity is a first-order approximation to MMI
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Results – Single example
±20% bounds for larger flows Average error ~11% Fast (< 5 seconds) Scales:
O(100) nodes
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More results
tomogravitymethod
simpleapproximation
>80% of demands have <20% error
Large errors are in small flows
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Other experiments
Sensitivity Very insensitive to lambda Simple approximations work well
Robustness Missing data Erroneous link data Erroneous routing data
Dependence on network topology Via Rocketfuel network topologies
Additional information Netflow Local traffic matrices
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Dependence on Topology
0
5
10
15
20
25
30
0 1 2 3 4 5 6 7 8 9 10 11unknowns per measurement
rela
tive
err
ors
(%)
randomgeographicLinear (geographic)
clique
star (20 nodes)
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Additional information – Netflow
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Local traffic matrix (George Varghese)
for referenceprevious case
0%1%5%10%
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Conclusion
We have a good estimation method Robust, fast, and scales to required size Accuracy depends on ratio of unknowns to
measurements Derived from principle
Approach gives some insight into other methods Why they work – regularization Should provide better idea of the way forward
Additional insights about the network and traffic Traffic and network are connected
Implemented Used in AT&T’s NA backbone Accurate enough in practice
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Results
Methodology Use netflow based partial (~80%) traffic matrix Simulate SNMP measurements using routing sim, and
y = Ax Compare estimates, and true traffic matrix
Advantage Realistic network, routing, and traffic Comparison is direct, we know errors are due to
algorithm not errors in the data Can do controlled experiments (e.g. introduce known
errors)
Data One hour traffic matrices (don’t need fine grained data) 506 data sets, comprising the majority of June 2002 Includes all times of day, and days of week
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Robustness (input errors)
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Robustness (missing data)
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Point-to-multipoint
We don’t see whole Internet – What if an edge link fails?Point-to-point traffic matrix isn’t invariant
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Point-to-multipoint
Included in this approach Implicit in results above Explicit results worse
Ambiguity in demands in increased
More demands use exactly the same sets of routes
use in applications is better
Point-to-point Point-to-multipoint
Link failure analysis
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Independent model
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Conditional independence
Internet routing is asymmetric A provider can control exit points for traffic going
to peer networks
peer links
access links
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Conditional independence
peer links
access links
Internet routing is asymmetric A provider can control exit points for traffic going
to peer networks Have much less control of where traffic enters
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Conditional independence
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Minimum Mutual Information (MMI)
Mutual Information I(S,D)=0 Information gained about S from D
I(S,D) = relative entropy with respect to independence Can also be given by Kullback-Leibler information
divergence
Why this model In the absence of information, let’s assume no
information Minimal assumption about the traffic Large aggregates tend to behave like overall network?
ds dDpsSp
dDsSpdDsSpDSI
, )()(
),(log),(),(
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Dependence on Topology
Unknowns per Relative
Errors (%)
NetworkPoPs Links
measurement
Geographic Random
Exodus* 17 58 4.69 12.6 20.0
Sprint* 19 100 3.42 8.0 18.9
Abovenet*
11 48 2.29 3.8 11.7
Star N 2(N-1) N/2=10 24.0 24.0
Clique N N(N-1) 1 0.2 0.2
AT&T - - 3.54-3.97 10.6* These are not the actual networks, but only estimates made by Rocketfuel
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Bayesian (e.g. Tebaldi and West) J(x) = -log(x), where (x) is the prior model
MLE (e.g. Vardi, Cao et al, …) In their thinking the prior model generates extra
constraints Equally, can be modeled as a (complicated) penalty
function• Uses deviations from higher order moments predicted by
model
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Acknowledgements
Local traffic matrix measurements George Varghese
PDSCO optimization toolkit for Matlab Michael Saunders
Data collection Fred True, Joel Gottlieb
Tomogravity Albert Greenberg and Nick Duffield