Modeling Distances in LargeScale Networks by Matrix Factorization

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Modeling Distances in Large-Scale Networks by
Matrix Factorization

• Yun Mao
• CIS dept, University of Pennsylvania
• October, 2004
• Joint work with Prof. Lawrence Saul

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Motivation

• Network distances are useful
• Content distribution networks
• Peer-to-peer networks
• Overlay routing, multicast
• Network games
• Measurement comes w/ cost
• time, bandwidth,
• Is there a map or something to estimate without
  measurement?

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Sorry, no global routing maps (yet)
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Our goal
Node1 location1
Node2 location2
Node3 location3
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Outline

• What has been done?
• Matrix factorization model
• Internet Distance Estimation Service
• Evaluation
• Conclusions

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What is network distance?

• Round trip latency
• Symmetric
• Relatively stable
• Triangle inequality violation
• Bandwidth, loss rate
• Not really distance, but useful
• Asymmetric
• This talk RTT (unless specified)

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State of the art

• Euclidean Embedding
• Each host has a coordinate
• Network distances are estimated as Euclidean
  distances
• Known systems
• GNP (IMW01, INFOCOM02)
• Simplex downhill
• ICS, Virtual Landmark (IMC03)
• Lipschitz PCA
• Vivaldi (HotNets03,SIGCOMM04)
• spring energy simulation, height extension
• Many others..

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Shared limitations

• Triangle inequality violations in RTT metrics
• Symmetric constraint
• Not suitable for complicated metrics
• Even if triangle inequality and symmetry
  properties hold, increasing dimensionality
  doesnt help to improve accuracy in many cases

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Simple topologies that dont have exact embeddings
(0.5,0.5)
(-0.5,0.5)
1

H1
H2
H1
H2
The estimated distance between H1 and H4 is 1.414
while the real distance is 2.0 Extra dimensions
dont help
(0,0)
1
1
H3
H4
H3
H4
1

(0.5,-0.5)
(-0.5,-0.5)
One Possible 2-D Embedding
Is there a better model?
Another tree topology example
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An algebraic perspective
Distance Matrix Dij is the distance from host i
to host j
Each host i has two vectors Xi and Yi Distance
function is the dot product.
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Comparison
(outgoing vector)
(coordinate)
(incoming vector)
(Euclidean distance)
(dot product)
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Questions

• How to factorize a matrix (D) into two (much)
  smaller matrices (X,Y)?
• Accuracy? d?
• How to build a system to assign outgoing/incoming
  vectors to Internet-scale networks?

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Algorithm 1 Singular Value Decomposition (SVD)
Simple, and can find a global minimum of
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Algorithm 2 Nonnegative Matrix Factorization
(NMF)

• Can handle distance matrix with missing elements
  with our simple extension.
• Non-negative constraint
• Iterative method converges to local minima
  quickly
• See paper for details.

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Do they work?

• Data sets
• NLANR AMP
• 100x100. Hosts were mostly in US, distance is
  minimum RTT ping time.
• P2PSim
• 1000x1000. Hosts were obtained from an
  Internet-scale Gnutella trace. Distances were
  collected by the King method IMW02
• Comparison
• SVD and NMF based on matrix factorization model
• LipschitzPCA based on Euclidean embedding model
  (used in ICS and VL)
• Error function

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Accuracy comparison
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So far

• Matrix factorization model seems promising
• Accurate, flexible
• How to build a scalable system on this model?

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IDES Internet Distance Estimation Service
(Xnew,Ynew)?
(Xnew2,Ynew2)?
The distances are not measured, but can be
predicted
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Vectors of ordinary hosts

• We hope
• So, minimize the least-squares

(for all landmark i)
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Practical concerns

• What if some landmarks fail?
• How to reduce the load on the landmarks?
• How many dimensions?
• How scalable, robust is IDES?
• Answered in our paper

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Evaluation Efficiency accuracy

• Datasets
• Landmarks were selected randomly
• Comparison
• IDES/SVD and IDES/NMF
• Euclidean embedding
• GNP
• ICS

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Efficiency
Environment Pentium 4 3.2GHz CPU, 2G memory GNP
is obtained from the original release written in
C. IDES and ICS are implemented in Matlab.
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Accuracy
P2PSim dataset (d8)
AMP dataset (d8)
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Conclusions

• A new model based on matrix factorization
• Simple to implement
• No constraints on network distances
• Two algorithms SVD, NMF
• Internet Distance Estimation Service
• Efficient, accurate, robust.

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Thank you!Questions?