Graph Drawing Using Sampled Spectral Distance Embedding (SSDE)

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Graph Drawing Using Sampled Spectral Distance
Embedding(SSDE)

• Ali Civril, Malik Magdon-Ismail,
• Eli Bocek-Rivele

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Spectral Graph Drawing

• Goals
• Create aesthetically pleasing structure
• Be able to do it quickly and efficiently
• Considering the case of straight-line edge
  drawings of connected graphs
• Spectral Approach! Some Examples

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Algebraic Multigrid Computation of Eigenvectors
(ACE)

• Minimizes Halls Energy Function
• Extension of the barycenter method
• Exploits multi-scaling paradigm
• Runtime and aesthetic quality may depend on the
  type of graph it is given

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High Dimensional Embedding (HDE)

• Find a drawing in high dimensions, reduce by PCA
• Comparable results and speed to ACE

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Classic Multidimensional Scaling (CMDS)
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Classic Multidimensional Scaling (CMDS)

• Its downfall?
• Huge matrices
• Matrix multiplication is slow
• Our work is an extension of this approach
• Have vertex positions that reproduce the distance
  matrix

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Intuition Behind SSDE

• Distance matrices contain redundant information
• Johnson-Lindenstrauss lemma
• Represent distances approximately in
  (practically constant) dimensions
• Based on approximate matrix decompositions DKM06

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Pick a column C from matrix of distances
Suppose C is a basis for L
Now Choose C-transpose
We can now show
Linear Time!
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The Algorithm

• Sample C
• Compute pseudo-inverse of
• Find spectral decomposition of L
• Power iteration only multiplies L and a vector v
  repeatedly, hence linear time

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The Algorithm in Pseudo Code
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The Sampling in More Depth

• Two approaches
• Random Sampling
• Greedy Sampling (more fun)

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Regularization

• Must do this to prevent numerical instability
• This is since the small singular values which are
  close to zero should be ignored
• Else huge instability is possible in

Our experiments revealed that is good
enough for practical purposes where is the
largest singular value
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Results
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CMDS (SDE) versus SSDE
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Some Huge Graphs
Finan512 V 74,752 E 261,120 Total Time
.68 Seconds
Ocean V 143,473 E 409,953 Total Time
1.65 Seconds
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And now what youve all been waiting for

• The Cow

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The Cow
SSDE
HDE
ACE
Cow V 1,820 E 7,940
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Conclusion

• SSDE sacrifices a little accuracy for time
  (versus CMDS)
• May use results as a preliminary step for slower
  algorithms

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Questions?

• You have them, I want them!
• (so long as theyre easy)