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Graph Drawing Using Sampled Spectral Distance Embedding (SSDE)

Graph Drawing Using Sampled Spectral Distance Embedding (SSDE). Ali Civril, Malik Magdon-Ismail, Eli Bocek-Rivele. Spectral Graph Drawing…. Goals: Create “aesthetically pleasing” structure Be able to do it quickly and efficiently

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Graph Drawing Using Sampled Spectral Distance Embedding (SSDE)

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  1. Graph Drawing Using Sampled Spectral Distance Embedding(SSDE) Ali Civril, Malik Magdon-Ismail, Eli Bocek-Rivele

  2. 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…

  3. Algebraic Multigrid Computation of Eigenvectors (ACE) • Minimizes Hall’s 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

  4. High Dimensional Embedding (HDE) • Find a drawing in high dimensions, reduce by PCA • Comparable results and speed to ACE

  5. Classic Multidimensional Scaling (CMDS)

  6. 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

  7. Intuition Behind SSDE • Distance matrices contain redundant information • Johnson-Lindenstrauss lemma • Represent distances approximately in (practically constant) dimensions • Based on approximate matrix decompositions [DKM06]

  8. 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!

  9. 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

  10. The Algorithm in Pseudo Code

  11. The Sampling in More Depth • Two approaches • Random Sampling • Greedy Sampling (more fun)

  12. 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

  13. Results

  14. CMDS (SDE) versus SSDE

  15. 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

  16. And now what you’ve all been waiting for… The Cow…

  17. The Cow SSDE ACE HDE Cow |V| = 1,820 |E| = 7,940

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

  19. Questions? You have them, I want them! (so long as they’re easy…)

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