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: l l l ¡ 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…
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 ¡ ¡
High Dimensional Embedding (HDE) Find a drawing in high dimensions, reduce by PCA ¡ Comparable results and speed to ACE ¡
Classic Multidimensional Scaling (CMDS)
Classic Multidimensional Scaling (CMDS) Its downfall? l l Huge matrices Matrix multiplication is slow Our work is an extension of this approach ¡ Have vertex positions that reproduce the distance matrix ¡
Intuition Behind SSDE ¡ ¡ Distance matrices contain redundant information Johnson-Lindenstrauss lemma Represent distances approximately in (practically constant) dimensions Based on approximate matrix decompositions [DKM 06]
Pick a column C from matrix of Suppose Now. We Choose Ccan is anow C-transpose basis show for L… distances Lin ea r Ti me !
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
The Algorithm in Pseudo Code
The Sampling in More Depth ¡ Two approaches l l Random Sampling Greedy Sampling (more fun)
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
Results
CMDS (SDE) versus SSDE
Some Huge Graphs Finan 512 |V| = 74, 752 |E| = 261, 120 Total Time: . 68 Seconds Ocean |V| = 143, 473 |E| = 409, 953 Total Time: 1. 65 Seconds
And now what you’ve all been waiting for… The Cow…
The Cow SSDE ACE Cow |V| = 1, 820 |E| = 7, 940 HDE
Conclusion SSDE sacrifices a little accuracy for time (versus CMDS) ¡ May use results as a preliminary step for slower algorithms ¡
Questions? You have them, I want them! (so long as they’re easy…)
- Slides: 19