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 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

n

ji jiij xxwE1,

2)(21

High Dimensional Embedding (HDE)

Find a drawing in high dimensions, reduce by PCA

Comparable results and speed to ACE

Classic Multidimensional Scaling (CMDS)

....2,1,for ,|||| njiDxx ijji 222 *2 ijjiji Dxxxx

]1,...,1[1 ],||||,...,||[|| ],,...,[

:as defined be 1 and Q, X,Let 22

11

n

Tnn

Tn

T xxQxxX

LXXQQ TTn

T 211n

Tnnn n

111I :be matrix projectionLet

LXXQQ TTn

T 211n

LXX T

21))((

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

Intuition Behind SSDE

Distance matrices contain redundant information

Johnson-Lindenstrauss lemma Represent distances approximately in

(practically constant) dimensions

Based on approximate matrix decompositions [DKM06]

)/)(log( 2nO

Pick a column C from matrix of distances

C

TC

iL

Suppose C is a basis for L…Now Choose C-transposeWe can now show

ii CL

iiC T

TCCL

T1CT1CCL

Linear Time!TiC

*

iL C

1n kn

1k

TC

CL

)...( 1 n

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

),,(dPhiComputeCan),(:1 cmethodGC )(),,(:2 T SVDVU

),(Regularize:3

TUV:4

),,tion(PowerItera:5 CreturnY

The Sampling in More Depth

Two approaches 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

22 / ii

ii

in entry diagonal theis Where thi i

31 Our experiments revealed that is good enough for

practical purposes where is the largest singular value1

Results

CMDS (SDE) versus SSDE

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

And now what you’ve all been waiting for…

The Cow…

The Cow

SSDE HDE

Cow|V| = 1,820|E| = 7,940

ACE

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…)