1. Graph Drawing by Stress Majorization Authors: Emden R. Gansner, Yehuda Koren and Stephen North Presenter: Kewei Lu

2. Outline Motivation The Stress Majorization method –Uniform Edge Length –Weighting Edge Length Results

3. Motivation How to draw a graph with best layout? The most common criteria: –Minimizing the number of edge crossing –Minimize the total area of the drawing –Maximizing the smallest angle formed by consecutive incident edges –Maximizing the display of symmetries

4. Motivation

5. Stress Majorization Stress Function Decompose

6. Stress Majorization n*n weighted laplacian L w : Constant

7. Stress Majorization The third term: By use the Cauchy-Schwartz inequality: So given any n*d matrix Z, we have So we can bound the third term as:

8. Stress Majorization

9. Combine all before: Thus, we have Differentiate F z (X) by X and find the minima of F z (X) are given by solving When solve the equation, fix X 1 to be 0. Why? Then we can remove the first row and column of L w, as well as the first row of L z Z, the resulting matrix is positive definite.

10. Stress Majorization Optimization process: Some initial layout X(t) Solve equation L w X(t+1) (a) =L X(t) X(t) (a), a=1,…d to get layout X(t+1) T Terminate F, let X(t)=X(t+1)

11. Stress Majorization Weighting Edge Lengths –In order to avoid the neighborhood of high degree nodes too dense –Set the length of each edge as Where N i ={j| ∈ E}

12. Results