New Techniques for Visualisation of Large and Complex Networks with

Directed Edges

Tim Dwyer1

Yehuda Koren2

1 Monash University, Victoria, Australia2 AT&T - Research

Papers

Tim Dwyer, Yehuda Koren

“DiG-CoLa: Directed Graph Layout through Constrained Energy Minimization”

IEEE Symposium on Information Visualization (2005) 65-72

Tim Dwyer, Yehuda Koren, Kim Marriott

“Stress Majorization with Orthogonal Ordering Constraints”

Graph Drawing (2005)

Directed graph drawing

Magnetic Springs – Sugiyama & Misue 1995

Augmentation of Force-directed layout for general graphs

Metaphor:– edges are “magnetised” to

align with a field force

Hierarchy Energy Carmel, Harel and Koren 2002

Edge i→j implies δij=1

Works well on nice, regular DAGs

Cycles – not so good.

Symmetric Nodes

Two nodes i and j are symmetric when there exists a permutation π such that: π(i)=j and π(j)=i and L=Lπ, b=bπ

Such i and j must have the same hierarchy energy Problematic symmetric nodes appear frequently in

cycles.

a b c d

a 2 -1

b -1 2 -1

c -1 2 -1

d -1 2

a b

cd

d c b a

d 2 -1

c -1 2 -1

b -1 2 -1

a -1 2

=

Layout by Stress Majorization

Stress function:

-w12 -w1n

-w2n

…

…

Σi≠1w1i

Σi≠2w2i…

Σi≠nwin

Constant terms

Quadratic coefficients

Linear coefficients

Layout by Stress Majorization

Iterative algorithm:

Take Z=Xt

Find Xt+1 by solving FZ(Xt+1)

t=t+1

Converges on local minimum of overall stress function

Stress function:

Stress Majorisation vs Kamada Kawai – Gansner et al. 2004

FM global minimisation leads to monotonic decrease in stress

KK can oscillate FM generally

converges faster Experiments suggest

FM handles weighted edges much better.

Our Contribution

Conjecture: – Hierarchy Energy provides a more “natural” mapping of

directed structure to levels than methods requiring cycle removal

We can overcome HE method’s problems with symmetric nodes using constrained graph drawing

We show that Stress Majorization (with it’s benefits over KK) is easily augmented with constraints

Other applications:– Directed Multi-Dimensional Scaling– Orthogonal order preserving layout

Quadratic Programming

At each iteration, in each dimension we solve:

xT A x – 2 xT AZ Z(a)

bT = 2 AZ Z(a)

xT A x – b xminx

subject to: C x ≥ d

Inducing levels from hierarchy energy

Inducing Level Constraints From Hierarchy Energy

c1

c2

c3

yi – c2 ≥ sep

c2 - yj ≥ sep

Stress Majorization with Level Constraints

Fz(x) is quadratic form Removing first row and column of matrices

(corresponding to y0) fixes y0 = 0 and forces positive-definite Laplacian

Remove y0 from any constraints, – ie. y0 – ci ≥ sep becomes ci ≥ sep

Can solve with any quadratic programming method– A standard optimisation toolkit (e.g. interior point - Mosek)– The simple form of the separation constraints means we

can design a very fast custom solver

Examples

Typical Sugiyama layout (dot) - preserves tree structure

Our method - preserves edge lengths

Dire

cted

Mul

ti-D

imen

sion

al S

calin

g

Edge Lengths

Edge Lengths

Crossing Counts

Running Time

Inducing level constraints from hierarchy energy

1. Compute optimiser of hierarchy energy: yH*

2. Create list of nodes sorted on increasing yH*

3. Scan list, create new level whenever: yi – yi-1>tol

4. Create |levels-1| dummy variables c1..|levels-1|

5. For each node j in each level i (except last) create constraint: yj – ci ≥ sep

6. For each node j in each level i (except first) create constraint: ci – yj ≥ sep