New Techniques for Visualisation of
Large and Complex Networks with
Directed Edges
Tim Dwyer1
Yehuda Koren2
1 Monash University, Victoria, Australia
2 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 Forcedirected 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
a
b
a
d
c
d
c
d
d
2 -1
b -1
c
b
d
2 -1
-1
=
2 -1
-1
2
a
b
a
2 -1
c -1
b
c
2 -1
-1
2 -1
-1
2
Layout by Stress Majorization

Stress function:
Σi≠1w1i -w12 …
Σi≠2w2i
-w2n
…
Constant terms
-w1n
Linear coefficients
Σi≠nwin
Quadratic coefficients
Layout by Stress Majorization

Stress function:

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 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:
min
x
subject to:
xT A x – b2 xT AZ Z(a)
Cx ≥d
bT = 2 AZ Z(a)
Inducing levels from hierarchy energy
Inducing Level Constraints From
Hierarchy Energy
c1
yi – c2 ≥ sep
c2
c2 - yj ≥ sep
c3
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 positivedefinite 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
Directed Multi-Dimensional Scaling
Edge Lengths
Edge Lengths
Crossing Counts
Running Time
Inducing level constraints from
hierarchy energy
1.
2.
3.
4.
5.
6.
Compute optimiser of hierarchy energy: yH*
Create list of nodes sorted on increasing yH*
Scan list, create new level whenever: yi – yi-1>tol
Create |levels-1| dummy variables c1..|levels-1|
For each node j in each level i (except last) create
constraint: yj – ci ≥ sep
For each node j in each level i (except first) create
constraint: ci – yj ≥ sep