1

H3: Laying Out Large Directed Graphs in 3D Hyperbolic Space

Andrew ChanCPSC 533C

March 24, 2003

2

H3

Image from: http://graphics.stanford.edu/papers/h3/fig/nab0.gif

3

Ideas behind H3

Creating an optimal layout for a general graph is tough

Creating an optimal layout for a tree is easier

Often it is possible to use domain-specific knowledge to create a hierarchical structure from a graph

4

Stumbling Blocks

The deeper the tree, the more nodes; exponential growth

You can see an overview, or you can see fine details, but not both

5

Solution

A layout based on hyperbolic space, that allows for a focus + context view

H3 used to lay out hierarchies of over 20 000 nodes

6

Related Work

H3 has its roots in graph-drawing and focus+context work

7

2D Graph and Tree Drawing

Thinking very small-scale Frick, Ludwig, Mehldau created

categories for graphs; # of nodes ranged from 16 in the smallest category, to > 128 in the largest

8

2D Tree Drawing (cont’d)

MosiacG SystemZyers and Stasko

Image from:http://www.w3j.com/1/ayers.270/paper/270.html

9

3D Graph DrawingSGI fsn file-system viewer

Image from:

http://www.sgi.com/fun/images/fsn.map2.jpg

10

3D Graph Drawing (cont’d)

Other work centered around the idea of a mass-spring system– Node repel one another, but links

attract– Difficulty in converging when you try

to scale the systems Aside: Eric Brochu is doing similar

work in 2D - http://www.cs.ubc.ca/~ebrochu/mmmvis.htm

11

3D Tree Drawing

Cone Trees, Robertson, Mackinlay, CardImage from: http://www2.parc.com/istl/projects/uir/pubs/items/UIR-1991-06-Robertson-CHI91-Cone.pdf

12

Hyperbolic Focus+Context

Hyperbolic Tree Browser,

Lamping, Rao

Image from:

http://www.acm.org/sigchi/chi95/Electronic/documnts/papers/jl_figs/strip1.htm

13

Alternate Geometry

Information at: http://cs.unm.edu/~joel/NonEuclid/

Euclidean geometry– 3 angles of a triangle add up to?– Shortest distance between two points?

Spherical geometry– How we think about the world– Shortest way from Florida to Philippines?

14

Alternate Geometry (cont’d)

Hyperbolic Geometry / Space– Is important to the Theory of Relativity– The “fifth” dimension– Can be projected into 2-D as a pseudosphere

– Key: As a point moves away from the center towards the boundary circle, its distance approaches infinity

15

H3’s Layout

Image from: http://graphics.stanford.edu/papers/h3/fig/nab0.gif

16

Finding a Tree from a Graph

Most effective if you have domain-specific knowledge

Examples:– File system– Web site structure– Function call graphs

17

Tree Layout

Cone tree layout versus H3 Layout

Image from: http://graphics.stanford.edu/papers/h3/html/node12.htm#conefig

18

Sphere Packing

Need an effective way to place information

Cannot place spheres randomly Want to have a fast algorithm

19

Sphere Packing (cont’d)

Image from: http://graphics.stanford.edu/papers/h3/fig/incrhemi.gif

20

Demo

21

Strengths

Can easily see what the important structures are and the relationships between them

Can let you ignore “noise” in data Animated transitions Responsive UI

22

Weaknesses

Starting view only uses part of the sphere

Moving across the tree can disorient you; cost of clicking on the wrong place is high

Labels not present if node too far from center

23

Questions?