1. Kai Xu Middlesex University, UK
Graph Visualisation
Kai Xu
Middlesex University, UK

2. Outline Introduction Trees and Hierarchies General Graphs
Beyond this talk

3. Outline Introduction Trees and Hierarchies General Graphs
Beyond this talk

4. Graphs

5. Graph Visualisation Many related fields. What's touched:
Vision, Perception, and Psychology
Applied
Many related fields.
Not possible to include everything.
Try to cover the basics.
Assume you know nothing about graph visualisation.
What's touched:
Visual Representation (trees).
Layout algorithms (general graphs).
Evaluation
Applications and libraries.
What's not covered: a lot
Human Computer Interaction
Sociology
Information Visualisation
Theoretical
Graph Drawing
Algorithms

6. Visualisation and Cognition
Information visualisation: all about external cognition.
Working memory capacity: 7 +/- 2.
Can’t really “think” without cognitive tools: writing, math symbols, diagrams, interactive visualisation, etc.
Brain and cognitive tools have to work together.
Computer is good at addition, very good at it, but that's about it.
Human brain is a very good pattern recogniser.
Visualisation matters only if user cares.
All visualisation methods should be evaluated with users.

7. Outline Introduction Trees and Hierarchies General Graphs
Beyond this talk

8. Terminology What's a “tree”? What's a “hierarchy”?
A (connected) undirected graph without cycles.
What's a “hierarchy”?
A (connected) directed graph without cycles.
Very important in Information Visualisation
Very common in the real world.
Simpler (less edges) than general graphs.

9. Most Famous Tree Visualisation Is?
Indented Layout
Places all items along vertically spaced rows
Uses indentation to show parent child relationships
Example: Windows explorer
Problems:
Only showing part of the tree
Bad aspect ratio (not space efficient)

10. Layered Layout
Radial drawing

11. Dendrogram Essentially a layered drawing
Orthogonal edges
Layering are done according to the leaves:
All the leaves are on the same layer
Commonly used in bioinformatics to represent
The result of hierarchical clustering
Phylogenetic trees

12. Balloon Trees A variation of radial layout
Children are drawn in a circle centered at their parents.
Effective on showing the tree structure
At the cost of node details

13. Hyperbolic Tree Simulate the distortion effect of fisheye lens.
Enlarge the focus and shrink the rest.
Focus+context.
Can be combined with different layout.
3D hyperbolic tree:
Similar to project a graph onto a sphere.
This example uses balloon tree drawing.

14. 3D Tree Visualization - Cone Tree
Cone trees are a 3D extension of the 2D layered tree drawing method.
Parent at the tip of a cone, and its children spaced equally on the bottom circle of the cone
Either horizontal or vertical
The extension to 3D does not necessarily means more information can be displayed
Occlusion problem
Couple with interaction is essential

15. Other 3D Tree Visualizations
3D poly-plane tree visualization
Put subtrees on planes.
Arrange these planes in 3D to reduce occlusion.
In this example, layered drawing is used within each plane.
3D layered tree
Only one cone.
Layers are the parallel circles on the surface.
Example: color indicate the layer.

16. Space-Filling Methods - Treemap
Treemap uses containment to show the hierarchy.
It partitions the space recursively according to the size of subtrees
It is space-efficient compare to node-link diagram
It is effective in showing the leaf nodes;
But difficult to see the non-leave nodes

17. Variations of Treemap Cushion treemap Voronoi treemap
Use shading to help identify the levels in a treemap
Voronoi treemap
Similar idea but uses voronoi diagram as partition
The space does not have to be rectangle.

18. Beamtree A variation of treemap in 3D.
Using overlap instead of nesting to show the hierarchy
3D version: representing each node as a beam
A bigger example

19. Space-Filling Tree Layout
Try to use as much screen space as possible.
Layout a tree according to the recursive partition of the screen space.
The area allocated to a subtree is proportional to its size.
A bigger example: nodes.
Use all the screen space.
Not the most effective on showing tree structure.

20. Icicle Trees Edges implied by adjacency and spatial relationship.
A bigger example (from infovis toolkit)

21. Information Slice and Sunburst Diagrams
Also a space-filling method.
Radial version of icicle trees.
Node size is proportional to the angle swept by a node.
Sunburst
Information Slice combines with focus+context.
Details are shown outside or inside the ring.

22. Elastic Hierarchies Hybrid of node-link diagrams and Treemaps.
Example:
Treemap nested within a node-link diagram.
Not the other way around.
Node-link diagram inside a treemap
Lots of crossings

23. TreeViewer
Visualizes trees in a form that closely resembles botanical trees
The root is the tree stem
Non-leave nodes are branches
Leave nodes are “bulbs” at the end of branches
Example: Unix home directory.

24. Collapsible Cylindrical Trees
Telescope metaphor
A set of nested cylinders
All cylinders of level 1 nodes are shown in a horizontal fashion,
Like being put on a stick.
A cylinder is constructed for the children of a node, and has a smaller radius.
This cylinder is nested and hidden within the cylinder contain the parent
It can be pulled out or collapsed as necessary.

25. Tree Representation Evaluation
“User Experiments with Tree Visualization Systems”, InfoVis 2004
Compared 6 tree visualisation systems:
TreeMap
Cushion TreeMap
BeamTrees
Hyperbolic tree
TreeViewer
Windows Explorer (base-line)
Data: eBay taxonomy
5 levels and 5799 nodes.
Shallow and wide.
15 tasks:
About tree structure and node attributes
Measurements:
Accuracy
Completion time
User satisfaction
Result:
Windows Explorer scores best
Only TreeMap achieves the similar level as Explorer, others perform worse.
Possible causes:
No tight integration with other tools.
Missing functionality.

26. Tree Representation Evaluation
“Qualities of Perceived Aesthetic in Data Visualization”, CHI 2007

27. Outline Introduction Trees and Hierarchies General Graphs
Beyond this talk

28. Graph Layout

29. Graph Drawing Aesthetics
Aesthetics are the graphic properties layout algorithms try to optimise.
Crossings
Aspect ratio
Edge length (several variations)
Angular resolution
Symmetry

30. Which Aesthetic is the most important?
The relative importance among aesthetics
Including 5 aesthetics:
minimizing edge crossings,
minimizing bends,
symmetry.
minimum angle
orthogonality
Purchase, H.C (1997)
Dataset
Planar graphs with 16 nodes and 28 edges
5 aesthetics and 10 drawings
2 drawings for each aesthetics: representing a strong or weak presence.
b: bends, c: crossings, m: minimal angle, o: orthogonality, s: symmetry

31. Which Aesthetic is the most important?
Tasks
Shortest path: between two nodes;
Connections between nodes: number of nodes to disconnect two nodes;
Connections between subgraphs: number of nodes to disconnect two subgraphs.
Results
Most important: reducing the number of crossing;
Less effective: minimizing the number of bends and maximizing symmetry;
Not obvious: maximizing the minimum angle and orthogonality.

32. Directed Graph – Layered Layout
Sugiyama method
Cycle removal: if there is directed cycles, temporarily reverses the direction of some to make the graph acyclic;
Layer assignment: nodes are assigned to horizontal layers, and thus determines their y-coordinate;
Crossing reduction: within each layer the nodes are ordered to reduce the number of crossings;
Horizontal coordinate assignment: the x-coordinates of each vertex is determined.

33. Undirected Graphs – Force Directed Methods
Use a physical analogy to draw graphs
A graph as a system of objects with forces acting between them.
Vertex
Object of the system;
Interacting with each other based on “some” force(s).
Edge
A different type of object;
Not interacting with each other;
Add new force(s) to vertex object.
Assumption:
A balanced system gives a good layout
Equilibrium state:
A system configuration with minimal energy level:
Locally, the sum of the forces on each object is zero.
Globally, the system has a minimal total energy.
Many force-directed methods.
Model: a force system defined by vertices and edges
Algorithm: technique for finding an equilibrium state

34. Spring Embedder Vertex: Edge: 1. 2.
Electrically charged particles; repel each other.
Ensure vertices not too close to each other
Edge:
Spring that connects particles;
Attraction force when longer than the natural length; repulsion force otherwise.
Ensure connected vertex distance is about the natural spring length. 1. 2.

35. Springs and Electrical Forces
The force on a vertex v :
fu,v : force on v by the spring between u an v
gu,v : Electrical repulsion exerted on v by vertex u
x component of the force on v
d(p,q) : Euclidean distance between points p and q
(xv, yv): position of vertex v
l0: natural length (zero energy) of the spring between u and v.
k1 : stiffness of the spring between u and v
k2 : the strength of the electrical repulsion between u and v

36. Finding an Equilibrium State
Initially place nodes at random location.
At each iteration:
Force F(v) on each vertex is computed
Each vertex v is moved in the direction of F(v) by a small amount proportional to the magnitude of F(v)
Stops when equilibrium is achieved or some conditions are met.
Not the fastest, but allow smooth animation.
Calculating attractive forces only between neighbors: O(|E|)
Calculating repulsive forces between all pair of vertices: O(|V|2)
Bottleneck of the algorithm in general.

37. An Example

38. Faster Force Directed method - FADE
It is feasible to use
a spring method, then
a geometric clustering method
To obtain a good graph clustering.
Clustered
Graph
Graph

39. Quadtree A tree data structure:
Each internal node has exactly four children.
Often used to partition a two dimensional space.
By recursively subdividing into four quadrants.

40. Barnes-Hutt Method Computing forces between stars.
Use Quadtree to cluster the stars.
Use forces between clusters to approximate forces between individual stars.
root a TL BR c b BL a b d d c e f e f

41. Barnes-Hutt method
The contents of a subtree of can be approximated by a mass at the centroid.
root e f a d c b s s TL BR c BL a b d e f

42. Barnes-Hutt method
The force that the subtree s exerts on the star x can approximate the sum of the forces that the nodes in s exert on x.
root a TL BR c s b s BL a b d d c e f e f

43. Layout vs. Graph Readability
Comparing different layout methods.
A planar graph of 17 nodes and 29 edges .
maximum node degree is 4:
For orthogonal drawing ;
A quite strong constraint.
3 layout algorithms:
Force-directed: 3 variations;
Planar orthogonal grid drawing: 2 variation;
Planar grid drawing: 3 variations.
Purchase, H.C. (1998)

44. Layout vs. Graph Readability
Tasks:
Shortest path between two nodes;
Disconnect two nodes;
Disconnect two subgraphs.
Results:
One planar grid drawing method (SEIS) produced significantly more errors than the rest;
For the rest, the average response times were not significantly different.
So there is not much difference between layout algorithms!

45. Does 3D Help? Comparing various visualisations in 2D and 3D
2D: orthographic (parallel) projection
Static Perspective: perspective projection
Stereo: shutter glasses
Passive rotation: automatic
Hand coupled rotation: mouse-controlled;
Head coupled perspective: head-controlled;
Stereo, head coupled perspective.
“Stereo, head coupled perspective” setup

46. Test Setup
Task: whether two nodes are connected by a path of length 2.
Dataset: randomly laid out graph.
Ware, C., Franck, G. (1996)

47. Results:
A static perspective is only slightly better than a 2D diagram;
3D motion and stereo viewing both help but not particularly important;
Both are more significant than stereo cues.
Stereo viewing alone increases the understandable graph size by a factor of 1.6;
Head coupling alone increases by a factor of 2.2;
Combine the two (head-coupled stereo viewing) increases by a factor of 3;

48. Beyond Node-Link Diagram
Matrix Representation
A graph can be represented by a connectivity matrix.
Advantage:
No edge crossing.
Disadvantage:
Large empty space for sparse graph.
Ghoniem, M. et al. (2004)

49. Node-Link Diagram vs. Matrix
Tasks:
estimating the number of nodes;
estimating the number of links;
finding the most connected node;
finding a node with a given label;
finding a link between two specified nodes;
finding a common neighbor between two specified nodes;
finding a path between two nodes.
Dataset
Random graph of size: 20, 50, and 100 nodes;
For each size, different link density: 0.2, 0.4 and 0.6.

50. Results
When graphs are bigger than 20 vertices, matrix outperforms node-link diagrams on most tasks.
Only path finding is consistently in favor of node-link diagrams.
For small graphs:
Node-link diagrams are always more readable than matrices;
For larger graphs:
Matrices are 30% more accurate;
Matrices have comparable or better answer time.
For more complex tasks such as “path finding”, interaction is needed:
For example, displaying all the possible paths after selecting two nodes;
For matrix, path can be displayed by connecting cells using curves (mix matrix with node-link diagram).

51. yEd Graph Editor

52. Graphviz - Graph Visualization Software
Open source.
More a command line tool.
Several types of layout algorithms.
Hierarchical
Force directed
Radial
Circular
Fast for large graphs.
Sophisticated algorithms.

53. Outline Introduction Trees and Hierarchies General Graphs
Beyond this talk

54. Graph Visualisation is much more …
What's not covered
Interaction.
Temporal/dynamic graph visualisation.
Geological graph visualisation.
Graph visualisation in specific domains.
Social network analysis (centralities).
Graph algorithms (such as clustering).
Multiple graph visualisation.
Multivariate graph visualisation.
And many more …
Graph Visualisation

55. Possible research problems
Improve existing methods:
Another variation of TreeMap?
Another variation of force-directed methods?
Methods based on new ideas:
A new visual representation of trees and hierarchies?
A layout algorithm based on a new natural analogy?
A less well studied area:
Many of those on the last slides.
… but do we really need them?

56. Applied Approach More problems than available resources can address.
Work on “important” problems.
A discovery is “important” only because it is useful
Not how difficult to discover it.
Talk to the users (such as anyone uses graphs)
Yes, it will take time,
Yes, some requirements may seem unreasonable,
But this is the best way to find “important” problems.
Once has the problem, start working on it.
Use external cognitive tools.
Implement existing methods and apply them to real data: you will find new research problems.

57. Contribute to the Community
You are what you contribute to the community.
Not just how many papers you publish.
Code, dataset, reviewing, organisation, promoting, …
Missing a Information Visualisation/Visual Analytics Toolkit
Visualisation Toolkit (VTK) for Scientific Visualisation.
Require lots of efforts, but everyone can contribute.
Take the extra step to make your code actual reusable by others (and documentation).
Make your tool part of a larger package.
GraphVisToolkit  InfoVisToolkit  VisualAnalyticsToolkit.

58. Gephi – Graph Visualisation Platform and Library

59. Think big we are only limited by our imaginations
And, I am always happy to help!