Envisioning Information
Lecture 5 – Connections
Ken Brodlie
kwb@comp.leeds.ac.uk
ENV 2006
5.1
Outline
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Connections
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Applications – networks of information
Graphs and trees
Layout algorithms
Treemaps
Hyperbolic trees
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5.2
Networks of Information
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In multivariate visualization (lectures 3&4)
we are looking among the observations for
relations between the variates
In other applications, we know the
structure of the data – for example, the
observations might be connected in a
graph structure
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2
3
eg directory tree
The challenge is to visualize these
connections – sometimes called network
visualization
A graph is a set of nodes or vertices (V)
and a set of edges (E),
in which an edge joins a pair of vertices.
In a directed graph, the edges
have an associated direction
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5.3
Examples of Networks of Information
My Windows2000
filestore
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Automobile web site
- visualizing links
5.4
Some More Applications
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Genealogy
http://www.genealogy.com
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Business management
– PERT charts
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Phylogenetic trees showing
evolution of life forms
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Can you suggest some more?
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http://aleph0.clarku.edu/~djoyce/java/Phyltree/cover.html
5.5
Online Communities
http://jheer.org/vizster/
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5.6
Think About …
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Suppose we have a collection of
documents…
.. How could we derive links
between these documents?
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5.7
Drawing Large Graphs
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Layout of large graphs is a
challenging problem
What are the basic principles
we want to follow in laying out
graphs?
1. ?
2. ?
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5.8
Force Directed Placement
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Analogy with electrically
charged atomic particles leads
to a solution
– Too close, repel
– Too far apart, attract
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Nodes become particles with
forces between them
determining their position
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Fruchterman and Reingold
developed a famous force-directed
placement algorithm
Suppose d = distance between two
nodes, and k = ‘ideal’ distance
between nodes
Attraction force:
– fa(d) = d2 / k
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Repulsion force:
– fr(d) = - k2 / d
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Ref: Fruchterman and Reingold,
Software Practice and Experience, 1991
http://citeseer.ist.psu.edu/fruchterman91graph.html
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Ideal distance:
– k = C*sqrt(area/number_of_nodes)
C is a constant (found experimentally)
5.9
Forces versus distance
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Notice how combined effect of
attractive and repulsive forces
works
Equilibrium at k – does this
make sense?
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5.10
Force-directed Placement Algorithm
• Assign nodes to random positions
• Iterate the following
– For each node, calculate force due to repulsion by every other node
(fr(d) = - k2 / d)
– For each link, calculate force due to attraction of one node by the
other (fa(d) = d2 / k)
– For each node, sum all the attractive and repulsive forces
– Determine displacement by Hooke’s Law for springs: displacement
proportional to force
– Limit displacement of any node by:
• Boundary of region
• A maximum step (‘temperature’) which decreases as optimum
is reached
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5.11
Force-directed Placement - Cooling
Rapid initial cooling - quenching
ENV 2006 Slower subsequent cooling –
simmering
5.12
Hierarchical Information
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Important special case is where information is hierarchical
– Graph structure can be laid out as a tree
http://www.nist.gov/dads/HTML/tree.html
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5.13
Recursive Algorithm for Binary Trees
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Famous algorithm by ReingoldTilford (1981)
Driven by aesthetics including:
– Nodes at same level lie along a
straight horizontal line
– Left son to left, right son to right
– Father centred over sons
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Recursively from bottom-up:
– Place roots of left and right
subtrees on top of each other
– Push roots apart a certain
distance
– Work down levels, pushing any
nodes apart as needed
– Draw parent one level above,
midway between children
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5.14
More general trees
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Extends to more general trees
David Duke
http://www.comp.leeds.ac.uk/djd/graphs/
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5.15
Tree Maps
• In many tree-drawing applications, it is the size of the node that
we wish to visualize
– For example, directory structure on hard disc
• Tree maps is a screen filling method which uses a hierarchical
partitioning of the screen into regions depending on attribute
values
• Alternate partitioning parallel to X and Y axes
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5.16
Tree Map of Filestore
Suppose user has
three subdirectories:
A, B and C
First partition in X
according to total
size of each subdirectory
A
B
C
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5.17
Tree Map of Filestore
Then within each
subdirectory, we
can partition in Y
by the size of
individual files,
or further
subdirectories
A
B
C
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5.18
Treemap Example
Usenet news
groups
For history of
treemaps see:
www.cs.umd.edu/
hcil/treemap-history
Developed over many years by Ben Schneiderman and colleagues
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5.19
Cushion Treemaps
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Treemaps can be hard to
interpret
Can you decipher the structure
here?
Cushion treemaps is an idea
suggested by Jark van Wijk and
colleagues
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5.20
Cushion Tree Maps – The Idea
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Imagine in 1D first, with a flat map
Suppose we have a binary
subdivision
Add a ‘bump’ for each division and
accumulate these recursively
This gives the top curve on right
Use this with a lighting effect to give
the lower image
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5.21
Cushion Tree Maps – A Filesystem
Download SequoiaView from:
http://www.win.tue.nl/sequoiaview/
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5.22
Hyperbolic Trees
• This is a popular method of
displaying hierarchical
structures
Triangle in
Hyperbolic
space
• Based on ideas of hyperbolic
geometry
– As Euclidean axioms –
except given a point and a
line, there are infinite
number of lines passing
through the point which are
parallel to the line
– See : wikipedia
– http://www.geom.uiuc.edu/java/t
riangle-area/
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Boundary of disc represents
infinity
Notice how angle of triangle
on boundary tends to zero
5.23
Visualizing Structure of Web Pages
• Hyperbolic trees have been successfully used to visualize web
site structures (Lamping et al, 1995)
• Place home page in centre
– with linked pages connected by hyperbolic arcs
– further arcs link to further links
– see: www.acm.org/sigchi/chi95/proceedings/papers/jl_bdy.htm
Escher woodcut…
.. inspired:
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5.24
Hyperbolic Trees
Automobiles
web site
Home page
in centre
Click on link
you want ...
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5.25
Hyperbolic Trees
Auto History
moves to
centre of screen
Click on next
link...
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5.26
Hyperbolic Trees
Henry Ford
is now at the centre
and so on...
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5.27
Hyperbolic Trees
Also works
for family
trees...
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5.28
Star Trees
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Now commercially available
from Inxight
– Spun out of Xerox PARC
– www.inxight.com
– Now called ‘star trees’ ;)
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5.29
Further Reading
• Excellent review article:
• Graph Visualization and Navigation in Information Visualization:
a Survey (Herman, Melancon, Marshall)
• IEEE Trans Vis and Computer Graphics, Vol 6, 2000
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5.30