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Graphs, Networks, Trees
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Examples for in class• Visual Web
▫Website as graphs See Brad’s
http://www.aharef.info/static/htmlgraph/?url=http://www.ils.unc.edu/bmh/
▫Touchgraph does websites, click on demos, then enter your home URL under Google box.
• BlogWorld (interactive hyperbolic tree)• Viszter
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Graphs, Networks, Trees• The two terms "graph" and "network" are both used in several
different ways. Following Dave Winer, the term "graph" is used to refer to (amongst other things):▫a visual representation of the variation of one variable in
comparison with that of one or more other variables▫a mathematical concept of a set of nodes connected by links
called edges▫a data structure based on that mathematical concept
• The term "network" is also used in several ways, including:▫an interconnected system of things (inanimate objects or
people)▫a specialized type of graph (the mathematical concept)
• Trees are subsets of graphs/networks.
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Trees are Limited Version of Graphs• Subcase of general graph• No cycles• Typically directed edges• Special designated root vertex
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Tree Hierarchies in the World• Pervasive
▫Family histories, ancestries▫File/directory systems on computers▫Organization charts▫Animal kingdom: Phylum,…, genus,…▫Object-oriented software classes▫...
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Trees• Hierarchies often represented as trees
▫Directed, acyclic graph• Two main representation schemes
▫Node-link▫Space-filling
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Node-Link Representations
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Node-Link Diagrams• Root at top, leaves at bottom is very common
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Sample Representation
From: Johnson & Shneiderman, ‘91
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Examples• Good for
▫Search• Bad for
▫Understanding Structure
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Why Put Root at Top?• Root can be at
center with levels growing outward too
• Can any node be the root?
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Drawing a Tree
• How does one draw this?▫DFS▫Percolate requirements upward
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Potential Problems• For top-down, width of fan-out uses up
horizontal real estate very quickly▫At level n, there are 2n nodes• Tree might grow a lot along one particular
branch▫Hard to draw it well in view without
knowing how it will branch
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InfoVis Solutions• Techniques developed in Information
Visualization largely try to assist the problems identified in the last slide• Alternatively, Information Visualization
techniques attempt to▫ show more attributes of data cases in
hierarchy ▫or focus on particular applications of trees
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SpaceTree• Uses conventional 2D layout techniques with
some clever additions
Grosjean, Plaisant, BedersonInfoVis ‘02
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Characteristics• Vertical or horizontal• Subtrees are triangles
▫Size indicates depth▫Shading indicates number of nodes inside• Navigate by clicking on nodes
▫Strongly restrict zooming
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Design Features•Make labels readable•Maximize number of levels opened• Decompose tree animation• Use landmarks• Use overview and dynamic filtering
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3D Approaches• Add a third dimension into which layout can go• Compromise of top-down and centered
techniques mentioned earlier• Children of a node are laid out in a cylinder
“below” the parent▫Siblings live in one of the 2D planes
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Cone Trees• Developed at Xerox PARC• 3D views of hierarchies such as file systems
Robertson, Mackinlay, CardCHI ‘91
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Alternate Views
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Cone TreesPositive Negative• More space
available to lay out tree• Aesthetically
pleasing (?)
• As in all 3D, occlusion obscures some nodes• Is it really more
efficient? For what tasks/users/contexts?
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Hyperbolic Browser• Focus + Context Technique
▫Detailed view blended with a global view• First lay out the hierarchy on the hyperbolic
plane• Then map this plane to a disk• Start with the tree’s root at the center• Use animation to navigate along this
representation of the planeLamping and Rao,JVLC ‘96
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2D Hyperbolic Browser• Approach: Lay out the
hierarchy on the hyperbolic plane and map this plane onto a display region.• Comparison
▫A standard 2D browser 100 nodes (w/3 character text
strings)▫Hyperbolic browser
1000 nodes, about 50 nearest the focus can show from 3 to dozens of characters
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Clicking on the bluenode brings it intofocus at the center
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Key Attributes• Natural magnification (fisheye) in center• Layout depends only on 2-3 generations from
current node• Smooth animation for change in focus• Don’t draw objects when far enough from
root (simplify rendering)
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Problems• Orientation
▫Watching the view can be disorienting▫When a node is moved, its children don’t
keep their relative orientation to it as in Euclidean plane, they rotate
▫Not as symmetric and regular as Euclidean techniques, two important attributes in aesthetics
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How about 3D?• Can same hyperbolic transformation be
applied, but now use 3D space?
Munzner,IEEE CG&A ‘98
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Performance• Handle much larger graphs, i.e. >100,000
edges• Support dynamic exploration & interactive
browsing•Maintain a guaranteed frame rate
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Old School• After all the interest in 3D and hyperbolic
techniques in the ’90’s, recently, there has been renewed interest in the old 2D methods (just done better)▫SpaceTree presented earlier▫Next 3 papers…
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Degree-of-Interest Trees• Problem
▫Trees quickly degrade into line• Approach
▫Use fisheye-like focus & context ideas to control how a tree is drawn
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Approach• Combine multiple ideas
▫Expanded DOI computation▫Logical filtering to elide nodes▫Geometric scaling▫Semantic scaling▫Clustered representation of
large unexpended branches▫Animated transition
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Example Operations
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Compression• For nodes: compress to fit (compress in X or in Y)
• Within-node compression▫Data deletion▫Word abbreviation▫Node rotation
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Better View of Org Chart
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FlexTree• Horizontally-drawn tree with compression
along vertical dimension• One focus is on showing decision trees well• Contextual multi-foci view• Basic idea: Push all nodes down as far as you
can
Song, Curran & SterrittInformation Visualization ‘04
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Example
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UNCBar Chart and Partial Views
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Node-link Shortcomings• Difficult to encode more variables of data
cases (nodes)▫Shape▫Color▫Size▫…but all quickly clash with basic node-link
structure
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Space-Filling Representations• Each item occupies an area• Children are “contained” under parent
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Treemap• Space-filling representation developed by
Shneiderman and Johnson, Vis ‘91• Children are drawn inside their parent• Alternate horizontal and vertical slicing at
each successive level• Use area to encode other variable of data
items
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Example
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Example
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Example
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Treemap?
http://blog.wired.com/wiredscience/2008/06/awesome-infogra.html
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Treemap Affordances• Good representation of two attributes beyond
node-link: color and area• Not as good at representing structure
▫What happens if it’s a perfectly balanced tree of items all the same size?
▫Also can get long-thin aspect ratios▫Borders help on smaller trees, but take up
too much area on large, deep ones
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Aspect ratios
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UNCEarly Treemap Applied to File System
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UNCA Good Use of TreeMaps and Interactivity
www.smartmoney.com/marketmap
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UNCTreemaps in Peets site
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Variation: “Cluster” Treemap• SmartMoney.com Map of the Market
▫Illustrates stock movements▫“Compromises” treemap algorithm to avoid
bad aspect ratios▫Basic algorithm (divide and conquer) with
some hand tweaking▫Takes advantage of shallow hierarchy
WattenbergCHI ‘99 http://www.smartmoney.com/marketmap
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SmartMoney Review• Tufte-esque micro/macro view• Dynamic user interface operations add to
impact• One of better applications of InfoVis
techniques we’ve seen
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Summary of Variations
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The World of TreemapsMaryland HCILwebsite devotedto Treemaps
Workshop in2001 there ontopic
www.cs.umd.edu/hcil/treemap-history/
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Another Technique•What if we used a radial rather than a
rectangular space-filling technique?▫We saw node-link trees with root in center
and growing outward already...•Make pie-tree with root in center and children
growing outward▫Radial angle now corresponds to variables
rather than area
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UNCSunburst:
Demonstration of System
http://www.cc.gatech.edu/gvu/ii/sunburst/
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SunBurst• Root directory at center, each successive level
drawn farther out from center• Sweep angle of item corresponds to size• Color maps to file type or age• Interactive controls for moving deeper in
hierarchy, changing the root, etc.• Double-click on directory makes it new root
http://www.cc.gatech.edu/gvu/ii/sunburst/
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SunBurst Negative• In large hierarchies, files at the periphery are
usually tiny and very difficult to distinguish
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InterRing• Follow-on to Sunburst that provides fixes and
new operations….
Yang, Ward & RudensteinerInfoVis ‘02
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UNCSummary: NodeLink vs SpaceFilling• Node-link diagrams or space-filling
techniques?• It depends on the properties of the data
▫Node-link typically better at exposing structure of information structure
▫Space-filling good for focusing on one or two additional variables of cases
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Network Graphs• http://apps.asterisq.com/mentionmap/ • Explores your twitter network
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UNCNetwork Graphs show Connections• Connections throughout our lives and the
world▫Circle of friends▫Delta’s flight schedules▫ …•Model connected set as a Graph
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What is a Graph?• Vertices (nodes) connected by• Edges (links)
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Graph Terminology• Graphs can have cycles• Graph edges can be directed or undirected• The degree of a vertex is the number of edges
connected to it▫In-degree and out-degree for directed
graphs• Graph edges can have values (weights) on
them (nominal, ordinal or quantitative)
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Graph Uses• In information visualization, any number of data
sets can be modeled as a graph▫US telephone system▫World Wide Web▫Distribution network for on-line retailer▫Call graph of a large software system▫Semantic map in an AI algorithm▫Set of connected friends• Graph/network visualization is one of the oldest
and most studied areas of InfoVis
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Graph Visualization Challenges• Graph layout and positioning
▫Make a concrete rendering of abstract graph• Navigation/Interaction
▫How to support user changing focus and moving around the graph
• Scale▫Above two issues not too bad for small
graphs, but large ones are much tougher
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Vertex Issues• Shape• Color• Size• Location• Label
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Edge Issues• Color• Size• Label• Form
▫Polyline▫ straight line▫ orthogonal, grid▫curved▫Planar▫upward/downward….
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UNCAesthetic Considerations• Crossings
▫minimize towards planar• Total Edge Length
▫minimize towards proper scale• Area
▫minimize towards efficiency• Maximum Edge Length
▫ minimize longest edge• Uniform Edge Lengths
▫minimize variances• Total Bends
▫minimize orthogonal towards straight-line
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Which Matters?• Various studies examined which of the aesthetic
factors matter most and/or what kinds of layout/vis techniques look best▫Purchase, Graph Drawing ’97▫Ware et al, Info Vis 1(2)▫Ghoniem et al, Info Vis 4(2)▫van Ham & Rogowitz, TVCG ‘08▫…• Results mixed: Edge crossings do seem important
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Shneiderman’s NetViz Nirvana• Every node is visible• For every node you can count its degree• For every link you can follow it from source to
destination• Clusters and outliers are identifiable
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Layout Heuristics• Layout algorithms can be
▫planar▫grid-based▫orthogonal▫curved lines▫hierarchies▫circular▫...
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Common Layout Techniques• Force-directed• Circular• Geographic-based• Clustered• Attribute-based•Matrix
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Scale Challenge•May run out of space for vertices and edges
(turns into “ball of string”)• Can really slow down algorithm• Often use clustering to help
▫Extract highly connected sets of vertices▫Collapse some vertices together
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Navigation/Interaction Issues• How do we allow a user to query, visit, or
move around a graph?• Changing focus may entail a different
rendering
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Graph Drawing Resources• Book
▫Di Battista, Eades, Tamassia, and Tollis,
Graph Drawing: Algorithms for the Visualization of Graphs, Prentice Hall, 1999
• Tutorial (talk slides)▫http://www.cs.brown.edu/people/rt/papers
/gd-tutorial/gd-constraints.pdf• Web links
▫http://graphdrawing.org▫http://www.graphviz.org
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Graph Drawing Uses•Many domains and data sets can benefit
significantly from nice graph drawings• Let’s look at some examples…
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Social Analysis• Facilitate understanding of complex
socioeconomic patterns• Social Science visualization gallery (Lothar
Krempel): http://www.mpifg.de/~lk/netvis/substanz.html• Next slides: Krempel & Plumper’s study of
World Trade between OECD countries, 1981 and 1992
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1981
http://www.mpi-fg-koeln.mpg.de/~lk/netvis/trade/WorldTrade.html
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Social Network Visualization• Social Network Analysis• Is obesity contagious?
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3 Subway Diagrams• Geographic landmarks largely suppressed on
maps, except water (rivers in Paris, London) and asphalt (highways in Atlanta)▫Rather fitting, no?• These are more graphs than maps!
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Airline flights
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Big Graphs• 20,000 - 1,000,000 Nodes•Works well with 50,000• Projects
▫Software Engineering▫Web site analysis▫Large database correlation▫Telephone fraud detection▫All Email traffic
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Challenge• Senators (Touchgraph example)
▫Party afflilation▫Religious association• Political Action Companies (PACs)• Networks
▫Senators co-sponsoring bills (# per year)▫PACs support of senators ($$ per year)• How would you visualize?
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Interaction• One of the key ways we move beyond graph
layout to graph visualization• (InfoVis) is interaction with the graph
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MoireGraph• Uses radial layout not terribly unlike
hyperbolic tree, but no hyperbolic geometry• Impose levels on graph by doing min span tree
from some node• Put root at center, nodes at subsequent levels
further out radially, with descreasing space for each• Interaction is key
Jankun-Kelly & MaInfoVis ‘03
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Navigation and interaction…
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Focus of Graph• Particular node may be focus, often placed in
center for circular layout• How does one build an interactive system that
allows changes in focus?▫Use animation▫But intuition about changes not always right
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Recent Trends in GraphViz• Attributes of nodes influence geometric
positioning▫Not just some arbitrary layout• Utilize graph statistical analysis too• Largely driven by interest in social network
analysis
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Vizster• Visualize social networking sites like friendster,
myspace, facebook• Implementation
▫crawled 1.5 million members (Winter 2003)▫written in Java using the prefuse tookit (
http://prefuse.sourceforge.net)• Oppose Shneiderman’s mantra. Instead:
▫“Start with what you know, then grow.”Heer & boydInfoVis ‘05
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Visualization
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SocialAction• Combines graph structural analysis (ranking)
with interactive visual exploration•Multiple coordinated views
▫Lists by ranking for analysis data▫Basic force-directed layout for graph vis
Perer & ShneidermanTVCG ‘06
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Social Network Attributes• Bary center – total shortest path of a node to all other
nodes• Betweenness centrality – how often a node appears on
the shortest path between all other nodes• Closeness centrality – how close a node is compared to
all other nodes• Cut-points – the subgraph becomes disconnected if the
node is removed• Degree – number of connections for node• HITs – “hubs and authorities” measure• Power centrality – how linked a node is to rest of network