Jose Rodrigues, Agma J M Traina, Christos Faloutsos, Caetano Traina Jr (2006) SuperGraph Visualization In: 8th IEEE International Symposium on Multimedia 227-234 IEEE Press. @inproceedings { DBLP:conf/ism/RodriguesTFT06, title = "SuperGraph Visualization", year = "2006", author = "Jose Rodrigues and Agma J M Traina and Christos Faloutsos and Caetano Traina Jr", booktitle = "8th IEEE International Symposium on Multimedia", pages = "227-234", publisher = "IEEE Press", doi = "10.1109/ISM.2006.143", url = "http://www.icmc.usp.br/~junio/PublishedPapers/RodriguesJr_et_al-ISM2006.pdf", urllink = "http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4061172", abstract = "Given a large social or computer network, how can we visualize it, find patterns, outliers, communities? Although several graph visualization tools exist, they cannot handle large graphs with hundred thousand nodes and possibly million edges. Such graphs bring two challenges: interactive visualization demands prohibitive processing power and, even if we could interactively update the visualization, the user would be overwhelmed by the excessive number of graphical items. To cope with this problem, we propose a formal innovation on the use of graph hierarchies that leads to GMine system. GMine promotes scalability using a hierarchy of graph partitions, promotes concomitant presentation for the graph hierarchy and for the original graph, and extends analytical possibilities with the integration of the graph partitions in an interactive environment.", keywords = "Application software , Bipartite graph , Computer networks , Computer science , Data structures , Scalability , Technological innovation , Tree graphs , Visualization , Web pages"}

1. 1
SuperGraph Visualization
http://www.icmc.usp.br/~junio/PublishedPapers/RodriguesJr_et_al-ISM2006.pdf
José F. Rodrigues Jr., Agma J. M.
Traina, Caetano Traina Jr.
University of São Paulo
Computer Science Department
ICMC-USP
Brazil
Christos Faloutsos
Carnegie Mellon University
Computer Science Department
USA

2. 2
Outline
Problem and Principle
SuperGraphs and the Graph-Tree
Connectivity
Performance
Conclusions

3. 3
Problem
• Large graphs
– Hundred-thousand nodes or more
– Million edges magnitude
– web graphs, computer communication graphs ,
recommendation systems, whotrusts-whom
networks, bipartite graphs of web-logs
• Visual exploration limits
– Prohibitive processing power requirements for
interactive visualization
– Excessive number of graphical items in screen

4. 4
Problem
• Large graphs
– Hundred-thousand nodes or more
– Million edges magnitude
– web graphs, computer communication graphs ,
recommendation systems, whotrusts-whom
networks, bipartite graphs of web-logs
• Visual exploration limits
– Prohibitive processing power requirements for
interactive visualization
– Excessive number of graphical items in screen

5. 5
Current Line of Research
• Draw graph according to the modular
decomposition theory

6. 6
Current Line of Research
• Draw graph according Limitation
to the modular
decomposition theory
Graphs represented like this are limited:
- What is the relation between a given group of nodes
and another group of nodes?
- How many edges connect these two groups?
- Which are they?
- Which are the graph nodes from other groups that connect to a
graph node of interest?
The graph hierarchy is dead and the original is graph is lost.

7. 7
Extending the idea
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?
?
SuperNodes
connectivity
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?
?
?
?
??
SuperEdges
?
Graph nodes
connectivity

8. 8
Extending the idea
?
?
?
SuperNodes
connectivity
?
?
?
?
?
??
SuperEdges
?
Graph nodes
connectivity

9. 9
Principle
• Utilize compartmented processing and
presentation
• Utilize a structured partitioned version of
the graph to be analized
• Add interaction for a richer experience

10. 10
Principle
• Utilize compartmented Our proposal
processing and
presentation
Introduce a theory and a data structure to allow the use of the
hierarchical graph partition representation without loosing the
original • Utilize graph information.
a structured partitioned version of
the graph to be analized
Do this on the context of visualization, interaction and scalability
• Add interaction for a richer experience

11. 11
Outline
Problem and Principle
SuperGraphs and the Graph-Tree
Connectivity
Performance
Conclusions

12. 12
SuperGraphs and the Graph-Tree
• Given a graph G={V,E} a SuperGrap is a recursive
partitioning of G
• A GraphTree is a SuperGraph structured as a tree
• Graph nodes are kept at the leaf nodes of the tree
• Graph edges are distributed along the tree structure

13. 13
Example

14. 14
Example
SuperGraph GraphTree
SuperNodes
LeafSuperNodes

15. 15
Building a Graph-Tree
Open
Open
Open
1
2
Open
3
4
Open
Open
Open
5
6
Open
7
8
1
4
5
7
Open
1
4
5
7

16. 16
Graph-Tree – LeafSuperNode
id file parent id nodes
open nodes SuperEdges

17. 17
Graph-Tree - SuperNode
id parent id
sons
open nodes SuperEdges

18. 18
Graph-Tree
• A tree of graph partitions or a hierarchical
partitioning of a graph
• A new data structure for graphs
• Benefits: novel graph storage + structured graph
partitions
• Provides: on demand processing/presentation +
inter partitions edges information + spatial search
tree (natural R-Tree properties)

19. 19
Outline
Problem and Principle
SuperGraphs and the Graph-Tree
Connectivity
Performance
Conclusions

20. 20
Outline
Problem and Principle
SuperGraphs and the Graph-Tree
Connectivity
Performance
Conclusions

21. 21
Graph Nodes Connectivity
• Theorem: if a graph node v is an open node for a
SuperNode V, then its set of parent (Parents(V)) embody
all the SuperEdges that hold edges connected to v.
Open
1
2
Open
3
4
Open
5
6
Open
7
8
Open
1
4
Open
5
7
Open
(2,3)
(2,4)
(1,5)
(1,7)
(4,7)

22. 22
Graph Nodes Connectivity
• Theorem: if a graph node v is an open node for a
SuperNode V, then its set of parent (Parents(V)) embody
all the SuperEdges that hold edges connected to v.
Open
1
2
Open
3
4
Open
5
6
Open
7
8
Open
1
4
Open
5
7
Open
(2,3)
(2,4)
(1,5)
(1,7)
(4,7)
SuperNode V = FindParentOf(v);
While(v in OpenNodes(V)){
V = Parent(V);
Scan SuperEdges of V;
}

23. 23
SuperNodes Connectivity
• Connectivity: the set of edges (SuperEdge)
between two SuperNodes
• Connectivity between siblings: part of the tree
• Connectivity between non-siblings: use open
nodes information
• Important for SuperNode-to-SuperNode analysis

24. 24
All possible edges
• The open nodes information specifies all the
nodes of a given SuperNode that connect to
nodes from other SuperNodes
• Theorem: given two SuperNodes vi and vj, the Cartesian
product OpenNodes(vi) x OpenNodes(vj) determines the
set of all possible edges between SuperNodes vi and vj.

25. 25
Actual connecting edges
• Theorem: the set of edges that actually connect any
two SuperNodes vi and vj in a Graph-Tree is a subset of
the unique SuperEdge ekl є FirstCommonParent(vi,vj).
vi
ekl={(4,12), (7,16),...}
(12,4)
(16,7)
vj
(4,12)
(7,16)
(12,4)
(16,7)
(4,12)
(7,16)
x
vk vl

26. 26
SuperNodes Connectivity
All possible edges
Actual connecting edges

27. 27
Outline
Problem and Principle
SuperGraphs and the Graph-Tree
Connectivity
Performance
Conclusions

28. 28
Performance

29. 29
Performance

30. 30
Short Demonstration

31. 31
Outline
Problem and Principle
SuperGraphs and the Graph-Tree
Connectivity
Performance
Conclusions

32. 32
Conclusion
• A new data structure for graphs
– hierarchical management of graph partitions
(More SuperNodes)
at http://www.cs.cmu.edu/~junio
– the original graph information is not lost
– relationship (SuperEdges) between groups of nodes
instead of nodes only
– scalability for visualization and interaction
• GMine - A new graph visualization tool

33. 33
End