1

SuperGraph Visualization

José F. Rodrigues Jr., Agma J. M. Traina, Caetano Traina Jr.

University of São PauloComputer Science Department

ICMC-USPBrazil

Christos Faloutsos

Carnegie Mellon UniversityComputer Science Department

USA

http://www.icmc.usp.br/~junio/PublishedPapers/RodriguesJr_et_al-ISM2006.pdf

2

OutlineProblem and PrincipleSuperGraphs and the Graph-TreeConnectivityPerformanceConclusions

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

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

Current Line of Research

• Draw graph according to the modular decomposition theory

6

Current Line of Research

• Draw graph according to the modular decomposition theory

Limitation

Graphs represented like this are limited:- What is the relation between a given group of nodesand 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

Extending the idea

??

??

??

??

??

?Graph nodesconnectivity

SuperNodesconnectivity

SuperEdges

8

Extending the idea

??

??

??

??

??

?Graph nodesconnectivity

SuperNodesconnectivity

SuperEdges

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

Principle

• Utilize compartmented processing and presentation

• Utilize a structured partitioned version of the graph to be analized

• Add interaction for a richer experience

Our proposal

Introduce a theory and a data structure to allow the use of the hierarchical graph partition representation without loosing the original graph information.

Do this on the context of visualization, interaction and scalability

11

OutlineProblem and PrincipleSuperGraphs and the Graph-TreeConnectivityPerformanceConclusions

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

Example

14

Example

SuperNodes

GraphTreeSuperGraph

LeafSuperNodes

15

Building a Graph-Tree

1

2

Open

3

4

Open

5

6

Open

7

8

Open

1

4

Open

5

7

Open

Open

1

2

Open

3

4

Open

5

6

Open

7

8

Open

1

4

Open

5

7

Open

Open

16

Graph-Tree – LeafSuperNode

id file parent id nodes

open nodes SuperEdges

17

Graph-Tree - SuperNode

id parent id

open nodes SuperEdges

sons

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

OutlineProblem and PrincipleSuperGraphs and the Graph-TreeConnectivityPerformanceConclusions

20

OutlineProblem and PrincipleSuperGraphs and the Graph-TreeConnectivityPerformanceConclusions

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.

1

2

Open

3

4

Open

5

6

Open

7

8

Open

1

4

Open

5

7

Open

Open

(2,3)(2,4)

(1,5)(1,7)(4,7)

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.

1

2

Open

3

4

Open

5

6

Open

7

8

Open

1

4

Open

5

7

Open

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

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

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

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

vj

(12,4) (16,7)

(4,12) (7,16)

(12,4) (16,7)

(4,12) (7,16)

x

vk vl

ekl={(4,12), (7,16),...}

26

SuperNodes Connectivity

All possible edges

Actual connecting edges

27

OutlineProblem and PrincipleSuperGraphs and the Graph-TreeConnectivityPerformanceConclusions

28

Performance

29

Performance

30

Short Demonstration

31

OutlineProblem and PrincipleSuperGraphs and the Graph-TreeConnectivityPerformanceConclusions

32

Conclusion

• A new data structure for graphs– hierarchical management of graph partitions

(SuperNodes)– 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

More at http://www.cs.cmu.edu/~junio

33

End