gSpan: GraphBased Substructure Pattern Mining

1
gSpan Graph-Based SubstructurePattern Mining

• Xifeng Yan and Jiawei Han
• Presented by
• Quang Lam Nguyen

2
Agenda

• Motivation Basics
• gSpan - Algorithm
• Example
• Experimental Results
• Conclusion

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Mining frequent connected sub-graphs

• Given a minimum support minSup and graph G find
  all the graphs in the database that have G as a
  subgraph
• If the number of such graphs is no less than
  minSup than G is a frequent subgraph

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Basics - MinSup
(1) (2) (3)
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gSpan - Algorithm

• Overview
• Graph, DFS Tree, DFS Code, Minimum DFS Code
• DFS Search Tree
• Pseudo Pseudo Code

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gSpan Overview (1)

• 3 concepts
• DFS Lexicographic Order of sub-graphs
• Minimum DFS Codes
• DFS (Depth First Search)
• Avoiding bad memory and time consumption problems
• Candidate Generation
• BFS
• Output Set of frequent substructure patterns
  within a graph dataset

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gSpan Overview (2)

• gSpan builds tree, returning the largest frequent
  sub-graphs.

-
0 edges
1 edge
2 edges

• New edges are only added if the new child
  represents a frequent sub-graph in the given
  graph data set.
• The algorithm ensures that childs representing
  the same sub-graph never are built twice

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Graph, DFS-Tree, DFS-Code

• One graph can have several DFS-Trees

Backward edge
Forward edge
a)
b)
c)

• Each DFS Tree is represented by a sequence of
  edges the DFS code

DFS Code
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DFS Code
0
X
0 (0,1,X,a,Y) 1 (1,2,Y,b,X) 2 (2,0,X,a,X) 3 (2,3,X
,c,Z) 4 (3,1,Z,b,Y) 5 (1,4,Y,d,Z)
a
a
1
Y
d
4
b
Z
b
2
X
c
--gt a graph can have several DFS Codes - must
choose canonical form
3
Z
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Minimum DFS Code Graph Isomorphism

• Sort DFS Code Edges (lt_e) and sort DFS Codes
  (lt_l)
• According to lt_l-order of DFS Codes for a graph
  there is a min(G) canonical form
• min(G) is unique
• Two graphs G and H are isomorphic if and only if
• min(G) min(H)

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DFS Search Tree (1)

• Each node a graph, a DFS Tree

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DFS Search Tree (2) - Pruning
Not minimal
Pruning
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DFS Search Tree (3)

• Each node represents a DFS Tree (thus a graph),
  and a nodes children are DFS trees grown one
  edge.
• The tree is built depth-first,
• And whenever expanding a node, the
  lexicographical ordering lt_L is applied to the
  nodes children.
• That is, a nodes leftmost child is always the
  smallest, and is searched first.
• This ensures that the smallest motifs always are
  visited first.

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Example (1)

• Dataset Graph a)
• Step 1 Clean graph according to minSup -gt b)

MinSup 2

• Step 2 Find all frequent single-edged
  graphs/patterns

(0,1,a,c) -gt (a_5,c_3),(a_6,c_1)
(0,1,b,c) -gt (b_2,c_3),(b_4,c_1)
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Example (2)

• Sort graphs and go depth-first
• Expand if children are frequent sub-graphs
• Else Backtrack, prune if not minimal

• Return Pattern (a,b,c) and instances

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Experimental Results

• Synthetic Data
• Chemical Data

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Synthetic Data
Number of graphs 10 000 Number of
frequent sub-graphs 200 minSup
100 N Number of labels I Size of
sub-graphs T Size of graphs
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Chemical Data
340 molecules 66 atom types and 4 bond types as
labels on average only 27 vertices with 28
edges
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Conclusion

• Lower memory requirements
• No Candidate Generation
• False Positives Pruning
• Lexicographic Ordering minimizes search tree

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Appendix

• Sources
• DFS Edge Order lt_e
• DFS Lexicographic Order lt_l

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Sources

• Other presentations on gSpan
• download.informatik.uni-freiburg.de/lectures/ML/20
  04-2005WS/Misc/Slides/16_1_Toxicology_4up.pdf
• www.informatik.uni-freiburg.de/ml/teaching/ws04/l
  m/20041109_gSpan_Guetlein.ppt
• Nice text on gSpan and similar algorithms
• http//www.diva-portal.org/ntnu/abstract.xsql?dbid
  1112
• Webpage for graph mining
• http//hms.liacs.nl/graphs.html

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DFS Codes DFS Edge Order - lt_e

• Recall from class that we can define an order
  over the edges of such a graph with respect to
  the DFS tree
• An edge is always listed (vi, vj) such that I lt j
• Given edges e (vi,vj) and e (vi,vj)
• e lt e if
• j lt j or (i gt i and j j) when both are
  forward edges
• i lt i or (j lt j and i i) when both are
  backward edges
• j lt i when e is forward and e is backward
• i lt j when e is backward and e is forward

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DFS Lexicographic Order - lt_l