gSpan: Graph-based
substructure pattern
mining
Authors: Xifeng Yan and Jiawei Han
Presented by: Ahmed R. Nabhan
University of Vermont
1
Copyright note:



This presentation was originally provided by Prof.
Xifeng Yan upon request from student
Citation:
Xifeng Yan and Jiawei Han. gSpan: graph-based
substructure pattern mining. In IEEE
International Conference on Data Mining (ICDM),
2002
2
Outlines
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Background
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Problem Definition
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Authors Contribution
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Concepts behind gSpan
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Experimental Result
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Conclusion
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Background
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Frequent Subgraph Mining is an extension to
existing frequent pattern mining algorithms
A major challenge is to count how many instances
of a pattern are in the dataset
Counting instances might be easy for sets, but
subtle for graphs
Recall the graph isomorphism problem
4
Background
G1=(V1,E1,L1)
1
2
X
U
3
5
W
4
(a)
G2=(V2,E2,L2)
V
1
V
Y
2
Y
3
4
W
U
X
(b)
f(V1.1)
f(V1.2)
f(V1.3)
f(V1.4)
f(V1.5)
=
=
=
=
=
V2.2
V2.5
V2.3
V2.4
V2.1
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(c)
Two Isomorphic graph (a) and (b) with their mapping function (c)
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Two graphs are isomorphic if one can find a mapping of nodes of the
first graph to the second graph such that labels on nodes and edges
are preserved.
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Problem: Finding Frequent
Subgraphs
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Problem setting: similar to finding frequent itemsets for
association rule discovery
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Input: Database of graph transactions
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Undirected simple graph (no loops(?), no multiples edges)
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Each graph transaction has labeled edges/vertices.
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Transactions may not be connected
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Minimum support thresholds
Output: Frequent subgraphs that satisfy the support
threshold, where each frequent subgraph is connected.
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Finding Frequent Subgraphs
Xifeng Yan
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Authors Contribution
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Representing graphs as strings (like TreeMiner)
No candidate generation!
“It combines the growing and checking of frequent
subgraphs into one procedure, thus accelerates the mining
process.”
Really fast, still a standard baseline system that most rivals
compare their systems to.
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Concepts behind gSpan
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The idea is to produces a Depth-First Search (DFS) codes
for each edge in graphs
Edges are sorted according to lexicographic order of codes
Yan and Han proved that graph isomororphism can be
tested for two graphs annotated with DFS codes
Starting with small graph patterns containing 1-edge,
patterns are expanded systemically by the DFS search
Employ anti-monotonic property of graph frequency
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Anti-Monotonicity of graph
frequency
The frequency of a super-pattern is less than or equal to
the frequency of a sub-pattern. Copyright SIGMOD’08
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Lexicographic Ordering in Graph
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It can tell us the order of two graphs.
The design can help us build a similar hierarchy.
The design should guarantee easy-growing from one level
to the lower level and easy-rolling-up from low level to
higher level.
It may be difficult to have such design that no two nodes
in this tree are same for graph case.
It can tell us whether the graph has been discovered.
And more, the most important, if a graph has been
discovered, all its children nodes in the hierarchy must
have been discovered.
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Lexicographic Ordering in Graph
1-edge
...
2-edge
...
...
...
...
3-edge
...
...
...
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DFS code and Minimum DFS code
Depth First Tree and Forward/Backward Edge Set
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DFS code and Minimum DFS code
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We use a 5-tuple (vi, vj, l(vi), l(vj), l(vi,vj)) to
represent an edge. (it may be redudant, but much
easier to understand.)
Turn a graph into a sequence whose basic element
is 5-tuple. Form the sequence in such an order:
 to extend one new node, add the forward edge
that connect one node in the old graph with this
new node.
 Add all backward edge that connect this new
node to other nodes in the old graph
 repeat this procedure.
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DFS code
a
b
X
a
Y
b
X
c
Z
d
v0
v1
v2
a
Z
b
v3
v4
X
Y
b
X
c
Z
e0: (0,1,x,y,a)
a
d
Z
e1: (1,2,y,x,b)
e2: (2,0,x,x,a)
e3: (2,3,x,z,c)
e4: (3,1,x,y,b)
e5: (1,4,x,z,d)
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Minimum DFS code
Each Graph may have lots of DFS code (why?):
one smallest lexicographic one is its Minimum DFS Code
Edge no.
(B )
(C )
(D )
0
( 0 ,1 ,x ,y ,a )
( 0 ,1 ,y ,x ,a )
( 0 ,1 ,x ,x ,a )
1
( 1 ,2 ,y ,x ,b )
( 1 ,2 ,x ,x ,a )
( 1 ,2 ,x ,y ,b )
2
( 2 ,0 ,x ,x ,a )
( 2 ,0 ,x ,y ,b )
( 0 ,1 ,y ,x ,a )
3
( 2 ,3 ,x ,z ,c )
( 2 ,3 ,x ,z ,c )
( 2 ,3 ,y ,z ,a )
4
( 3 ,1 ,z ,y ,b )
( 3 ,0 ,z ,y ,b )
( 3 ,1 ,z ,x ,c )
5
( 1 ,4 ,x ,z ,d )
( 0 ,4 ,y ,z ,d )
( 2 ,4 ,y ,z ,d )
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Graph Parent and its Children
?
a
?
X ? ?
c
?
Y a
?
b
Z
Z
?
Given a DFS code
c0=(e0,e1,…,en)
if c1=(e0,e1,…,en,ex)
if c0<c1, then
c0 is c1’s parent,
c1 is c0’s child.
X
?
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DFS Code Tree
1-edge
...
2-edge
...
...
...
...
3-edge
...
...
...
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Theorem
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1. Given two graph G0 and G1, G0 is isomorphic
to G1 iff min_dfs_code(G0)=min_dfs_code(G1).
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2. DFS Code Tree covers all graphs although
some tree nodes may represent the same graph.
(Covering)
3. Given a node in DFS Code Tree, if its DFS
code is not its minimum DFS code, prune this
node and its all descendants won’t change
“Covering”.
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Algorithm
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Algorithm
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Experimental Result
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Experimental Result
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Conclusion
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No Candidate Generation and False Test
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Space Saving from Depth First Search DFM
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Good Performance: using “memory Pool” and one
major counting improvement, it seems the
performance will be improved 5 times more. (but
need more testing).
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Questions?
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Exam Questions
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Q1) Compare gSpan to Apriori-based algorithms
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Answer:
Unlike Apriori-based algorithms, gSpan does not generate candidate patterns and
tests for false positive pruning. This feature of gSpan is both time and space
efficient. Apriori-based algorithms must generate a candidate and then test for
isomorphism against graph dataset to calculate support. This test is costly. On
the other hand, gSpan does not test for isomorphism!
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Q2) What are the main concepts behind gSpan
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Answer:
- Using Depth-First-Search (DFS) codes to label graph edges
- Employing anti-monotonic property of sub-graph frequency
- Pattern growths and pruning
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Exam Questions (cont.)
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Q3) Please similar and different features of gSpan
and TreeMiner.
Answer:
- Both algorithms employ string representation of graphs
- TreeMiner generates candidate patterns and then find support, while
gSpan expand frequent patterns directly
- gSpan is generally more applicable (can handle both trees and graphs)
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