1. Association Analysis (7) (Mining Graphs)

2. Frequent Subgraph Mining
Extend association rule mining to finding frequent subgraphs
Useful for Web Mining, computational chemistry, spatial data sets, etc
Homepage
Teaching
Databases
Data Mining

3. Bio/Chem-Informatics
Each year, new chemical compounds are designed.
We know that structure of a compound plays a big role in its chemical properties.
However, it is difficult to establish their exact relationship.
Frequent subgraph mining can aid by identifying the substructures commonly associated with certain properties of known compounds.

4. Web mining E.g. Mining the DBLP Web Graph A mined subgraph
Garcia-Molina
Widom
Jeff Ullman
Alfred Aho
Lenzerini
Calvanese
Vardi
Kuperferman
A mined subgraph
Two examples of matches

5. Graph Definitions

6. Mining Subgraphs

7. The Exhaustive Way…Listing all...

8. Apriori-Like Approach
Support:
number of graphs that contain a particular subgraph
Apriori principle still holds
Level-wise (Apriori-like) approach:
Vertex growing:
k is the number of vertices
Edge growing:
k is the number of edges

9. Apriori-Like Algorithm
Generate candidate
Merge pairs of frequent (k - 1)-subgraphs to obtain a candidate k-subgraphs.
Prune candidates
Discard all candidate k-subgraphs that contain infrequent (k - l)-subgraphs.
Count support
Counting the number of graphs in DB that contain each candidate.
Discard all candidate subgraphs whose support counts are less than minsup.

10. Vertex Growing r
The resulting matrix is the first matrix, appended with the last row and last column of the second matrix. The remaining entries of the new matrix are either zero or replaced by all valid edge labels connecting the pair of vertices.

11. Edge Growing
Edge growing inserts a new edge to an existing frequent subgraph during candidate generation.
Doesn’t necessarily increase the number of vertices in the original graphs.

12. Topological equivalence
Two vertexes are topologically equivalent if they have:
The same label and
The same number and label of edges incident to them.
v1,v4 are topologically equivalent
v2,v3 are topologically equivalent
No topologically equivalent
vertexes
v1,v2,v3,v4 are topologically equivalent

13. Multiplicity of Candidates
Case 1a: v  v’ , v1v2 (Topologically in the (k-2)-graphs) + a b d c q p r v v’ e v1 v2 e a b d c q p r
Core: The (k-2)-edge subgraph that is common between the joint graphs
We try to map the cores.

14. Multiplicity of Candidates
Case 1b: v  v’ , v1=v2 (Topologically in the (k-2)-graphs) e a b c q p r + a b e c q p r v v’ v1 v2 a b e c q p r

15. Multiplicity of Candidates
Case 2a: v  v’ , v1v2 (Topologically in the (k-2)-graphs) + a b d c q p r v v’ e v2 v1 e a b d c q p r

16. Multiplicity of Candidates
Case 2b: v  v’ , v1=v2 (Topologically in the (k-2)-graphs) e a b c q p r + a b e c q p r v v’ v1 v2 e a b c q p r

17. Multiplicity of Candidates
Case 2c: v  v’ (Topologically in the (k-2)-graphs) e a b d q r p + a b d q r v v’ e p e a b d q r p
We try to map the cores, and there two ways to do this.

18. Multiplicity of Candidates
Case 2d: v  v’ (Topologically in the (k-2)-graphs) e a b q r p + a b e q r v v’ p e a b q r p
We try to map the cores, and there two ways to do this.

19. Multiplicity of Candidates
More than two topologically equivalent vertexes b a c a a a a a b c b a c + a a a a a a a a a a c a
Core: The (k-2) subgraph that is common between the joint graphs a b a

20. Adjacency Matrix Representation
B(5)
B(6)
B(7)
B(8) 1
A(1)
A(2)
A(3)
A(4)
B(5)
B(6)
B(7)
B(8) 1
The same graph can be represented in many ways

21. Graph Isomorphism
A graph G1 is isomorphic to another graph G2, if G1 is topologically equivalent to G2
Test for graph isomorphism is needed:
During candidate generation, to determine whether a candidate can be generated
During candidate pruning, to check whether its (k-1)-subgraphs are frequent
During candidate counting, to check whether a candidate is contained within another graph, we should use more specialized algorithms (possibly using indexes with each frequent (k-1) sub-graph)

22. Codes
A(1)
A(2)
A(3)
A(4)
B(5)
B(6)
B(7)
B(8) 1
Code =
A(1)
A(2)
A(3)
A(4)
B(5)
B(6)
B(7)
B(8) 1
Code =

23. Graph Isomorphism Use canonical labeling to handle isomorphism
Map each graph into an ordered string representation (known as its code) such that two isomorphic graphs will be mapped to the same canonical encoding
Example:
Choose the string representation with the lowest
Lexicographical value
Then, the graph isomorphism problem can be solved by string matching.