1. Mining Graphs

2. Frequent Subgraph Mining
Extend association rule mining to finding frequent subgraphs
Useful for computational chemistry, Web Mining, spatial data sets, etc.
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3. Example
In drug discovery, the goal is to identify common parts in molecules sharing similar chemical properties.
Use the two dimensional atom-bond structure of molecules.
The database is searched for subgraphs that appear at least in a certain number of molecules.
A famous example for a frequent molecular fragment is the so called AZT, which is a well-known HIV-1 inhibitor (see Figure on the right)

4. Graph Definitions

5. Mining Subgraphs

6. 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

7. 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.

8. 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.

9. 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.

10. Topological equivalence
Two vertices 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
vertices
v1,v2,v3,v4 are topologically equivalent

11. 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.

12. 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

13. 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

14. 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

15. 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.

16. 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.

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

18. 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

19. 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)

20. 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 =

21. 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.