Mining Graphs

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Mining Graphs
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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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Teaching
Databases
Data Mining
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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)

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Graph Definitions
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Mining Subgraphs
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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

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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.

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Vertex Growing
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Edge Growing
Edge growing inserts a new edge to an existing
frequent subgraph during candidate
generation. Doesnt necessarily increase the
number of vertices in the original graphs.
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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
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Multiplicity of Candidates
Case 1a v ? v , v1?v2 (Topologically in the
(k-2)-graphs)
Core The (k-2)-edge subgraph that is common
between the joint graphs
We try to map the cores.
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Multiplicity of Candidates
Case 1b v ? v , v1v2 (Topologically in the
(k-2)-graphs)
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Multiplicity of Candidates
Case 2a v ? v , v1?v2 (Topologically in the
(k-2)-graphs)
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Multiplicity of Candidates
Case 2b v ? v , v1v2 (Topologically in the
(k-2)-graphs)
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Multiplicity of Candidates
Case 2c v ? v (Topologically in the
(k-2)-graphs)
We try to map the cores, and there two ways to do
this.
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Multiplicity of Candidates
Case 2d v ? v (Topologically in the
(k-2)-graphs)
We try to map the cores, and there two ways to do
this.
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Multiplicity of Candidates

• More than two topologically equivalent vertexes

Core The (k-2) subgraph that is common
between the joint graphs
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Adjacency Matrix Representation

• The same graph can be represented in many ways

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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)

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Codes
Code 1 10 011 1000 01001 001010 0001011
Code 1011010010100000100110001110
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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.