Mining Frequent Subgraphs

1
Mining Frequent Subgraphs

• EECS435
• Fall 2008

2
Overview

• Introduction
• Finding recurring subgraphs from graph databases.
  
• gSpan
• FFSM

3
Labeled Graph

• We define a labeled graph G as a five element
  tuple G V, E, ?V, ?E, ? where
• V is the set of vertices of G,
• E ? V ?V is a set of undirected edges of G,
• ?V (?E) are set of vertex (edge) labels,
• ? is the labeling function V ? ?V and E ? ?E
  that maps vertices and edges to their labels.

4
Frequent Subgraph Mining
Input A set GD of labeled undirected graphs

• ? 2/3

Output All frequent subgraphs (w. r. t. ?) from
GD.
5
Finding Frequent Subgraphs

• Given a graph database GD G0,G1,,Gn, find
  all subgraphs appearing in at least ? graphs.
• Isomorphic subgraphs are considered the same
  subgraph.
• Apriori approaches
• Generation of subgraph candidates is complicated
  and expensive.
• Subgraph isomorphism is an NP-complete problem,
  so pruning is expensive.

6
gSpan

• DFS without candidate generation
• Relabels graph representation to support DFS.
• Discovers all frequent subgraphs without
  candidate generation or pruning.
• DFS Representation
• Map each graph to a DFS code (sequence).
• Lexicographically order the codes.
• Construct a search tree based on the
  lexicographic order.

7
Depth-First Search Tree
(a)
(b)
(c)
(d)
8
DFS Codes

• Given ei (i1,j1), e2 (i2,j2) e1 lt e2 if
• i1 i2 j1 lt j2
• i1 lt j1 j1 i2
• code(G,T) edge sequence of ei lt ei1

(a)
(b)
(c)
(d)
9
DFS Lexicographic Order

• ? code(G?,T?) (a0,a1,,am)
• ß code(Gß,Tß) (b0,b1,,bn)
• ? ß iff (1) or (2)
• (1)
• (2)
• Minimum DFS code
• The minimum DFS code min(G), in DFS lexicographic
  order, is the canonical label of graph G.
• Graphs A and B are isomorphic if min(A) min(B).

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DFS Codes Parents and Children

• If ? (a0,a1,,am) and ß (a0,a1,,am,b)
• ß is the child of ?.
• ? is the parent of ß.
• A valid DFS code requires that b grows from a
  vertex on the rightmost path.

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DFS Code Trees

• Organize DFS code nodes as parent-child.
• Pre-order traversal follows DFS lexicographic
  order.
• If s and s are the same graph with different DFS
  codes, s is not the minimum and can be pruned.

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gSpan

• D is the set of all graphs.
• S is the result set.

Algorithm 1 GraphSet_Projection(D,S) 1 sort
labels in D by frequency 2 remove infrequent
vertices and edges 3 relabel remaining vertices
and edges 4 S all frequent 1-edge graphs in
D 5 sort S in DFS lexicographic order 6 S
S 7 foreach edge e in S do 8 s graph
defined by e 9 s.D subgraphs in D containing
e 10 Subgraph_Mining(D,S,s) 11 D D -
e 12 if D lt minSup 13 break
Subprocedure 1 Subgraph_Mining(D,S,s) 1 if s !
min(s) 2 return 3 S S U s 4 s 1-edge
children of s in s.D 5 foreach child c of s
do 6 if support(c) minSup 7 Subgraph_Mining
(Ds,S,c)
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Runtime Synthetic
Runtime (sec)
14
Runtime Chemical
Apriori (FSG)
gSpan
15
gSpan Advantages

• Lower memory requirements.
• Faster than naïve FSG by an order of magnitude.
• No candidate generation.
• Lexicographic ordering minimizes search tree.
• False positives pruning.
• Any disadvantage?

16
FFSM Fast Frequent Subgraph Mining -- An
Overview

• How to solve graph isomorphism problem?
• A Novel Graph Canonical Form CAM
• How to tackle subgraph isomorphism problem
  (NP-complete)?
• Incrementally kept embeddings
• How to enumerate subgraphs
• An Efficient Data Structure CAM Tree
• Two Operations CAM-join, CAM-extension.

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Adjacency Matrix

• Every diagonal entry of adjacency matrix M
  corresponds to a distinct vertex in G and is
  filled with the label of this vertex.
• Every off-diagonal entry in the lower triangle
• part of M1 corresponds to a pair of vertices in
  G and is filled with the label of the edge
  between the two vertices and zero if there is no
  edge.

1for an undirected graph, the upper triangle is
always a mirror of the lower triangle
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Code

• A Code of n ? n adjacency matrix M is defined as
  sequence of lower triangular entries (including
  the diagonal entries) in the order
• M1,1 M2,1 M2,2 Mn,1 Mn,2 Mn,n-1 Mn,n

Code(M1) aybyxb0y0c00y0d gt Code(M2)
aybyxb00yd0y00c gt Code(M3) bxby0d0y0cyy00a
a
b
y
b
x
y
y
d
0
0
c
0
y
0
0
M2

• The Canonical Adjacency Matrix is the one
  produces the maximal code, using lexicographic
  order.

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MP Submatrix

• For an m ? m matrix M, an n ? n matrix N is Ms
  maximal proper submatrix (MP Submatrix), iff N is
  obtained by removing the last non-zero entry from
  M.

• We define a CAM is connected iff the
  corresponding graph is connected.
• Theorem I A CAMs MP submatrix is CAM
• Theorem II A connected CAMs MP submatrix is
  connected

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CAM Tree Subgraphs
b
d
c
a
b
b
c
y
b
x
a
b
a
a
b
y
b
x
b
y
b
y
c
y
0
d
0
y
b
x
0
b
x
0
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CAM Tree Frequent Subgraphs
? 2/3
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How to Enumerate Nodes in a CAM Tree?

• Two operations to explore CAM tree
• CAM-Join
• CAM-Extension
• Augmenting CAM tree with Suboptimal CAMs
• Objectives
• none false dismissal
• no redundancy
• Plus We want to this efficiently!

23
Suboptimal Tree
We define a Suboptimal CAM as a matrix that its
MP submatrix is a CAM.
d
b
c
a
b
b
a
d
y
b
x
b
y
24
Summary

• Theorem
• For a graph G, let CK-1 (Ck) be set of the
  suboptimal CAMs of all the size (K-1) (K)
  subgraphs of G (K 2). Every member of set CK
  can be enumerated unambiguously either by joining
  two members of set CK-1 or by extending a member
  in CK-1.

25
Experimental Study

• Predictive Toxicology Evaluation Competition
  (PTE)
• Contains 337 compounds
• Each graph contains 27 nodes and 27 edges on
  average
• NIH DTP Anti-Viral Screen Test (DTP CA/CM)
• Chemicals are classified to be Confirmed Active
  (CA), Confirmed Moderate Active (CM) and
  Confirmed Inactive (CI).
• We formed a dataset contains CA (423) and CM
  (1083).
• Each graph contains 25 nodes and 27 edges on
  average

26
Performance (PTE)
Support Threshold ()
Support Threshold ()
27
Performance (DTP CACM)
Support Threshold ()
Support Threshold ()