# Mining Tree Queries in a Graph

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```					Mining Tree-Query Associations
in a Graph

Bart Goethals
University of Antwerp, Belgium
Eveline Hoekx
Jan Van den Bussche
Hasselt University, Belgium
Graph Data

A (directed) graph over a set of nodes N is a set G of
edges: ordered pairs ij with ij  N.

Snapshot of a graph representing the complete metabolic pathway of a
human.

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Graph Mining

Transactional category
– dataset: set of many small graphs (transactions)
– frequency: transactions in which the pattern occurs (at least once)
– ILP: Warmr
[AGM, FSG, TreeMiner, gSpan, FFSM]

Single graph category
– dataset: single large graph
– frequency: copies of the pattern in the large graph
[Subdue, Vanetik-Gudes-Shimony, SEuS, SiGraM, Jeh-Widom]

Focus on pattern mining, few work on association rule mining!
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Our work

• Single graph category
• Pattern + association rule mining
• Patterns with:
– Existential nodes
– Parameters
• Occurrence of the pattern in G is any
homomorphism from the pattern in G.
• So far only considered in the ILP (transactional)
setting

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Example of a pattern

frequency   x   z 5z  G  z8 G  zx 
G
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Patterns are conjunctive queries.

select distinct G3.to as x
from G G1, G G2, G G3
where G1.from=5 and G1.to=G2.from
and G1.to=G3.from and G2.to=8

frequency   x   z 5z  G  z8 G  zx 
G
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Example of an Association Rule

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Features of the presented algorithms

•   Pattern mining phase + association mining phase
•   Restriction to trees => efficient algorithms
•   Equivalence checking
•   Apply theory of conjunctive database queries
•   Database oriented implementation

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Outline rest of talk

•   Formal problem definition
•   Algorithms:
1. Pattern Mining
•   Overall approach
•   Outer loop: incremental
•   Inner loop: levelwise
•   Equivalence checking
2. Association Rule Mining
• Result management
• Experimental results
• Future work
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Formal definition of a tree pattern.

A tree pattern is a tree P whose nodes are called variables,
and:
1. some variables marked as existential 
2. some variables are parameters (labeled with a constant)
3. remaining variables are called distinguished

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Formal definition of a tree query.

A tree query Q is a pair (H,P) where:
1. P is a tree pattern, the body of Q
2. H is a tuple of distinguished variables and parameters of
P. All distinguished variables of P must appear at least
once in H, the head of Q

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Formal definition of a matching

A matching of a pattern P in a graph G is a homomorphism
h: P  G, with hza, for parameters labeled a.

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Example: Matching

z y   z x

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Example: Matching

z y   z x

14
Example: Matching

z y    z x

h 0    8   4

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Example: Matching

z y    z x

h 0    8   4
h 0    8   8

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Example: Matching

z y    z x

h 0    8   4
h 0    8   8
h 0    8   4

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Example: Matching

z y    z x

h 0    8   4
h 0    8   8
h 0    8   4
h4 0    8   5

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Example: Matching

z y    z x

h 0    8   4
h 0    8   8
h 0    8   4
h4 0    8   5
h5 0  8 8

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Formal definition of frequency

We define the answer set of Q in G as follows:
QGf(H)|f is a matching of P in G

The frequency of Q in G is #answers in the answer set.

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Example: Matching

z y    z x

h 0    8   4
   h 0    8   8
h 0    8   4

h4 0    8   5
h5 0  8 8

frequency 

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Problem statement 1: Tree query mining

Given a graph G and a threshold k, find all tree queries
that
have frequency at least k in G, those queries are called
frequent.

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Formal definition of an association
rule

An association rule (AR) is of the form Q1  Q2 with Q1 and Q2
tree queries. The AR is legal if Q2  Q1. The confidence of the
AR in a graph G is defined as the frequency of Q2 divided by
the
frequency of Q1.

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Problem statement 2: Association rule mining

• Input: a graph G, minsup, a tree query Qleft frequent in
G, minconf
• Output: all tree queries Q such that Qleft  Q is a legal
and confident association rule in G.

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Outline rest of talk

•   Formal problem definition
•   Algorithms:
1. Pattern Mining
•   Overall approach
•   Outer loop: incremental
•   Inner loop: levelwise
•   Equivalence checking
2. Association Rule Mining
• Result management
• Experimental results
• Future work
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Pattern Mining Algorithm
Outer loop:                                            x1
Generate, incrementally, all possible
x2
trees of increasing sizes. Avoid
generation of isomorphic trees.                 x3        x4

Inner loop:
For each newly generated tree, generate all queries based
on that tree, and test their frequency.

             5            x

                         
...
x1       x2   x1       x2   5        

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Outer loop

• It is well known how to efficiently generate all trees
uniquely up to isomorphism

• Based on canonical form of trees.

• [Scions, Li-Ruskey, Zaki, Chi-Young-Muntz]

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Inner loop: Levelwise approach

• A query Q is characterized by
 Q set of existential nodes
 Q set of parameters
– Labeling Qof the parameters by constants.

• Q   specializes Q   if  ,   
and  agrees with  on .

• If Q specializes Q then freqQ  freqQ

• Most general query: T = (, , )

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Inner loop: Candidate generation

•   CanTab is a candidate query
FreqTabis a frequent query

•   Q’=’’ is a parent of Q= if either:
   ’ and  has precisely one more node than ’, or
   ’ and  has precisely one more node than ’

•   Join Lemma:
Each candidacy table can be computed by taking the
natural join of its parent frequency tables.

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Inner loop: Frequency counting

• Each candidacy table can be computed by a single SQL
query. (ref. Join lemma).

• Suppose: Gfromto table in the database, then each
frequency table can be computed with a single SQL query.

 
» formulate in SQL and count
  
» formulate   in SQL E
» natural join of E with CanTab
» group by 
» count each group

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Inner loop: Example

x
x x
x0 x8

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Inner loop: Example

x
x x
x0 x8

• Join expression:

CanTab{x}{x,x} = FreqTabxx⋈ FreqTab   xx⋈ FreqTabx x

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Inner loop: Example

x
x x
x0 x8

• SQL expression E for x  

select distinct G1.from as x1, G2.to as x3,
G3.to as x4
from G G1, G G2, G G3
where G1.to = G2.from and G3.from = G2.from

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Inner loop: Example

x
x x
x0 x8

• SQL expression for filling the frequency table:
select distinct E.x1, E.x3, count(E.x4)
from E, CanTab{x2}{x1,x3} as CT
where E.x1 = CT.x1 and E.x3 = CT.x3
group by E.x1, E.x3
having count(E.x4) >= k

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Equivalent queries

Queries Q and Q are equivalent if same answer sets on all
graphs G (up to renaming of the distinguished variables)

•   2 cases of equivalent queries:
1. Q1 has fewer nodes than Q2
2. Q1 and Q2 have the same number of nodes

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Equivalence theorem

Two queries are equivalent if and only if there are containment
mappings between them in both directions.

A containment mapping from Q to Q is a h: QQ that
maps distinguished variables of Q one-to-one to distinguished
variables of Q, and maps parameters of Q to parameters of Q,
preserving labels

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Case : Q fewer nodes than Q2
Redundancy lemma:
Let Q be a tree query without selected nodes. Then Q has a
redundancy if and only if it contains a subtree C in the form of a
linear chain of  nodes (possibly just a single node), such that the
parent of C has another subtree that is at least as deep as C.

Redundant
Q1     x
subtree
x

x

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Case : Q and Q same number of nodes

• Q and Q must be isomorphic.

• Canonical form of queries: refine the canonical ordering of
the underlying unlabeled tree, taking into account node
labels.

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Association Mining Algorithm

• Input: a graph G, minsup, a tree query Qleft frequent in
G, minconf
• Output: all tree queries Q such that Qleft  Q is a legal
and confident association rule in G.

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Containment mappings

• For each tree query, generate all containment mappings
from Qleft to Q, ignoring parameter assignments.

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Instantiations

• For each containment mapping, generate all parameter
assignments such that Qleft  Q is frequent and
confident.

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Equivalent Association rules

• Equivalence checking of association rules is as
hard as general graph isomorphism testing.

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Outline rest of talk

• Result management
• Experimental results
• Future work

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Result management

• Output: frequency tables stored in a relational database.

• Browser

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Experimental results: Real-life datasets

• Food web nodes54 edges0

frequency = 176

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Experimental results: Real-life datasets

• Food web nodes54 edges0
(x1,x2,x3,x4,x5)       (x1,x2,x4,x2,x5)

x1                    x1

x2        x4               x2

x3        x5          101 x4 x5

confidence = 11%

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Experimental results: Performance

• Fully implemented on top of IBM DB2
• Preliminary performance results:
– pattern mining algorithm:
• huge number of patterns
• constant overhead per discovered pattern
– association mining algorithm:
• very fast
• constant overhead per discovered rule

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Future work

• Applications: scientific data mining
• Loosen restriction to trees

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References

• Bart Goethals, Eveline Hoekx and Jan Van den Bussche,
Mining Tree Queries in a Graph, in Proceedings of the
eleventh ACM SIGKDD International conference on
Knowledge Discovery and Data Mining, p 61-69, ACM
Press 2005
• Eveline Hoekx and Jan Van den Bussche, Mining for Tree-
Query Associations in a Graph, to appear in Proceedings of
the 2006 IEEE International Conference on Data Mining
(ICDM 2006)

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