Mining%20Tree-Query%20Associations%20in%20a%20Graph

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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

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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.from5 and G1.toG2.from
• and G1.toG3.from and G2.to8

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 gt 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
• Pattern Mining
• Overall approach
• Outer loop incremental
• Inner loop levelwise
• Equivalence checking
• 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
• some variables marked as existential ???
• some variables are parameters (labeled with a
  constant)
• 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
• P is a tree pattern, the body of Q
• 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

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Example Matching
z? y z? x
h? 0 1 8 4
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Example Matching
z? y z? x
h? 0 1 8 4
h? 0 1 8 8
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Example Matching
z? y z? x
h? 0 1 8 4
h? 0 1 8 8
h? 0 2 8 4
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Example Matching
z? y z? x
h? 0 1 8 4
h? 0 1 8 8
h? 0 2 8 4
h? 0 2 8 5
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Example Matching
z? y z? x
h? 0 1 8 4
h? 0 1 8 8
h? 0 2 8 4
h? 0 2 8 5
h? 0 2 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 1 8 4
h? 0 1 8 8
h? 0 2 8 4
h? 0 2 8 5
h? 0 2 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
• Pattern Mining
• Overall approach
• Outer loop incremental
• Inner loop levelwise
• Equivalence checking
• Association Rule Mining
• Result management
• Experimental results
• Future work

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Pattern Mining Algorithm

• Outer loop
• Generate, incrementally, all possible trees of
  increasing sizes. Avoid generation of isomorphic
  trees.

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

• Join expression
• CanTabx?x?,x? FreqTab?x???x????
  FreqTab?x????x?? ? FreqTab????x???x??

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Inner loop Example
?????x?? ?????x?? x?? ?????x????? x????

• 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????? x????

• SQL expression for filling the frequency table

select distinct E.x1, E.x3, count(E.x4) from E,
CanTabx2x1,x3 as CT where E.x1 CT.x1 and
E.x3 CT.x3 group by E.x1, E.x3 having
count(E.x4) gt 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
• Q1 has fewer nodes than Q2
• 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 subtree
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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????? ?edges?????

frequency 176
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Experimental results Real-life datasets

• Food web ??nodes????? ?edges?????

confidence 11
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Experimental results Performance

• Fully implemented on top of IBM DB2
• Preliminary performance results
• pattern mining algorithm
• adequate performance
• 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)