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Mining for Tree-Query Associations in a Graph Jan Van den Bussche Hasselt University, Belgium joint work with Bart Goethals (U Antwerp, Belgium) and Eveline Hoekx (U Hasselt, Belgium)

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2 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 metabolic pathway of a human. Applications: life sciences, biology, social sciences, WWW,...

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3 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, Horvath-Ramon-Wrobel] 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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4 Tree-Query Pattern powerful tree-shaped pattern inspired by conjunctive database queries special features: –existential nodes –parameterized nodes occurrence of the pattern in G is any homomorphism from the pattern in G frequency: x z: 0 z G z 8 G z x G

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5 Association rules fully fledged associations over tree-query patterns example:

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6 Experimental results: Real-life datasets Food web nodes edges frequency = 176 confidence = 89%

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7 Experimental results: Real-life datasets Food web nodes edges frequency = 176 confidence = 89%

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8 Experimental results: Food web nodes edges 45%55%

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9 Experimental results: Real-life datasets Protein interactions graph nodes edges confidence = 10%

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10 Experimental results: Protein interaction graph nodes edges 90%

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11 Outline rest of the talk Formal problem definition Algorithm –overall approach –levelwise generation of tree patterns –generation of containment mappings –generation of parameter assignments Equivalent association rules Certhia Performance and Experimental results Future work

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12 Tree pattern

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13 Tree pattern

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14 Tree pattern

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15 Tree pattern

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16 Tree pattern 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

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17 Matching zz yzz x

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18 Matching zz yzz x

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19 Matching zz yzz x h1h1 0184

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20 Matching zz yzz x hh 0184 hh 0188

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21 Matching zz yzz x hh 0184 hh 0188 hh 0284

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22 Matching zz yzz x hh 0184 hh 0188 hh 0284 hh 0285

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23 Matching zz yzz x hh 0184 hh 0188 hh 0284 hh 0285 hh 0288

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24 Frequency zz yzz x hh 0184 hh 0188 hh 0284 hh 0285 hh 0288 frequency = 3

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25 Tree Query P, body H, head Q = (H,P)

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26 Association Rule AR: Q 1 Q 2 Confidence (AR) = freq(Q 2 )/freq(Q 1 ) Q 2 Q 1 { (x 1,x 2,x 3 ) | Q 1 (x 1,x 2,x 3 ) G} { (x,x,6) | Q 2 (x,x,6) G }

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27 Examples of Association Rules (1)(2)

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28 Association Rule AR: Q 1 Q 2 Confidence (AR) = freq(Q 2 )/freq(Q 1 ) Q 2 Q 1 { (x 1,x 2,x 3 ) | Q 1 (x 1,x 2,x 3 ) G} { (x,x,6) | Q 2 (x,x,6) G }

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29 Containment Mapping containment mapping

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30 Containment Mapping containment mapping

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31 Containment Mapping containment mapping

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32 Containment Mapping containment mapping

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33 Containment Mapping containment mapping Q Q containment mapping from Q to Q

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34 Problem statement: Mining tree queries 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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35 Problem statement: Association rules Input: –a graph G –minsup –Q left frequent in G –minconf Output: All association rules Q left Q –frequent in G –confident in G.

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36 Algorithm: mining tree queries 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.... x1x1 x4x4 x3x3 x2x2 x2x2 x1x1 x2x2 x1x1 xx

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37 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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38 Inner loop: Levelwise approach A query Q is characterized by – Q set of existential nodes – Q set of selected nodes –Labeling Q of the selected nodes 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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39 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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40 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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41 Inner loop: Example x x x x x

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42 Inner loop: Example x x x x x Join expression: CanTab {x }{x ,x } = FreqTab x x ⋈ FreqTab x x ⋈ FreqTab x x

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43 Inner loop: Example x x x x x Join expression: CanTab {x }{x ,x } = FreqTab x x ⋈ FreqTab x x ⋈ FreqTab x x

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44 Inner loop: Example x x x x x Join expression: CanTab {x }{x ,x } = FreqTab x x ⋈ FreqTab x x ⋈ FreqTab x x

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45 Inner loop: Example x x x x x Join expression: CanTab {x }{x ,x } = FreqTab x x ⋈ FreqTab x x ⋈ FreqTab x x

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46 Inner loop: Example x x x x x Join expression: CanTab {x }{x ,x } = FreqTab x x ⋈ FreqTab x x ⋈ FreqTab x x

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47 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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48 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, 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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49 Algorithm: Mining association rules Loop 1: Generate incrementally all possible trees T of increasing sizes. Loop 2: For each T, generate all frequent tree patterns P based T. Loop 3: For each P, generate all containment mappings f from P left to P. Loop 4: For each f, generate Q=(f(H left ),P) and all parameter instantiations for Q left Q.

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50 Pattern database For each P a table FreqTab P, that contains all frequent parameter instantiations. Pattern Database

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51 Loop 3: Generation of containment mappings Efficiently solvable, due to tree shape.

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52 Loop 4: Generation of parameter instantiations single relational algebra expression (SQL)

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53 Example: Loop 4

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54 Example: Loop 4

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55 Example: Loop 4 select freqQleft.x1, freqQleft.x4, freqP.x1, freqP.x4, freqP.x5, freqP.freq, freqP.freq/freqQleft.freq from freqP, freqQleft where freqQleft.x1=freqP.x1 and freqQleft.x4=freqP.x4 and freqP.freq/freqQleft.freq >= minconf

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56 Equivalent queries Queries Q and Q are equivalent if same result sets on all graphs G (up to renaming of the distinguished variables) 2 cases of equivalent queries: 1.Q 1 has fewer nodes than Q 2 2.Q 1 and Q 2 have the same number of nodes

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57 Equivalence theorem 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 selected nodes of Q to selected nodes of Q , preserving labels Two queries are equivalent if and only if there are containment mappings between them in both directions.

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58 Case : Q fewer nodes than Q 2 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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59 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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60 Equivalent Association Rules (1) (2)

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61 Equivalence detection for rules Many cases efficiently checked. But worst case still as hard as general graph isomorphism checking. Fast heuristics for graph isomorphism checking i.e. Nauty

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62 Certhia Loop 1 + Loop 2: preprocessing step Pattern database Loop 3 + Loop 4: interactive browsing tool Certhia Demo session

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63 Experimental results: Performance Fully implemented on top of IBM DB2 Preliminary performance results: –adequate performance –huge number of patterns –constant overhead per discovered pattern

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64 Performance: Association rules Loop 3 and Loop 4: –very fast –constant overhead per rule

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65 Future work Serious scientific data mining Loosen restriction to trees

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66 Publications B. Goethals, E. Hoekx, J. Van den Bussche, “Mining tree queries in a graph”, KDD’05, p 61–69. E. Hoekx, J. Van den Bussche, “Mining tree-query associations in a graph”, ICDM’06 regular paper. “Certhia: Tree-query mining in large graphs”, ICDM’06 software demo.

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