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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 ij with ij N. Snapshot of a graph representing the complete metabolic pathway of a human. 2 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! 3 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 4 Example of a pattern frequency x z 5z G z8 G zx G 5 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 5z G z8 G zx G 6 Example of an Association Rule 7 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 8 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 9 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 10 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 11 Formal definition of a matching A matching of a pattern P in a graph G is a homomorphism h: P G, with hza, for parameters labeled a. 12 Example: Matching z y z x 13 Example: Matching z y z x 14 Example: Matching z y z x h 0 8 4 15 Example: Matching z y z x h 0 8 4 h 0 8 8 16 Example: Matching z y z x h 0 8 4 h 0 8 8 h 0 8 4 17 Example: Matching z y z x h 0 8 4 h 0 8 8 h 0 8 4 h4 0 8 5 18 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 19 Formal definition of frequency We define the answer set of Q in G as follows: QGf(H)|f is a matching of P in G The frequency of Q in G is #answers in the answer set. 20 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 21 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. 22 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. 23 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. 24 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 25 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 26 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] 27 Inner loop: Levelwise approach • A query Q is characterized by Q set of existential nodes Q set of parameters – Labeling Qof the parameters by constants. • Q specializes Q if , and agrees with on . • If Q specializes Q then freqQ freqQ • Most general query: T = (, , ) 28 Inner loop: Candidate generation • CanTab is a candidate query FreqTabis 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. 29 Inner loop: Frequency counting • Each candidacy table can be computed by a single SQL query. (ref. Join lemma). • Suppose: Gfromto 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 30 Inner loop: Example x x x x0 x8 31 Inner loop: Example x x x x0 x8 • Join expression: CanTab{x}{x,x} = FreqTabxx⋈ FreqTab xx⋈ FreqTabx x 32 Inner loop: Example x x x x0 x8 • 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 33 Inner loop: Example x x x x0 x8 • 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 34 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 35 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: QQ 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 36 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 37 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. 38 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. 39 Containment mappings • For each tree query, generate all containment mappings from Qleft to Q, ignoring parameter assignments. 40 Instantiations • For each containment mapping, generate all parameter assignments such that Qleft Q is frequent and confident. 41 Equivalent Association rules • Equivalence checking of association rules is as hard as general graph isomorphism testing. 42 Outline rest of talk • Result management • Experimental results • Future work 43 Result management • Output: frequency tables stored in a relational database. • Browser 44 45 Experimental results: Real-life datasets • Food web nodes54 edges0 frequency = 176 46 Experimental results: Real-life datasets • Food web nodes54 edges0 (x1,x2,x3,x4,x5) (x1,x2,x4,x2,x5) x1 x1 x2 x4 x2 x3 x5 101 x4 x5 confidence = 11% 47 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 48 Future work • Applications: scientific data mining • Loosen restriction to trees 49 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) 50

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