CLAN: An algorithm for mining closed cliques from large dense graph database

1

• CLAN An algorithm for mining closed cliques from
  large dense graph database
• ICDE 2006

2
Outline

• Introduction
• Problem definition
• CLAN
• Empirical results
• Conclusion

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Introduction

• Most previously proposed frequent graph mining
  algorithms mine the complete set of frequent (or
  closed) subgraphs without considering the
  underlying topology of these subgraphs.
• Mining complete set of all frequent subgraphs
  wastes considerable computing power and space.
• Most of this research only studies mining clique
  patterns from a single graph (ex Maximum clique
  or enumerating all the cliques) , which is
  different from the problem studied in this paper,
  that is, mining frequent closed cliques from a
  set of graph transactions.
• We develop a new algorithm, CLAN, to mine the
  frequent closed cliques.
• Several effective pruning methods are proposed to
  prune the search space, while the clique closure
  checking scheme is used to remove the non-closed
  clique patterns.

4
Problem definition
back1 back2
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Problem definition
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Problem definition

• D graph transaction database which consists of
  a set of undirected graph transactions.
• G graph transaction, a tuple G V ,E ,LV ,FV
  , LV is the set of vertex labels, and
• FV V -gt LV maps the vertices to their
  labels.
• C clique, a set of fully connected labeled
  vertices and denoted by a tuple C V ,LV ,
  FV ,a clique of size n is also called a
  n-clique.
• A clique C1 V1 , LV1 , FV1 is isomorphic to
  another clique C2 V2 , LV2 , FV2 iff
• V1 V2 and there exists a bijection f V1
  -gt V2 s.t.
• If a clique C is isomorphic to a subgraph of
  another clique C, C is called a subclique of C
  and C is called a superclique of C.
• Embedding exist a fully connected subgraph g of
  a graph G can form a clique and g is isomorphic
  to a clique C, we call g an embedding of C in G.

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Problem definition

• C is supported by a graph G, if there is at least
  one embedding of C in G.
• supD(C) the number of graph transactions in
  graph database D that support C.
• If supD(C) min_sup, C is called a frequent
  clique.
• If there does not exist another clique C s.t. C
  C and supD(C) supD(C) , C is called a
  closed clique

fig1
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CLAN

• Canonical form
• Definition 4.1 (Canonical form)
• Given a clique C, its canonical form is defined
  as the minimum string among all its possible
  strings and denoted by CFC
• If two strings canonical forms are identical,
  they are isomorphic.
• A string Sa a1a2an is called a substring of
  another string Sb b1b2 bm(denoted by Sa
  Sb), if there exist integers 1 i1 lt i2 lt lt
  in m s.t. a1bi1 ,, anbin.

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CLAN

• Lemma 4.1 (subclique relationship test)
• Given any two cliques, Ca and Cb , with
  canonical forms CFCa and CFCb , respectively. Ca
  Cb holds iff CFCa CFCb holds.
• The subclique relationship between two cliques
  can be converted to a substring relationship
  between their corresponding canonical forms.

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CLAN

• Lattice-like structure
• canonical formsupport
• Edge mean a direct subclique
• relationship.
• Mining frequent cliques becomes how to traverse
  the lattice-like structure to enumerate frequent
  cliques
• DFS
• a2,ab2, abc2, abcd2,abd2,
• abcd2, ac2, abc2, abcd2,

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CLAN

• Structural redundancy pruning
• Lemma 4.2 (Prefix closure property of canonical
  form)
• Given any clique C and its canonical form CFC,
  any non-empty prefix of CFC represents the
  canonical form of a certain clique.
• EX
• For the current prefix clique C with canonical
  form CFC , let the last vertex label of CFC be c.
  
• Grow C in order to mine larger cliques, we
  require C be extended only with vertices whose
  labels are lexicographically no smaller than c.
• If the current prefix clique is ac2, we can
  only grow it with vertices whose labels are from
  c, d, e

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CLAN

• The 19 frequent cliques in running example will
  be mined in such an order
• a2, ab2, abc2, abcd2, abd2, ac2, acd2,
  ad2, b2,bc2, bcd2, bd2, bde2, be2, c2,
  cd2, d2, de2, e2
• Pseudo low-degree vertex pruning
• Observation 4.1 (Low-degree vertex pruning)
• No vertex with a degree lower than (k-1) can be
  contained in a k-clique.

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CLAN

• Clique closure checking scheme
• If there exists any vertex with a label ß ,
  which can be used to grow C to get a
  (k1)-clique, C s.t. supD(C) supD(C) , C
  must be non-closed.
• ß is called a new extension vertex if ß ak, or
  an old extension vertex if ßltak
• Lemma 4.3 (Clique closure checking)
• If there exists no new extension vertex nor old
  extension vertex w.r.t. a pre?x k-clique C, C
  must be closed, otherwise C must be non-closed.

fig4
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CLAN

• Let the current prefix clique be C, the number of
  vertices in C be k,
• Canonical form be CFC
• Assume there are totally m embeddings of C in the
  graph database G and the set of embeddings is
  denoted by EMB(C) ,
  
• For any embedding ,a vertex v is
  an extension to ,if there exists an
  edge between v and every vertex in
• Let denote the set of extension vertices
  of the ith embedding .
• A vertex v is called a fully
  connected vertex in , if there is an edge
  between vertex v
• and any vertex in
• The set of fully connected vertex labels in
  is denoted by
• and let
• If , and
  and we call a non-closed
  extension vertex label w.r.t. clique C.

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CLAN
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Empirical results

• The stock market data w.r.t. a certain period of
  time can be converted to a graph based on the
  cross correlations of price fluctuations.
• Each stock is a vertex whose label is the
  corresponding stock index
• If correlation coefficient , there is an
  edge.

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Empirical results
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Empirical results
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Empirical results
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Conclusion

• Its efficient for large dense graph databases
• Clique closure checking scheme
• Bipartite?