Lecture07-2-Out-of-Core-Cocain - Out-of-Core Coherent...

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Out-of-Core Coherent Closed Quasi-Clique
Mining from Large Dense Graph Databases
Jianyong Wang
Department of Computer Science and Technology
Tsinghua University, Beijing, P.R. China
Email: [email protected]
— Joint work with Zhiping Zeng, Lizhu Zhou, and George Karypis

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 Case study 1: an initial solution
- Frequent Closed Clique Mining
 The CLAN algorithm
 CLAN stands for Frequent closed CL ique p A tter N mining
 Case Study 2: a more complex solution
- Frequent Coherent Closed Quasi-Clique Mining
 The Cocain algorithm
 Cocain stands for Co herent c losed qu a si-cl i que mi n ing
 Case Study 3: an out-of-core solution
- Out-of-core Coherent Closed Quasi-Clique Mining
 The Cocain* algorithm
Frequent Coherent Closed Subgraph Mining
—
Case Studies

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Part 1: Closed Clique Mining

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CLAN: An Algorithm for Mining Closed
Cliques from Large Dense Graph Databases
Jianyong Wang
1
, Zhiping Zeng
2
, Lizhu Zhou
3
{
1
jianyong,
3
dcszlz}@tsinghua.edu.cn
2
[email protected]
Department of Computer Science and Technology
Tsinghua University, Beijing, P.R. China
— Proc. 2006 IEEE Int. Conf. on Data Engineering. (ICDE'06)

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Outline
 Problem definition and motivation
- Problem definition
- Motivation
 The CLAN solution
- Canonical form of a clique
- Efficient clique enumeration
 Low-degree vertex pruning
 Structural redundancy pruning
- Closed clique discovery
 Clique closure checking scheme
 Non-closed prefix pruning
- Integrated algorithm
 Empirical results
 Conclusions

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Problem Formulation

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Preliminaries
 Input database D: a set of undirected labeled input graphs.
 Undirected labeled input graph G
G={V, E, L
V
, F
V
}
V: the set of vertices
E: the set of edges,
L
V
: the set of vertex labels
 Cardinality of graph G: |G|= |V|
 Note: in this work, we do not consider the edge labels.
V V E  
V V
L V F  :

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Preliminaries
 Induced subgraph:
- An induced subgraph is a subset of the vertices of a graph together
with any edges whose endpoints are both in this subset.
 Examples:
- V(G
2
)={v
1
,v
2
,v
3
,v
4
,v
5
,v
6
}
- E(G
2
)={(v1,v2), … }
- L
v
(G
2
)={a, b, c, d, e}
- Card(G
2
)=6
u
4
u
3
u
5
Induced Subgraph of G
1
c
d b

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Preliminaries
 Clique: a clique C is a fully connected subgraph
 Clique Isomorphism :
- A clique C
1
={V
1
, L
1
, F
1
} is isomorphic to another clique C
2
={V
2
, L
2
, F
2
} iff
|V
1
|=| V
2
| and there exists a bijection f: V
1
 V
2
such that
 Subclique and Superclique :
- If a clique C is isomorphic to a subgraph of another clique C ’ , C is called a
subclique of C ’ , while C ’ is called a superclique of C. We use C C ’ or C C ’
(C C ’ but C≠C ’ ) to denote the subclique or proper subclique relationship.



)) ( ( ) ( ,
2 1 1
v f F v F V v   

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Preliminaries
 Embedding
- If a fully connected subgraph h of a graph G is isomorphic to a clique C, we call h an
embedding of C in G.