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2009/11/19
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
2
2009/11/19
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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2009/11/19
Part
1:
Closed Clique Mining
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2009/11/19
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)
5
2009/11/19
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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2009/11/19
Problem Formulation
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2009/11/19
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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2009/11/19
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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2009/11/19
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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2009/11/19
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.
