Mining Graphs with Constrains on Symmetry and Diameter

Mining Graphs with Constrains on Symmetry and Diameter

Natalia VanetikDeutsche Telecom Laboratories at

Ben-Gurion University

IWGD10 workshop July 14th, 2010 Jiuzhaigou, China1

Graph mining (1)Problem statement


Graph mining (2)Motivation• Graphs are everywhere

– Chemical compounds (Cheminformatics)– Protein structures, biological pathways/networks (Bioinformactics)– Program control flow, traffic flow, and workflow analysis – XML databases, Web, and social network analysis

• Graph is a general model– Trees, lattices, sequences, and items are degenerated graphs

• Diversity of graphs– Directed vs. undirected, labeled vs. unlabeled (edges & vertices), weighted, with

angles & geometry (topological vs. 2-D/3-D)

• Complexity of algorithms: many problems are of high complexity (NP complete or even P-SPACE !)


Graphs, graphs, everywhere

Aspirin Yeast protein interaction network

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Internet Co-author network

Constraints: diameter

• Diameter d(G) of a graph G is the maximum among minimal distances between pairs of its vertices.

• d(G)=1 implies that G is complete.• d(G)= implies that G is not connected.


d(G)=1 d(G)=2 d(G)=2 d(G)=

Constraints: symmetry• Symmetries of a graph G are determines by its automorphism

group Aut(G).• Aut(G) is a permutation group.• Largest possible automorphism group for a graph of size n is

Sn, which has order n!


Aut(G)=S5 Aut(G)=S3 Aut(G)=D5 Aut(G)=S5

Measuring symmetry and diameter (1)

• Graph diameter is computable in polynomial time.• Automorphism group of a graph is not likely to be

computable in polynomial time.– Best known algorithm: Nauty by B. McKay, outputs a set of generators

of Aut(G).

• Intuitively, graphs with smaller diameter and higher symmetry are more interesting.


d(G)=2 d(G)=3

Measuring symmetry and diameter (2)

• Symmetry is harder to measure.• Observation: maximum symmetry of a graph is achieved

when is automorphism group is the symmetric group of order equal to the size of a graph.

• Suggestion: measure symmetry of G as


| |

( )( )

G

Aut Gs G

S

s(G)=|S5|/|S5|=1 s(G)=|S3|/|S5|= 1/20 s(G)=|D5|/|S5|= 1/12

Tree decomposition of a graph

• Let G=(V,E) be a graph. Tree T is called a tree decomposition of G if– Nodes of T are subsets X1,…,Xn V such that X1…Xn=V

– If node vXiXj , then every node Xk of T on the path from Xi to Xj contains v as well.

– For every edge e=(v,u) there exists i so that u,v Xi.


1 2

4 3

G

1 23 4

1 2 4

2 34

T1={{1,2,3,4}, } T2={{1,2,4},{2,3,4}},}(} 1,2,4},{2,3,4{{){

Minimal tree decomposition

• Width of a tree decomposition T is (max i |Xi|)-1.

• Minimum width among all tree decomposition is called tree width of a graph.

• Tree width equals maximum clique size minus 1.• Tree decomposition of minimum width is called minimal tree

decomposition.• Computing minimal tree decomposition is NP-hard problem

as it contains the problem of finding all maximum cliques in a graph.


Different tree decompositions


1 2 3

5

4

67

8

1 2 8

2 46 8

2 3 4

8 7 6

4 5 6

1 2 8

8 7 6

2 6 8

2 46

2 3 4

4 5 6

Minimal tree decomposition

Non-minimal tree decomposition

Intuition behind the proposed algorithm1. Compute the finest tree decomposition possible for every DB

transaction under given time constraints.2. Use basic pattern growing algorithm, such as FSG or gSpan to

extend instances of frequent patterns.3. Every time an instance of a frequent pattern is extended by an

edge of a nodea. Compute its diameter and symmetry estimates based on pattern’s

position within tree decomposition of a DB transaction;b. if one of the estimates is lower than user-specified symmetry or

diameter constraints, remove patterns instance from instance list,c. otherwise, keep the instance in the list.d. If the count of instances is higher than support bound, this is a

frequent pattern.


How does it work?

• Let T be tree decomposition of DB graph transaction t.• Let Gt be an instance of a candidate pattern.• Let TG=(VG,EG)T be minimal subtree of T containing G.

Claim 1. d(G)d(TG).

Claim 2. s(G)≤(|LAut(TG)|X VG|X\EG|!e EG

|e|!)/|G|!

where LAut is automorphism group of TG viewed as

tree where each node X is labeled by |X|.


Example (1)


1 2 3

5

4

67

8

1 2 8

8 7 6

2 6 8

2 46

2 3 4

4 5 6

Pattern instance and corresponding subtree of minimal Tree decomposition

Diameter is at least 1Diameter is at least 2

Example (2)


1 2 3

5

4

67

8

1 2 8

8 7 6

2 6 8

2 46

2 3 4

4 5 6

Pattern instance and corresponding subtree of minimal Tree decomposition

Symmetry is at most 1Symmetry is at most 2*2!*1!*1!/4!=1/6

Properties of estimates


The algorithm


Correctness


Complexity concerns


Test results (symmetry)


Test results (diameter)


Mining Graphs with Constrains on Symmetry and Diameter

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