Partitioning a Graph in Alliances and its Application to Data Clustering

153 Pages·2004·1.55 MB·English

by  Khurram Hassan Shafique

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Electronic Theses and Dissertations, 2004-2019
2004
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Khurram Hassan-Shafique
University of Central Florida
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Partitioning a Graph in Alliances and its Application to Data
Clustering
by
Khurram Hassan Shafique
B.E. (Computer Systems Engineering)
N.E.D. University of Engineering and Technology
M.Sc. (Computer Science)
University of Central Florida
A dissertation submitted in partial fulfillment of the requirements
for the degree of Doctor of Philosophy
in the School of Computer Science
in the College of Engineering and Computer Science
at the University of Central Florida
Orlando, Florida
Fall Term
2004
Major Professor:
Ronald D. Dutton
c 2004 by Khurram Hassan Shafique
°
Abstract
Any reasonably large group of individuals, families, states, and parties exhibits the phe-
nomenon of subgroup formations within the group such that the members of each group
have a strong connection or bonding between each other. The reasons of the formation
of these subgroups that we call alliances differ in different situations, such as, kinship and
friendship (in the case of individuals), common economic interests (for both individuals and
states), common political interests, and geographical proximity. This structure of alliances is
not only prevalent in social networks, but it is also an important characteristic of similarity
networks of natural and unnatural objects. (A similarity network defines the links between
two objects based on their similarities). Discovery of such structure in a data set is called
clustering or unsupervised learning and the ability to do it automatically is desirable for
many applications in the areas of pattern recognition, computer vision, artificial intelligence,
behavioral and social sciences, life sciences, earth sciences, medicine, and information theory.
In this dissertation, we study a graph theoretical model of alliances where an alliance of
the vertices of a graph is a set of vertices in the graph, such that every vertex in the set
is adjacent to equal or more vertices inside the set than the vertices outside it. We study
the problem of partitioning a graph into alliances and identify classes of graphs that have
such a partition. We present results on the relationship between the existence of such a
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partition and other well known graph parameters, such as connectivity, subgraph structure,
and degrees of vertices. We also present results on the computational complexity of finding
such a partition.
An alliance cover set is a set of vertices in a graph that contains at least one vertex from
every alliance of the graph. The complement of an alliance cover set is an alliance free set,
that is, a set that does not contain any alliance as a subset. We study the properties of
these sets and present tight bounds on their cardinalities. In addition, we also characterize
the graphs that can be partitioned into alliance free and alliance cover sets.
Finally, we present an approximate algorithm to discover alliances in a given graph. At
each step, the algorithm finds a partition of the vertices into two alliances such that the
alliances are strongest among all such partitions. The strength of an alliance is defined as
a real number p, such that every vertex in the alliance has at least p times more neighbors
in the set than its total number of neighbors in the graph). We evaluate the performance of
the proposed algorithm on standard data sets.
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Dedicated to my daughter Aleesha
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Acknowledgments
I would first like to thank my thesis advisor, Professor Ronald D. Dutton, who introduced
me to the fields of graph theory, computational complexity and algorithms.
Special thanks to Professor Mubarak Shah for his support and encouragement during my
studies.
It is also a pleasure to thank Professor Robert Brigham, Professor Narsingh Deo, Pro-
fessor David Workman, and Professor Yue Zhao for serving as my committee members and
for their valuable comments and suggestions.
Finally I thank my wife, Wajiha Khurram, my sister, Huma Shafique, my parents Rehana
Shafique and M. Shafique Siddiqui, for their patience, and understanding during the entire
period of my tenure as a graduate student.
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Table of Contents
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
CHAPTER 2 ALLIANCES IN GRAPHS . . . . . . . . . . . . . . . . . . . . 8
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Types of Alliances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Alliance Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Basic Properties and Known Bounds on Alliance Numbers . . . . . . . . . . 16
2.4.1 Defensive Alliance Numbers . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.2 Global Defensive Alliance Numbers . . . . . . . . . . . . . . . . . . . 20
2.4.3 Offensive Alliance Numbers . . . . . . . . . . . . . . . . . . . . . . . 21
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2.4.4 Powerful Alliance Numbers . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
CHAPTER 3 PARTITIONING AGRAPHINTODEFENSIVEANDGLOBAL
DEFENSIVE ALLIANCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Satisfiability and Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Subgraph Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5 Satisfiability and Cardinality of Minimum Alliance . . . . . . . . . . . . . . 46
3.6 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.6.1 Satisfiability of Regular Graphs . . . . . . . . . . . . . . . . . . . . . 49
3.6.2 Satisfiability of Odd Graphs and Triangle free Eulerian Graphs . . . . 55
3.6.3 Satisfiability of Line Graphs . . . . . . . . . . . . . . . . . . . . . . . 57
3.7 Computational Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
CHAPTER 4 ALLIANCE FREE AND ALLIANCE COVER SETS . . . 67
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 Defensive k Alliance Free & Cover Sets . . . . . . . . . . . . . . . . . . . . 71
−
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4.4 Offensive k Alliance Free & Cover Sets . . . . . . . . . . . . . . . . . . . . . 80
−
CHAPTER 5 PARTITIONING AGRAPHINTODEFENSIVE0-ALLIANCE
FREE (COVER) SETS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.1 When G is not Partitionable . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 When a Block is Not Partitionable . . . . . . . . . . . . . . . . . . . . . . . 88
CHAPTER 6 GRAPH PARTITIONING AND DATA CLUSTERING . . 93
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.2 Graph Theoretical Techniques for Clustering . . . . . . . . . . . . . . . . . . 100
6.3 Clustering Using Maximum Satisfactory Minimum Cut . . . . . . . . . . . . 106
6.3.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.3.2 Semidefinite Relaxation of MSMC . . . . . . . . . . . . . . . . . . . . 110
6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.4.1 Zachary’s Karate Club Network . . . . . . . . . . . . . . . . . . . . . 115
6.4.2 Zoo Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.4.3 Networks of Fictional Characters . . . . . . . . . . . . . . . . . . . . 121
6.4.4 Other Standard Data Sets . . . . . . . . . . . . . . . . . . . . . . . . 122
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
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