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Edge Partitioning and Finding Community Structure Using Spectral Decomposition

Permanent Link: http://ufdc.ufl.edu/UFE0017534/00001

Material Information

Title: Edge Partitioning and Finding Community Structure Using Spectral Decomposition
Physical Description: 1 online resource (91 p.)
Language: english
Creator: Kim, Ung Sik
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: community, edge, eigen, spam, spectral
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Many systems take the form of networks, sets of nodes or vertices joined together in pairs by links or edges. These network structure can be found in diverse fields as engineering, social, economic, and biological systems. Due to the omnipresence of networks, many efforts have been made to uncover the organizing principles that govern the formation and the evolution of various complex networks. One of the important properties of the networks is that of community structure - nodes are often found to cluster into tightly-knit groups with a high density of within-group edges and lower density of between-group edges. This community structure of the networks performs an important role in the study of networks. We proposed a new method for detecting such community, using the spectral decomposition, and it overcomes shortcomings of the conventional spectral partitioning approaches such as min-cut, and max-cut. We show this method can be a powerful approach for finding the community structure in the networks. We apply this method to the computer generated networks and real-world networks and show the advantages of the proposed method. We analyze personal emails in the form of network data and proposed a new approach for classifying spam and non-spam emails based on graph theoretic approaches. The proposed algorithm can distinguish between unsolicited commercial emails, so called spam and non-spam emails using only the information in the email headers. We exploit the properties of social networks and spectral decomposition to implement our algorithm. In this study, we mainly used the community structure in social network to classify non-spam and proposed a new method for edge partition of networks. We tested our method on a users's mail box, and it classified 41% of all emails as spam or non-spam emails, with no error. And these results are obtained with only few subnetworks resulted from the proposed decomposition method. It requires no supervised training and solely based on the properties of networks, not on the contents of emails.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ung Sik Kim.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Boykin, P. Oscar.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0017534:00001

Permanent Link: http://ufdc.ufl.edu/UFE0017534/00001

Material Information

Title: Edge Partitioning and Finding Community Structure Using Spectral Decomposition
Physical Description: 1 online resource (91 p.)
Language: english
Creator: Kim, Ung Sik
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: community, edge, eigen, spam, spectral
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Many systems take the form of networks, sets of nodes or vertices joined together in pairs by links or edges. These network structure can be found in diverse fields as engineering, social, economic, and biological systems. Due to the omnipresence of networks, many efforts have been made to uncover the organizing principles that govern the formation and the evolution of various complex networks. One of the important properties of the networks is that of community structure - nodes are often found to cluster into tightly-knit groups with a high density of within-group edges and lower density of between-group edges. This community structure of the networks performs an important role in the study of networks. We proposed a new method for detecting such community, using the spectral decomposition, and it overcomes shortcomings of the conventional spectral partitioning approaches such as min-cut, and max-cut. We show this method can be a powerful approach for finding the community structure in the networks. We apply this method to the computer generated networks and real-world networks and show the advantages of the proposed method. We analyze personal emails in the form of network data and proposed a new approach for classifying spam and non-spam emails based on graph theoretic approaches. The proposed algorithm can distinguish between unsolicited commercial emails, so called spam and non-spam emails using only the information in the email headers. We exploit the properties of social networks and spectral decomposition to implement our algorithm. In this study, we mainly used the community structure in social network to classify non-spam and proposed a new method for edge partition of networks. We tested our method on a users's mail box, and it classified 41% of all emails as spam or non-spam emails, with no error. And these results are obtained with only few subnetworks resulted from the proposed decomposition method. It requires no supervised training and solely based on the properties of networks, not on the contents of emails.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ung Sik Kim.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Boykin, P. Oscar.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0017534:00001


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ddb224ed9548d6daf23a47c8f5148e94
d86ee2e18ce93806ba2438d8185d4a43c3551895







EDGE PARTITIONING AND FINDING COMMUNITY STRUCTURE USING
SPECTRAL DECOMPOSITION


















By
UJNG SIKK EIM


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007
































S2007 Ung Sik Eim




































To my lovely wife Jiwon, my parents-in-law, my son Min-suk, and my daughter

Hyun-kyung, for their love and patience.










ACKENOWLED GMENTS

I would like to thank my advisor Professor Oscar Boykin and ex-advisor Professor

.Jianho Gao for their guidance and their inspiration for my scientific research. Besides

my advisor, I would also like to thank the rest of my coninittee niembers: professors

William Ogle, Liuqing Yang and .John Harris for their interest in my study. I express my

appreciation to my colleagues: .Jan-Min Lee, and II Park. Especially, II Park helped me

revise papers and gave me some worthwhile -II__. -r;an-. Finally, I would like to share a

great deal of my achievement with my wife .Jiwon, parents-in-law, and my kids, Min-suk,

Hyun-kyung. Without their love and encouragement, this research could not have been

completed.











TABLE OF CONTENTS


page

ACK(NOWLEDGMENTS .......... . .. .. 4

LIST OF TABLES ......... ..... .. 7

LIST OF FIGURES ......... .... .. 8

ABSTRACT ......... ..... . 10

CHAPTER

1 INTRODUCTION ......... ... .. 12

1.1 Graph Theoretic Concept ......... ... 13
1.2 Classification Of Networks ......... .... 16
1.2.1 Random Graph ......... .. 16
1.2.2 Small-World Networks ........ .. .. 17
1.2.3 Scale-Free Networks ......... .... 17
1.3 Social Networks ......... . .. .. 19
1.4 Organization of This Study ......... .. 20

2 RELATED WORKS : CENTRALITIES, SPECTRAL PARTITIONING AND
COMMUNITY STRUCTURE ........ .. 23

2.1 Centrality ........ .. .. 23
2.2 Graph Partitioning ......... . .. 25
2.3 Finding Community Structure ....... .. .. 26

3 PROPOSED METHOD: SPECTRAL DECOMPOSITION OF COMPLEX NETWORKS
AND FINDING COMMUNITY STRUCTURE .... .. .. 33

3.1 Scale-Free Networks and Spectral Analysis .... .. 34
3.2 Proposed Algorithm ......... . 36
3.2.1 Spectral Decomposition . .... .. .. 37
3.2.2 Finding Community Structure in Subgraphs .. .. .. 51

4 APPLICATIONS ......... .. .. 64

4.1 Detecting Communities in Social Networks .... .... .. 64
4.1.1 K~arate Club Data ......... .. .. 64
4.1.2 The Social Network of Dolphins ..... .. . 67
4.2 Detecting Communities in Dynamic Networks ... ... .. 69
4.3 Personal Email Networks ......... .. 74

5 CONCLUSIONS ......... ... .. 83

REFERENCES ......._._.. ........... 86











BIOGRAPHICAL SK(ETCH ......... . .. 91










LIST OF TABLES


Table page

3-1 Absolute values of the difference for edges in each projection. .. .. .. .. 50

3-2 Clustering coefficient of two combined subgraphs. ... .. .. 57

4-1 Algorithm results for three data sets. . ..... .. .. 81











LIST OF FIGURES


Figure

1-1 Examples of graphs and their .Il11 Il-ency matrix and Laplacian matrix. .

1-2 Degree distribution (Eq.1-5) of the protein-protein interaction network .

1-3 Example of social network : A collaboration network of scientists. ....

2-1 Simple example of a graph partitioning. .....

2-2 Best division into equal-sized parts founded by the spectral partitioning.

2-3 Schematic representation of a network with coninunity structure. ....

2-4 Dendrogrant of the coninunity structure of K~arate club. .....

:3-1 Rank-ordered eigfenvalue spectrum with noise. .....

:3-2 Discrete spectrum of the classical random network. .....

:3-3 Discrete spectrum of a protein-protein interaction network (Fruit Fly). .

:3-4 Two main steps of the proposed algorithms .....

:3-5 Projection on the basis vector has :3 axes components .......

:3-6 Projections on eigenvectors (As qi). ......

:3-7 Example of the decomposition using the Laplacian matrix. .....

:3-8 Correlation matrix of subgraphs in a social network, which has :34 nodes.

:3-9 Finding the best matched subgraph of subgraph a. .....

:3-10 Comparison of two types of the vector representation of subgraph. ...

:3-11 Comparison of correlated values and clustering coefficients for each pair o



:3-12 Combine subgraph 4(a) and 1(b), Combine subgraph 4(a) and 5(c) ...

:3-13 Simple example of finding Coninunity Structure front 6 subgraphs. ...

:3-14 Block diagram of the proposed methods ......

4-1 Friendship network front Zachary's karate club study. ......

4-2 Coninunity dendrogrant of Zachary's karate club network ..... .

4-:3 1\odularity of Zachary's karate club network ......

4-4 Degree distribution of each coninunity. ......


page

. 15

. 18

. 20

. 25

. 26;

. 28

32

35

37

38

39

. 41

. 48

. 51

. 55

. 56

.. 57


subgraph.
58

. 58

. 61

. 6:3

. 65

66

.. 67

. 6;8











4-5

4-6

4-7

4-8

4-9

4-10

4-11

4-12

4-1:3

4-14

4-15

4-16

4-17

4-18

4-19

4-20

4-21

4-22

4-2:3


Clustering coefficient of each coninunity .....

Modified eigenvector centrality .....

Coninunity structures by adding subgraphs in K~arate club ..

Predicted coninunities by fast algorithm and our method ..

Social network of 62 dolphins .....

Modularity of dolphin social network. .....

Eigfenvalues and the number of edges. .....

Detected 4 Coninunity modules in the social network of 62 do

Artificial network. Z,,t, = 2 and Zin = 14. .....

Coninunity structures of an artificial network .....

Artificial network. Z,,t, = 7 and Zin = 9. .....

Fraction of vertices correctly classified as the number x,,t, is va

Subgraph resulting front an example for message which has M10

Rank ordered eigenvalue spectrum of entail network ......

Entire personal entail network ......

Cumulative number of links in subnetworks .....


. . 6

. 70

. 70

. 71

. 71

. 72

. 72

. . 7:3

.. 74

. 75

. 76

. . 76

. . 77

. 78

. 79

80

80

81

82


lphins .


ried. .

ID .


Spant entail network .....

Non-spant entail network ......

Cumulative links and subgraph size ......










Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

EDGE PARTITIONING AND FINDING COMMUNITY STRITCTITRE RISING
SPECTR AL DECOMPOSITION

By

I~ng Sik K~in

August 2007

Cl.! ny~: P. Oscar Boykin
Major: Electrical and Computer Engineering

?1 Iny: systems take the fornt of networks, sets of nodes or vertices joined together

in pairs by links or edges. These network structure can he found in diverse fields

as engineering, social, economic, and biological systems. Due to the oninipresence of

networks, many efforts have been made to uncover the organizing principles that govern

the formation and the evolution of various complex networks. One of the important

properties of the networks is that of coninunity structure nodes are often found to

cluster into tightly-knit groups with a high density of within-group edges and lower

density of between-group edges. This coninunity structure of the networks performs

an important role in the study of networks. We proposed a new method for detecting

such coninunity, using the spectral decomposition, and it overcomes shortconlings of

the conventional spectral partitioning approaches such as nmin-cut, and nmax-cut. We

show this method can he a powerful approach for finding the coninunity structure in

the networks. We apply this method to the computer generated networks and real-world

networks and show the advantages of the proposed method. We analyze personal entails in

the form of network data and proposed a new approach for classifying span1 and non-spant

entails based on graph theoretic approaches. The proposed algorithm can distinguish

between unsolicited coninercial entails, so called span1 and non-spant entails using only the

information in the entail headers. We exploit the properties of social networks and spectral

decomposition to intplenient our algorithm. In this study, we mainly used the coninunity










structure in social network to classify non-spam and proposed a new method for edge

partition of networks. We tested our method on a users's mail box, and it classified 41 of

all emails as spam or non-spam emails, with no error. And these results are obtained with

only few subnetworks resulted from the proposed decomposition method. It requires no

supervised training and solely based on the properties of networks, not on the contents of

emails.










CHAPTER 1
INTRODUCTION

We are facing the explosion of spam-unsolicited commercial email-everyd 11- and

having a spam wave-more like a tsunami. Recent study has shown that the volume of junk

mail on the Internet at large began -1:i-i, n1: 1:, Him; in 2006. A recent investigation revealed

that more than half percent of e-mail users ;?i- spam has impaired the thrust of e-mail

and 25' ?i- that the volume of spam has reduced their e-mail use [1]. This crisis has

demanded proposals for a broad range of potential solutions, such as the design of efficient

anti-spam tools, and calls for anti-spam laws.

For any anti-spam tool, it is especially important that the algorithm should generate

virtually no false negative, since each non-spam message that classified as a spam

undermines the confidence of the user, and decrease the likelihood that anti-spam filters

will be used universally. The ease of use of the anti-spam tool is also an important issue

and there is a strong preference for automated algorithms, which need little or no direct

intervention from individual users [2].

We propose an effective technique which can he easily implemented based on graph

theoretic methods and the spectral decomposition of networks. Networks are the most

common feature linking diverse systems ranging from the technological, biological,

economic, and social systems [3], [4], [6], [7]. As one example of technological systems,

the internet is a complex network of computers and routers connected by various

communication links. In a social network, nodes are human beings and edges represent

various social relationships between these nodes [8], [9].

The main ideas in this algorithm are based on the unique characteristics of social

networks and an eigen-projection of matrix which represent a network. In our social

activities, almost all our contractual decisions depend strongly on information provided hv

our networks of friends. The reliability of the decisions we made, then, depends heavily

on the trustworthiness of our social networks [2]. We seem to have developed interaction










strategies for the generation of a trustworthy network. The common rule is that trust is

built based not only on how well you know a person, but also on how well that person is

known to the other people in your existent network. This strategy results in community

structure that is one of the important issues in social network studies. It is also known

as one of the properties of a small-world network. This concept can be extended to the

cyberspace as well, and can be used to find some features for the spam filtering tool. The

emails originating from one of user's friend or friend's friends is trustworthy or can be

classified as non-spam. After construction of personal email network, then we can apply

many network analysis techniques to provide an effective and automated algorithm. We

also propose a new spectral decomposition based on the eigen-projection that will show

the advantages of this method compared to other methods.

In this chapter, we first present some background about the basic graph theory and

brief classification of networks. Then we briefly introduce the social networks. Finally we

outline the scope of this study.

1.1 Graph Theoretic Concept

Graph theory studies the properties of graphs in mathematics and computer science.

Definition of graphs vary in substance and style, depending on the level of abstraction

that is suitable for a particular application. Generally p. .1:;19 a graph G is a set V=

V(G) of p number of points together with a prescribed set X of q number of unordered

pairs of distinct points of V. The points p are called vertices or nodes connected by links

called edges. Each element of x = (u, v) of X represents a line of G, and x is said to join

u and v. We write x = av and ;?i that u and v are .Illi Il:ent points, point u and line x

are incident with each other, as are v and x. If two distinct lines x and y7 are incident with

a common point, then they are .Il11 Il:ent lines. A graph with p points and q lines is called

a (p, q) graph. The (1,0) graph is trivial [10]. When following conditions are satisfied, a

graph G= (V, E, f) is defined as an undirected graph.

*V is a set, whose elements are variously referred to as nodes, points, or vertices.










*E is a set, known as edgfes or lines.


*f is a function that maps each element of E to an unordered pair of distinct vertices
in V, referred to as the ends endpoints, or end vertices of the edge.

We consider an email network as an undirected graph. On the contrary, a directed

graph G is an ordered pair G= (V, A) subject to the following conditions:

V is a set, whose elements are variously referred to as nodes, points, or vertices.

A is set of ordered pairs of vertices, called arrows, or directed edges. An edge
e = (x, y) is said to be directed from x to y, where x is the tail of e and y is the head
of e.

A graph is usually written G(V, E), when V = 1 n is the set of a nodes, and E is

the set of edges. Each edge i, j is associated with a non-negative weight I, which shows

the .ll11 Il:ency of nodes i and j. For an unweighted graph, we assign I, = 0 for any

non- Il11 Il:ent pair of nodes and I, = 1 for any .Illi Il:ent pair of nodes. The 70<.i.c

matrix: of the graph G is the symmetric n x n matrix A(G)


A(G) = 0 if ii = j i ,--- (1-1)


The .Il11 Il:ency matrix of a graph is a matrix with rows and columns labeled by graph

vertices, with a 1 or 0 in position (I vj) according to whether I and vj are .ll11 Il:ent

or not. For a simple graph with no self-loops, the .Il11 Il:ency matrix must have O's on

the diagonal. For an undirected graph, the .Il11 Il:ency matrix is symmetric, I, = tr, .

Let us denote the neighborhood of i by NV(i) = {j|(i, j) E E}. The degree of node i is

deg(i1) AE Cj!) ". The Laplacian matrix L is another symmetric n x n2 matrix that

represented by the diagonal D and .Il11 Il:ency A components. The diagonal matrix shows

the number of connections of each vertex. Let us define the degrees matrix: as the a x n

diagonal matrix D that satisfies Dii = deg(i).


Le() = deg(i) if i = j i ,-- 12
-r-. if i / j









The Laplacian matrix has many useful features:


Laplacian is positive semi-definite matrix(for all i, eigenvalue As > 0), which has no
negative eigfenvalues.

Laplacian is a real symmetric and hence its n eigenvalues are real and its eigenvectors
are orthogonal.

1, n (1, 1, -, 1)T E R" is an eigenvector of Laplacian, with associated eigenvalue 0.








3' 3- 3


0 0 0 1 0 0 110111
I:Adjacency Matrix A 0 0 01 1 0 0 111 1 11 11 01 11 11



1 0 0 -1 2 0 -1 -1 3 -1 -1 -1
"0 1 0 -1 0 2 -1 -1 0 3 -1 -1
LapacanMari L0 0 1 -1 -1 -1 2 0 -1 -1 3 -1
-1 -1 -1 3 -1 -1 0 2 -1 -1 -1 3


Figure 1-1. Examples of graphs and their .Il11 Il:ency matrix and Laplacian matrix.

OtI;, i. ring is a common property of social networks. Cliques are formed, representing

circles of friends or acquaintances in which every member knows every other member

[11]. This inherent tendency to cluster can be quantified by the clustering coefficient [25].

Let us focus on a selected node i in a network, having ki edges which connect to ki other

nodes. If the nearest neighbors of the original node were part of a clique, there would be

ki(ki 1)/2 edges between them. The ratio between the number Ei of edges that actually

exist between these ki nodes and the total number ki(ki 1)/2 gives the value of the

clustering coefficient of node i,
2Ei
Ci (1-3)
k, (ki 1)










The clustering coefficient of the whole network is the average of all individual Cis.

Average path length can describe the degree of separation. Given two nodes I vj E V, let

dmin(i, j) be the shortest distance between these two nodes. The average path length of a

given node I is given by

L~r = din~i j)(1-4)
j=1
where n is the total number of nodes of the graph G. It should be noted that the average

path length is a topological measure, which is of interest to structural analysis of large

networks .

1.2 Classification Of Networks

Some discoveries as mentioned above have initiated a revival of network modeling

in the past few years, resulting in the introduction of three main classes of modeling

paradigms [11]. First, random graph, which are variants of the Erdiis-Riinyi model.

Second, motivated by the clustering phenomena, a class of models, collectively called

small-world models, has been proposed. Finally, the discovery of the power-law degree

distribution has led to the construction of various scale-free models that focused on the

network dynamics. We will briefly review each model of network.

1.2.1 Random Graph

A variant of the Erdiis-Riinyi model is still widely used in many fields and serves as

a benchmark for many modeling and empirical studies [11]. Traditionally, the study of

complex networks has been the territory of graph theory. While graph theory initially

focused on regular graphs, since the 1950s large scale networks with no apparent design

principles have been described as random graphs, proposed as the simplest and most

straightforward realization of a complex network. Random graphs were first studied by the

Hungarian mathematicians Paul Erdiis and Alfriis Riinyi. According to the Erdiis-Ri~nyi

model, we start with NV nodes and connect every pair of nodes with probability p,

creating a graph with approximately pNV(N 1)/2 edges distributed randomly. There

are C" -1(_)/2] graphs with NV nodes and n edges, forming a probability space in which










every realization is equiprobable. This model has guided our thinking about complex

networks for decades. But the growing interest in complex systems has forced many

scientists to reconsider this modeling paradigms and need to develop new concept deviated

from a random graph.

1.2.2 Small-World Networks

Real-world networks are likely to be small-world networks that demonstrate two hasic

properties [25]. The first property of small-world is that the separation between any two

randomly chosen nodes is very small. The separation is characterized by the notion of

path, which is defined as the shortest distance between nodes. The average path length

for a social network is likely to reflect a small degree of separation. In the domain of the

Internet [12], the web forms a kind of a small-world network with separation from page to

page around 19 clicks. Mathematically, such a separation can he described by an average

path length in Eq. (1-4). The second property of small-world networks is their high degree

of clustering. This can also be seen from our daily experience where, for example, our

friends are likely to be friends of each other as well, or in other words, social networks tend

to be clustered. This property is characterized by the clustering coefficient in Eq. (1-3). A

high degree of clustering is not a property of a random graph.

1.2.3 Scale-Free Networks

The degree of a node is the number of nodes it is .Illi Il ent to; or, equivalently, it

is the number of edges that are incident on it. A node with no edges (degree 0) is an

isolate. A node with degree 1 is called a pendant. And the nodes of a graph can he

characterized by the number of edges that they have the number of other nodes to which

they are .Il11 Il-ent. This property is called the node degree. In undirected networks, we

don't distinguish the in-degree, the number of edges that point toward the node, and

the out-degree, the number of edges that start at the node. Whereas the node degree

characterize individual nodes, the degree distribution is defined to quantify the diversity

of the whole network. The degree distribution P(k) gives the fraction of nodes that have


























The degree distribution of numerous networks, such as the World Wide Web [12], [5],

social networks [13] and molecular networks, follow the power-law [14], [15]. In Fig. 1-2,

we show that the degree distribution of the protein-protein interaction network of the

fruit fly, which follows the power-law property. This property indicates a high diversity

Degree distribution log-log plots


*
~

*

*
~
*
*






+* *

i I ,, I I
10'
degree


degree k and is obtained by counting the number of nodes NV(k) that have k = 1, 2, 3, .

edges and dividing it by the total number of nodes NV. The power-law is defined in

equation (16).


P(k) =( 6 C(k,, k))


P(k) = Ak Y


(1-5)


(1-6)


Degree distribution (Eq.1-5) of the protein-protein interaction network has
4637 nodes. Degree of this network follows the scale-free property.


Figure 1-2.


of node degrees and that there is no typical node in the network that could be used to

characterize the rest of the nodes. The absence of a typical degree is the reason these

networks are described as -I I!.--free" [16], [17], [18]










1.3 Social Networks

A social network is a set of people or groups of people with some pattern of

interactions between them [19]. The pattern of friendships between individuals, business

relationship between companies, and marriages between families are example of the social

network [20], [21], [22]. In respect of academic disciplines, the social sciences have the

longest history of the quantitative study of real-world networks [23], [24], [25]. The pattern

of a social network helps determine a network's usefulness to its individuals.

A simple network analysis consider social relationships in terms of nodes and links.

Nodes are the individual actors within the networks, and links are the relationships

between the actors. There are many kinds of links between the nodes.

As the simplest form, a social network is a map of all of the relevant links between

the nodes. The network can also be used to determine the social capital of individual

actors [19]. This concept can he di;11l li4I in a social network diagram, where nodes are

the points and links are the lines. Social network analysis, which is related to network

theory, has emerged as a key technique in modern sociology, anthropology, sociolinguistics,

geography, social psychology, information science and organizational studies, as well as

a popular topic of speculation and study [19]. Fig. 1-3 shows the largest component

of the collaboration network. Social network Analysts reason from whole to part;
from behavior to attitude from structure to relation to individual. They either study

whole networks, all of the links containing specified relations, or personal networks, the

links that specified people have, such as their personal communities. In summary, social

networks can he represented using an undirected or directed graph. Edges represent social

relationships such as frequency of conversation, rating of friendship, phone-calls, email,

and co-authorship [26], [27]. The structural properties of a social network also represent

human relationships-sstatus and positions. As an extended form of a social network,

hibliometrics includes any network generated by human, citation network, and www

hyperlinks. These concept of the social network can he extended to the cyberspace as well,

and can he used to find some important features for the spam filteringf tool.





















Mathematical
Ecology


Statistical Physics


of RNA


Figure 1-3. Example of social network : A collaboration network of scientists. After AI.
Girvan and 31. E. J. ?-. ein-~! 1 (2002) [3].


1.4 Organization of This Study

Several studies focused on the statistical properties of networked system such as

social networks and World-Wide Web, many efforts have been given particularly on a few

properties coninon to various networks: the sniall-world property, power-law property,

and transitivity of networks. In this study, we utilized the property of coninunity, in

which network nodes are joined together in tightly-knit groups between which there are

only looser connections, to find good features for classifying spant entails. There are

many algorithms to detect such coninunities and almost all of these methods are based










on a heuristic approach. Our spectral decomposition method gives not only a set of

subnetworks, but also foundations for detecting coninunity structure in networks. The

main contributions of this work can he suninarized as:

1. This method can decompose uniquely the entire complex network into a set of
subnetworks based on the spectral analysis.

2. In the rank-ordered eigenvalue spectrum of the complex network, we clearly observe
a straight line part in a double logarithm scale plot which indicates that some part
of the eigfenvalue spectrum follows the power law. In addition, this slope is robust
under significant amount of noise which is random addition and deletion of edges.
This el__o-r-;- that the eigfenvalues and eigfenvectors obtained front the proposed
method may be also considered robust. Therefore, high-noise network data can he
analyzed properly.

3. It shows that the personal entail network follows the power-law property. This
proposed method shows that:

The rank-ordered eigenvalue spectrum of the personal entail network follows the
power-law property.

The distribution of sunt of interactions in an each subnetwork also follows the
power-law distribution.

4. We propose a new method for finding coninunities, and we can find coninunity
structures in the network using subnetworks as the building blocks of the coninunity
structure. In this process, we utilize the clustering property and correlation matrix of
the subgraphs to find the coninunity structure.

5. We proposed an effective spant filtering algorithm hased on the property of the social
network and the spectral decomposition.

The rest of this proposal is organized as follows. In C'!s Ilter 2, we review some nietrics for

the analysis of the network and briefly review the conventional methods of the spectral

partitioning and finding coninunity structure. In ('! .pter 3, we introduce the proposed

algorithm, spectral decomposition of the complex network and study how to find the

coninunity structure in the network. In this chapter, we will explain main contribution of

our work, especially for suninary 4 in the above paragraph. As applications, in ('!s Ilter 4,

we apply the algorithm to the real-world networks and an artificial network and show that

the proposed algorithm can he an effective method to find the coninunity structures of the










entire network. Then we apply the spectral decomposition method to the personal email

network and use the property of the social network to show that the proposed algorithm

can he an effective anti-spam fighting tool. In this chapter, we show how the proposed

methods support claims 3, 4 and 5 in the above paragraph. Finally, we summarize our

work and discuss issues in ChI Ilpter 5.









CHAPTER 2
RELATED WORKS : CENTRALITIES, SPECTRAL PARTITIONING AND
COMMUNITY STRUCTURE

2.1 Centrality

For the analysis of the network data, especially for social networks, finding the

most important node (person) is one of the most frequently required and fundamental

measurement of the network structure. We usually use ~ ,le.i~ritl./ measures for this purpose

[12]. There are several types of the centrality measures,

1. degree centrality

2. eigenvector centrality

3. betweenness centrality

4. closeness centrality

The simplest centrality measure is degree ..u:i~~lil:;, also known as degree. The degree of a

node in a network is the number of edges attached to it. The degree ki of a node i is


kg = Asy(2-1)
j=1

Where, Aij is elements of .llli Il:ency matrix A.


Aij = 1 if i = j1 (2-2)
0 if i /j

Even it is simple, the degree is often a highly effective measure of the importance or

influence of a node. In many social situations, people with more connections tend to have

more power than other with less connections. A more complicated version of the same

idea is eigenvector .. t.1 .li;1: Contrary to the degree, eigenvector centrality acknowledges

that not all connections are equal. In general, connections to people who are themselves

influential will lend a person more influence than connections to less influential people. If

we denote the centrality of a node i by xi, then we can represent this effect by making xi










proportional to the average of the centralities of i's network neighbors.


2'i = Asyrs (2-)
j= 1

where A is a constant. Defining the vector of centralities :r = (111, xa ...), We 0811 TOWrite

this equation in a matrix form as:

AXr = Axr (2-4)

and hence we know that :r is an eigenvector of the .Il11 Il-ency matrix with an eigenvalue A.

The eigenvector centrality defined in this way accords each node a centrality depends both

on the number and the quality of its connections: having a larger number of connections

still counts to measure, but a node with a smaller number of high-quality contacts may

outrank one with a larger number of mediocre contacts. Two other centrality measures are

closeness .. ialierll.;; and betweenness .. ialitl.1;; Both are based on the concept of network

paths. A path in a network is a sequence of nods traversed by following edges from one

to another across network. A geodesic path is the shortest path, in terms of number of

edges traversed, between a specific pair of nodes. The closeness centrality of node i is the

mean geodesic distance from node i to every other node. Closeness centrality is lower for

vertices that are more central in the sense of having a shorter network distance on average

to other vertices. Some vertices may not he reachable from node i -two nodes can lie in

separate components of a network, with no connection between the component at all. The

betweenness centrality of a node i is the fraction of geodesic paths between other vertices

that i falls on. That is, we find the shortest path between every pair of nodes, and ask on

what fraction of those paths node i lies. Betweenness is a crude measure of the control

i exerts over the flow of information between others. If we imagine information flowing

between individuals in the network and ahr-l- .- taking the shortest possible path, then

betweenness centrality measures the fraction of that information that will flow through i

on its way to wherever it is going. In many social contexts a node with high betweenness

will exert substantial influence hv virtue of not being in the middle of the network but










of lying between other nodes in this way. It is in most cases only an approximation to

assume that information flows along geodesic paths.

2.2 Graph Partitioning

There is a long tradition of research in computer science on graph partitioning, a

problem that arises in a variety of contexts, but most prominently in the development

of computer algorithms for parallel or distributed computation. Fig. 2-1 shows a simple

graph partitioning. Suppose a computation requires the performance of some number














Figure 2-1. Simple example of a graph partitioning.

of a tasks, each to be carried out by a separate process, thread running, or program on

one of different computer processors. Typically, there is a desired number of tasks or

volume of work to be assigned to each of the processors. If the processors are identical, for

instance, and the tasks are of similar complexity, we may wish to assign the same number

of tasks to each processor so as to share the workload roughly equally. It is also typically

the case that the individual tasks require for their completion results generated during

the performance of other tasks, so tasks must communicate with one another to complete

the overall computation. The pattern of required communications can be thought of as a

network with a vertices representing the tasks and an edge joining any pair of tasks that

need to communicate, for a total of m edges. In theory the amount of communication

between different pairs of tasks could vary, leading to a weighted network, but we here

restrict our attention to the simplest unweighted case, which already presents interesting










challenges. Normally, communications between processors in parallel computers are slow

compared to data movement within processors, and hence we would like to keep such

communication to a minimum. In a network representation this means we would like

to divide the vertices of our network (the processes) into groups (the processors) such

that the number of edges between groups is minimized. This is the graph partitioning

problem. Fig. 2-2 shows clustering of the mesh network in [28]. Spectral clustering uses




















2.3 ~ ~ ~ ~~e Fidn Comuit Stutr



As wre m-. ent iione in the peviulsie sectiosvrlsuis fo cused on the speta atatisticg.alt






properties of the network have been given particularly on a few properties, which

common to various networks: the small-world property, the power-law property, and

the transitivity of networks. In this study, we mainly utilized the property of community

to find good features for classifying spam emails. Such structures have been discovered in

networks as diverse as power-grid networks, biological networks, technological networks,

and most notably in social networks [3]. There are many algorithms to detect such










communities and almost all of these methods are based on heuristic approach. In spite

of these efforts spanning several decades in this direction [:31], [:32], the identification of

the community structure in networks remains an open problem. The space of possible

partitions of even a small network is very large indeed. Several methods have been

proposed for findings meaningful partitions in networks of reasonable size. These methods

vary considerably from one another, not only in their general approach, but also in

sensitivity and computational effort [:33]. In general, more accurate methods tend to be

able to explore a large portion of the partition space, and are computationally expensive

[:34]. On the other hand, those methods which explode a smaller region of the partition

space tend to be faster, but as a consequence, less accurate [:35]. Fig. 2-3 shows a simple

example of community structure. Our spectral decomposition method gives not only a set

of subnetworks, but also the foundation for detecting community structure in networks.

The fundamental problem with all these spectral partitioning methods such as, min-cut,

max-cut, and ratio-cut are that cut sizes are simply not the right thing to optimize

because they do not accurately reflect the intuitive concept of network communities.

To quantify how good a particular partitions is, the modularity measure Q was

introduced [:36]. It has been widely used and many well-known algorithms for finding

community structure in networks are based on the maximization of the modularity. On

a predefined set of communities f in a network, a community connection matrix ef, is

defined, where each member the proportion of links from community f to community g.

The matrix is normalized, that is, each of the members of the matrix ef, = Lfs/Leotal, Lfg

being the number of links between community f and community g, and Ltamay is the total

number of links in the network [:36]. The proportion of links belonging to community f is

denoted af and is simply the sum, af = Cf eys The computation of Q is as follows









































Figure 2-3. Schematic representation of a network with coninunity structure. In this
network there are four coninunities of densely connected nodes, with a much
lower density of connections between them.


The modularity, Q, quantifies the difference between the intra-coninunity links and the

expected value for the same coninunities in a randonlized network. The modularity is

a relative value, and while it gives measure of how good a partition of the network is, it

cannot show us whether this partition is the best possible partition.

A good division of a network into coninunities is not simply one in which the number

of edges running between groups, intra-group links, is small. In fact, it is one in which

the number of edges between groups is smaller than expected, for a same size randonlized

network. Only if the number of between-group edges is significantly lower than would be

expected purely by chance can we correctly claim to have found a significant coninunity









structure. Equivalently, we can examine the number of edges within communities and

look for divisions of the network in which this number is higher than expected-the two

approaches are equivalent since the total number of edges is fixed and any edges that do

not lie between communities must be lie inside them [36]. These considerations give us a

modified benefit function Q, not based on pure cut size, defined as


Q = (number of edges within communities) (expected number of such edges) (2-6)


This benefit function is also called modularity [36]. It is a function of the particular

division of the network into groups, which indicates stronger community structure when it

has larger values. In principle, we can find good divisions of a network into communities

by optimizing the modularity over possible divisions.

The first term in Eq. (2-6) is easy to calculate. However, the second term is relatively

not clear and needs to be precisely defined before we can calculate the modularity. To find

the expected number of edges within community, we choose a null model against which to

compare the network. The definition of this modularity needs a comparison of the number

in some equivalent randomized model network in which edges are placed without regard to

community structure and the number of within-group edges in a real network. This is one

of the advantages of this modularity that can make the role of this null model clear and

explicit. This null model must have the same number of vertices n as the original network,

so that we can divide it into the same groups for comparison. There is a lot of freedom

to choose randomized models in which we specify the probability Pij for an edge to fall

between every pair of vertices i, j. Given Pij, the modularity can be defined as follows.

The actual number of edges falling between a particular pair of vertices i and j is Aij and

the expected number is Pij. Thus the actual minus expected number of edges between i

and j is Aij Pi and the modularity is the sum of this quantity over all pairs of vertices

belonging to the same community. Let us consider gi to be the community to which vertex









i belongs. Then the modularity can be written.


Q = Asy- Py1o(gs, gy) (2-7)


Where o-(f, g) = 1, if f = g and 0 otherwise and m is the number of edges in the network.

The scale factor 2 is needed for compatibility with the previous definition at Eq. (2-5).

When we consider only undirected networks, we know that Pij = Pji and Q = 0 when all

vertices are placed in a single group together. In this case, we set all gi in Eq. (2-7), and

find that CE [Aij Pij] = 0 as





There are many possible choice of null model and various null models have been

considered previously in the literature [34], [36], [37]. The simplest is the standard random

graph, in which edges appear with equal probability Pij = p between all vertex pairs, but

many authors have claimed [34], [36], [37] that this model is not a good representation

of most real-world networks. They usually consider the actual degree of the vertex in the

real network and assume the independence of the probabilities for the two ends of a single

edge. The expected degree of vertex i is given by Ci Pij = ki. The expected number of

edges Pij between vertices i and j is the product f (k )f (ky) of a separate function of the

two degrees, where the function must be the same since Pij is symmetric. Then


Pay f (s) f(ks)= ki(2-9)
j=1 j=1

for all i and hence f(ki) = Gki for some constant C. Combining with Eq. (2-8) results in


2m = P ay = Cr2 ) ksk; = (2mnC)2 (2-10)


and hence C = and
J~2m









This model has been studied in the past as a right model of a network, for example by

Chlungb and Lu [38]. It is also closely related to the configuration model, which has been

studied widely in the mathematics and physics literature [38], [39]. Let us consider an

algorithm for finding the community structure based on the modularity. If a particular

division gives no more within-community edges that would be expected by random chance

we will get Q = 0. Values other than 0 indicate deviations from randomness, and in

practice values greater than 0.3 appear to indicate significant community structure [36].

This fact also so----- -is an alternative approach to finding community structure [35]. If

a high value of Q represents a good community division, the best one can be found by

simple optimizing Q over all possible division. The problem is that the true optimization

of Q is very costly. The number of v- .--s to divide a vertices into g non-empty groups is

givenl by the Stirling nlumrber of the second k~ind S ), anld hlence the number of distinct

commnunityi division is C_ S ). This sumn is not known in a closed form, but wve observe

that for all n > 1, so that the sum must increase at least exponentially in n. To perform

an exhaustive search of all possible divisions for the optimal value of Q would therefore

take at least an exponential amount of time, and is in practice infeasible for systems

larger than twenty of thirty vertices [35]. Various approximate optimization methods are

available: genetic algorithm, simulated annealing, and so forth. ?-. Fin-I! 11 [35] considers

a scheme based on a standard greedy optimization algorithm. This algorithm falls in

the general category ofl ....1.. ... 1 .i Ive hierarchical clustering methods [19], [40]. Starting

with a state in which each node is the only member of one of a communities, they

repeatedly join communities together in pairs, choosing at each step the join that results

in the greatest increase (or small decrease) in Q. The progress of the algorithm can be

represented as a de; J~'lr..,u, a tree that shows the order of the joins. Fig. 2-4 shows

a simple example of a dendrogram. Cuts through this dendrogram at different levels

give division of the network into smaller or larger number of communities and we can

select the best cut by looking for the maximal value of Q. Since the joining of a pair of
























Figure 2-4. Dendrogrant of the coninunity structure of K~arate club. The numbers at the
bottom represent the nodes in the network, and the tree shows the order in
which they join together to form coninunities.


coninunities between which there are no edges at all can never result in an increase in

Q, we need only consider those pairs between which there are edges, of which there will

at any time he at most ni, where ni is again the number of edges in the graph. When we

consider a network that has been partitioned in some arbitrary way, the change in Q is

given by

dQ = efs + egy 2afa, = 2(ef, afa,) (2-12)

which can clearly be calculated in constant time. This can he interpreted as a measure

of affinity of coninunities f and y and can subsequently be used to find the two

coninunities which are most alike (highest dQ). Following a join, some of the matrix

elements ef, must he updated by adding together the rows and columns corresponding

to the joined coninunities, which takes worst case time complexity of O(n). Thus each

step of the algorithm takes worstcase time complexity of O(m + n). There are a nmaxiniun

of n 1 join operations necessary to construct the complete dendrogrant and hence the

entire algorithm runs in time O((m + n)n), or O(n2) On R sparse graph. We compare our

proposed algorithm for finding coninunity structure with this algorithm in application

part in Cl'I Ilater 4.










CHAPTER 3
PROPOSED METHOD: SPECTRAL DECOMPOSITION OF COMPLEX NETWORKS
AND FINDING COMMUNITY STRITCTIRE

The contribution of our work is the introduction of the spectral decomposition

algorithm that can produce good separation of networks in the projective space and a

method for finding coninunity structure in the network. This can he a good method for

real-world network data analysis, especially for the scale-free networks.

Front the early d on oOf spectral graph theory, matrix and linear algebra were used

to analyzed the matrix representation of graphs. There are many publications about on

spectral graph theory [41], [42], [43], [44], [45], [46], [47], [48]. Based on these results,

spectral clustering became a popular approach for data <1In-rh 11). which includes a class

of clustering methods that use eigenvectors of the Laplacian of the syninetric matrix

W = (I, .) containing the pairwise similarity between data objects i, j. The spectral

partitioning algorithms have two obvious shortconlings. First, it basically divides networks

into two clusters, it is extended to niulti-way clustering through recursive 2-way < In-1 hint l

for real-world networks can certainly have more than two clusters. Second, it makes use

only of the leading eigenvector of the Laplacian matrix and ignores all the others, which

throws away useful information contained in those other vectors [49].

Our method also can he used to find the coninunity structure in a network. There

are many algorithms to detect such coninunities and almost all of these methods are

based on a heuristic approach. Our spectral decomposition method gives not only a set of

subnetworks, but also the foundation for detecting coninunity structure in networks. Here

is a brief outline of this chapter. The first section begins with the brief explanation of the

spectral analysis and the power-law property of the scale-free network. Then, we introduce

a spectral decomposition method and an algorithm to find the coninunity structure in the

network, which utilizes the results front the spectral decomposition.










3.1 Scale-Free Networks and Spectral Analysis

The analysis of many real complex networks has shown the presence of several

typical properties, which can he found in various systems. The scale-free nature of degree

distribution P(k) is one of most relevant properties. It is defined as the probability that

a randomly chosen node has degree k. The scale-free property can he expressed by a

power-law function of the form. The power law property is one of well-known properties,

which originated front the random fractal theory. A power-law distribution, in the most

general sense, has the form in Eq. (3-1).


P(.r) a L(.r) .I-D (1


The empirical results demonstrate that many large networks are scale free, that is,

their degree distribution follows a power law [15], [12]. The rank-ordered eigenvalue

spectrum of many real-world network data also follows the power-law distribution.

The power-law eigenvalue spectrum thus provides an objective way to determine the

dimension of the network. The rank-ordered eigenvalue spectrum of the protein-protein

interaction network is shown in Fig. 3-1, where we clearly observe a straight line part in

a double logarithm scale plot which indicates that some part of the eigfenvalue spectrum

follows the power law. In addition, this slope is robust under significant amount of noise

which is random addition and deletion of edges. This -II- -- -; that the eigenvalues

and eigenvectors obtained front the proposed method may be also considered robust.

Therefore, highly noisy network data, such as protein-protein interaction network obtained

via high-throughput experiments, can he analyzed properly. Front this fact, we know

that the eigenvalues and eigenvectors of the network to be analyzed do not show a big

difference with noise. For this reason, we assume that the proposed algorithm can he

robust to noisy data. As we mentioned, the structure of a network can he completely

described by the associated .Il11 Il-ency matrix. The .Il11 Il-ency matrix of an undirected

graph is syninetric and its elements are equal to number of edges between the given











[he rank ordered elgenialue spectrunmnoise



05-




05


S-15-





-2 5C 40%/
60%/


-3 -

351 05 1 15 2 25 3
Index n [log10]

Figure 3-1. Rank-ordered eigenvalue spectrum with noise. This log-log plot shows the
power-law property of the PPI network. Note that the slope of the straight
part is resilient to random noise. From 211' up to I011' of total interactions are
replaced with random addition or deletion.


vertices. The eigenvalues of an .Il11 Il:ency matrix are related to basic topological invariants

of network such as, the diameter of a network [50], [51]. Recently, it was proposed to

consider spectra of eigenvalues of the .Il11 Il:ency matrix as a fingerprint of the networks

[52]. The characteristic polynomial det(A AI) of the .Il11 Il:ency matrix A of G is called

the characteristic polynomial of G and denoted by PG(x). The eigenvalues of A and the

spectrum of A, which consists of the n eigenvalues, are called the eigenvalues and the

spectrum of G, respectively: these notions are independent of vertex labeling. Clearly,

isomorphic graphs have the same spectrum. The .Il11 Il:ency matrix is symmetric and

non-negative in the case of undirected networks and accordingly has real eigenvalues Ay,

j = 1,...NV, being solutions of det(A AI) = 0. The relation between features of network










and properties of its spectral density with respect to its .Illi Il:ency matrix.

N32

j=1

Following Farkas et al [51] we define scaled variables p and A


A = A/ Np(1 l-: p)p = (p/ Np(1 p) (33)


where p = k/NV is the average number of links per node divided by the total number

of nodes. For random networks the density in Eq. (3-2) of states can be computed

analytically from random matrix theory and the result is the so-called Wigner's semicir-

cular law [53]. Fig. 3-2 shows the density for a random network which has same number

of interactions with the protein-protein interaction network in Fig. 3-3. Fig. 3-3 shows

p(A) for the protein-protein interaction network of fruit fly (Drosophila no, 1.; ..9. ;- /.l r)

[54] and has a distinct behavior, having the emergence of peaks at specific eigenvalues.

Using these properties, we may know how the prevalence of specific peaks in the discrete

spectrum of a network reflects the networks' topologfies and relates to other concepts like

the search for motifs [55], [56]. But there are problems in this approach. First, subgraphs

are not generally represented by their eigenvalues in the spectrum of the whole network.

Second, isospectral graphs, which are same spectral density, are not necessarily isomorphic

[50], [57]. This spectral method is a indirected approach and not effective to reveal the

structural property of a network.

3.2 Proposed Algorithm

As we mentioned previously, the main contributions of our work are the spectral

decomposing method for edge partitioning to make a set of subgraphs, which can be good

building blocks and the new approach of the node partitioning for finding the community

structure in network data. Our proposed algorithm is made of two parts, first step is

the spectral decomposition of the network, latter one is an algorithm for findings the

community structure in the network. At the first step, we can obtain the subgraphs, which










Classical Random Network(1-10) Spectrum


1 1.5







0.5




-4 -3 -2 -1 0 1 2 3 4
Eigen Values

Figure 3-2. Discrete spectrum of the classical random network, which are same size of the
protein-protein interaction network. The number of nodes is 4555


are computed front eigfenvalues and eigfenvectors of the Laplacian matrix. This process is

based on the orthogonal projection on the basis vector, as an additional accomplishment,

we also introduce a modified centrality measure using eigenvectors of the Laplacian

matrix. As the second step, we introduce the method for finding the coninunity structure

in the network with subgraphs that we obtain at the spectral decomposition. Figf. 3-4

shows two main steps of our method. We know that eigfenvectors corresponding to those

eigenvalues contain a lot of distinguished information on subnetworks. There is a study

that used eigenvalues and eigenvectors to successfully find protein coding sequence [58].

3.2.1 Spectral Decomposition

As we explained in previous section, there are many algebraic spectral approaches for

analysis of network data. Usually Laplacian eigenvalue spectra are used to compare and










FruitFly PPI Sepectrum


2.5








1 1.5







0.5




-5 -4 -3 -2 -1 0 1 2 3 4 5
Eigen Values

Figure :3-:3. Discrete spectrum of a protein-protein interaction network (Fruit Fly). It
shows some specific peaks. The number of nodes is 4555.


find structurally similar graphs. Two graphs are deemed to be isomorphic when they have

the same eigenvalue spectrum. This method is imperfect since cospectral non-isomorphic

graphs exist, in other words, even if two graphs have same Laplacian eigenvalue spectra,

these graphs can not he isomorphic graphs. For Laplacian spectra, the method fails less

than 10 to 15 percent of the cases. The topological properties of graphs can he analyzed

using spectral graph theory and the eigenvalues are related to the connectivity pattern of

the graph. Specifically, the second smallest eigenvalue is a measure of the compactness of

a graph. A large second eigenvalue indicates a compact graph, whereas a small eigenvalue

implies an elongated topology [:31]. We showed that the prevalence of specific peaks in

the discrete .Illi Il ency spectrum of a network reflects the network's topologies in the

previous section :3.1. The spectral analysis based on Laplacian spectra is a more popular

















Spectral Decomposition Finding Community
Using Laplacian Spectra Structure from subgraphs




Figure 3-4. Two main steps of the proposed algorithms. Spectral Edgfe partitioning and
Finding community structure.


approach than the analysis using the .Illi Il:ency spectrum. In summary, the conventional

eigen analysis of the complex networks uses the discrete spectrum to find indirect features,

the emergence of spectral peaks, and to relate the spectral properties to the networks'

topologfies in an indirect manner. Similar to other approaches of the spectral method,

the conventional spectral partitioning methods only use a few eigenvectors for clustering.

On the contrary, in the proposed method, we can obtain the subnetworks based on the

projection on every significant eigenvector in the network. It is possible that the entire

original complex network can be represented by the summation of subnetworks. There is

no missing or redundant edge in all subnetworks and these subnetworks are results of the

edge partitioning.

Our algorithm is based on the orthogonal projection of network data on basis

vectors, which are computed from eigfenvectors of the Laplacian matrix. We recast the

spectral decomposition of network data as an approximation of the entire network by

the summation of low-rank matrices, which have the same format with the matrix for

the entire network. If the .Il11 Il:ency matrix is used to represent the entire network, the

low-rank matrix also has the same structure with an .Il11 Il:ency matrix such as zero

diagonal terms.









Now let us consider an .ll11 Il:ency matrix to represent network data. Since A is a

symmetric matrix with real entries there exists an orthogonal matrix Q such that,


A = QT A Q (3-4)


is a diagonal matrix. Here A = diag(Ay, X2, Am), and the column of Q are corresponding

eigfenvectors which form an orthonormal basis of Rm.


AQ = QA (3-5)

QT = Q-l (3-6)

Q"AQ = A (3-7)


q{Aq = Ai if ii = j3 (3-8)


Eq. (3-7) and (3-8) are the spectral theorem in mathematics and the principal axis

theorem in geometry and physics [59]. The represented matrix of the network can be

consider as a transformation matrix, if we multiply this matrix on the left of a basis

vector, the answer is another vector that is transformed from it's original position, such as

Aq = Aq. It is the nature of the transformation that the eigfenvectors arise from.

M ii~ 3y
i= 1

Where, M~ is the rank of A. Eq. (3-9) is the great factorization QAQT, written in terms of

A's and q's. Eq. (3-8) represents the spectral decomposition and this decomposition also

can perform the dimensionality reduction if we use m, which is less than the rank of the

matrix A. Principal component r, al;, 7- (PCA) for data analysis usually uses the first few

largest eigenvalues and associated eigenvectors to reduce the dimensionality of the system.

In case of the conventional PCA, the matrix used for eigenvectors is computed from

the covariance matrix of data. In this study, we use an .Il11 Il:ency matrix or a Laplacian

matrix of network data for the spectral decomposition. First, we compute eigfenvectors



































Figure :3-5. Projection on the basis vector has :3 axes components, and the result is
expressed with el, e2, and e:3.


and eigenvalues of the matrix, then consider eigenvector as basis vectors. As in Eq. (:310),

projections of the matrix on the basis vector are perpendicular to each other because of

the property of eigenvector. We can express the matrix in terms of these perpendicular

eigenvectors, instead of the original matrix form. In Fig. :3-5, we can see the projection of

the matrix on a basis vector 17i-< on the vector and this is a just scaled basis vector, not

changing its direction.


A4q = X111, -492 = 2q2, ... Aq~z = ;zq~z (:310)


Each Eigenvalue tells whether the projected vector is stretched or shrunk or reversed

or left unchanged when it is multiplied by 4. From Eq. (:39) and (:310), the spectral

theorem for symmetric matrices ; -, that 4 is a combination of projection matrix. Pi =

. is a projection matrix. Each projection matrix can he considered as a low-ranked


Projection on the basis










matrix. Eq. (3-7) and (3-8) can be represented by the projection matrix P satisfied

P2 = P. Every symmetric matrix A = AT has the factorization QVQT with real diagonal

V and orthogonal matrix Q. The eigenvalues of a real symmetric matrix are real and

eigenvectors of a real symmetric matrix, when they correspond to different A, are ahr-7- .-

perpendicular.


Projection matrix Pi = qi qT (3-11)

P1 + P2 + P, = 1 P : OrthOnOrmal space (3-12)

A = Al-P1 + a2' P2++X "' nP, 3


To get the valid form of subgraphs from the spectral decomposition of the entire network,

we have to consider a fundamental issue:

*How can we decompose the original network into the summation of subnetworks that
also have the same form of the original network?

For the issue of this study, we consider the eigen decomposition of the matrix. Since, the

eigenvectors of the original .Il11 Il:ency matrix are orthonormal, and the space spanned

by the outer product, results in projection matrix, of themselves are also orthogonal

[59]. Hence, the weighted projection matrices defined by the outer product of each

eigenvector, multiplied by the eigenvalue, will tend to have exclusive values for elements

in the respective position. We utilize this exclusiveness directly and convert the weighted

projection matrix to the .Il11 Il:ency matrix of a subnetwork. Therefore the summation

of every .Il11 Il:ency matrix of the subnetwork becomes the .Il11 Il:ency matrix of the entire

network. Now let us show a simple example of the eigfen decomposition of the network.

011

A = 1 0 1 (3-14)

110

This matrix A is symmetric and has zero diagonal terms. Because of symmetry of the

.Il1i Il:ency matrix A, it has only real eigenvalues and eigenvectors of A can be chosen










orthonormal. If A is symmetric, the number of positive eigenvalues equals the number of

positive pivots and the pivots and the eigenvalues have the same sign. The summation of

the n eigenvalues equals the sum of the a diagonal entries which are zero.


At +X~ A2 n = trace(A) = sum( II.:1 .i..o.J term of A) = 0 (3-15)


Eigenvalues of this A are At = -1, X2 = 1 3 = 2, and projections of A on the

eigfenvectors are

0.7152 -0.3938 1.1547

AQ = 1-0.0167 0.8163 1.1547 (3-16)

-0.6987 -0.4225 1.1547

Projections for each eigenvector are Argy = [0.7152, -0.0167, -0.6987]

22=[-0.3938, 0.8163, -0.4225], and A3q3 = [1.1547, 1.1547, 1.1547], respectively.

Elements of each projection corresponds to el, e2, and e3 in Fig. 3-5. From these values,

we can find the eigfenvector basis vector, which is the 'I;__ -r contributor to each non-zero

element in the .Il11 Il:ency matrix A. To find this eigenvector, we compute the weighted

projection matrix of each eigenvector in Eq. (3-13).

-0.5116 0.0118 0.4997

X1PI = Xlgiff = 0.0118 -0.003 -0.0116 (3-17)

0.4997 -0.0116 -0.4882



-0.1551 0.3215 -0.1664

12P 229T 0.3215 -0.6664 0.3449 (-8

-0.1664 0.3449 -0.1785










0.6667 0.6667 0.6667

X3P3 = 3q3q3T = 10.6667 0.6667 0.6667 (3-19)

0.6667 0.6667 0.6667

Since, the non-diagonal elements in the .Il11 Il:ency matrix A are positive, we expect

connected nodes have same signed values in projections. We know that A = AzP1 +

X2P2 + 3P3, but for the eigen decomposition, each weighted projection matrix can

be converted to the similar form of the .Il11 Il:ency matrix, such as zero diagonal terms.

Let us consider P3. All of elements of this matrix have the same value (0.6667). When

we compare these values with the edge in the matrix A, same values are assign to two

different status, 1 and 0 in the original .Il11 Il:ency matrix A(A13 = 1, A33 = 0). With this

approach, we cannot decompose this matrix into proper subnetworks. From these facts,

when we use the .Il11 Il:ency matrix for the eigen decomposition, we choose eigenvalues,

which compensate each other to get approximated null diagonal terms. Now one can pick

those eigenvectors; one could, for example, use just two eigenvectors corresponding to +Ai;

or four eigenvectors corresponding to +A ,+A2, etC.

I positive II negative 1 positive Xnegative
pr ~ ~ -~ [ .llb + pl ]riu~rL (3-20)

r/2 r/2

i= 1 i= 1
| A positive negative|I not Jn- rII,~ (3-22)


* Using the Laplacian Matrix In Eq. (3-22), we know a positive eigenvalue and a

corresponding negative eigenvalue do not alr-ws- perfectly compensated each other.

Due to this imbalance problem, there are some errors in the reconstructed .Il11 Il:ency

matrix, when n i mank of A, in Eq. (3-13). We consider the Laplacian matrix for the

spectral decomposition to solve this problem. The Laplacian matrix also can be defined

from D and A as follows

L = D (3-23)










trace(L) = At +X~ A2 n = SUMMatfiO Of degree


(3-24)


The corresponding D and A values are as follows: Each column and row in the above

matrix corresponds to the graph's vertices. The corresponding .ll11 Il:ency matrix specifies

these connections explicitly. The Laplacian matrix L(G) of a graph G, where G = (NV, E)

is an undirected, unweighted graph without graph loops (i, i) or multiple edges from one

node to another, D is the degree matrix, NV is the vertex set, and E is the edge set, is an

|N|I x |N|I symmetric with one row and column for each node. Similar to the .Il11 Il:ency

matrix, the Laplacian matrix is also represented by the summation of weighted projection

matrices.

L = Al-P 1+ A2 P2+ + Xn o (3-25)

Lx = Ax (3-26)

As in case of the .Il11 Il:ency matrix, we can express the Laplacian matrix in terms of

perpendicular eigfenvectors such as


Lql = Xiql, Lq2 2 92, ..., Lq, = A,q, (3-27)


We already know that the projection matrix on a basis vector, eigenvector in this study,

1 on~ on the vector and it is a just scaled basis. From projections for each eigenvector in

Eq. (3-27), we can find the eigenvector basis vector, which is the N----- -1 contributor

to each non-diagonal element in the Laplacian matrix L. At this point, we introduce

modified eigenvector centrality. It is similar to the eigenvector .. cida~i;, which

is defined in the previous section 2.1. Contrary to the degree centrality, eigenvector

centrality acknowledges that not all connections are equal. In general, connections to

people who are themselves influential will lend a person more influence than connections

to less influential people. Let us define the modified eigenvector centrality, when denote

the centrality of node i by xi, then we call represent this effect by making xi proportional










to the average of the centralities of i's network neighbors


j= 1

Defining the vector of centralities x = (xl, x2,...). Since, the non-diagonal elements in the

Laplacian matrix are negative, we expect nodes, which have connections, have different

signs in the projection in Eq. (3-25). The modified eigenvector centrality defined in here

not only accords each node a centrality depending both on the number and the quality of

its connections but also provides information about the links such as two nodes must have

different signs to have connections each other. To understand this concept and method, we

consider again the previous simple example, this time using the Laplacian matrix.

2 -1 -1

L = -1 2 -1 (3-29)

-1 -1 2

Because this matrix L is a semi-positive definite matrix, eigfenvalues of L are At = 0, XA2

3, A3 = 3, and projections of L on the eigenvectors are

0 0.8018 2.3146

AQ = 0 -2.4054 -0.4629 (3-30)

0 1.6038 -1.8516

Projections for each eigenvector are Argy = [0, 0, 0], Xa22 = [0.8018, -2.4054, 1.6038],

and A3q3 = [2.3146, -0.4629, -1.8516], respectively. Element of each projection is

also corresponding to el, e2, and e3 in Fig. 3-5, respectively On the contrary to the

.Illi Il:ency matrix, we must consider the sign of nodes to find the eigenvector basis vector,

which is the N----- -1 contributor to each non-zero element in the network. For example,

node 1 and node 3 can't be linked in the projection, Xa22, because both nodes have

positive values in this projection. To find eigenvectors, which can contribute to edges in










the network, we compute the weighted projection matrix of each eigenvector in Eq. (3-13).

1.7860 -0.3571 -1.4286

12P 229T -0.3571 0.0714 0.2857 (-1

-1.4286 0.2857 1.1429



0.2140 -0.6400 0.4300

X3 3 3q393 -0.6400 1.9200 -1.2857 (-2

0.4286 -1.2857 0.8571

We expect nodes, which have connections, have the most negative values in the projection

matrix. In this case, there are multiple order eigenvalues (a2 = 3 = 3). We know that any

real symmetric matrix are diagonalizable and there are ahr-l-w enough eigenvectors (equal

to order of matrix)and these eigenvectors are orthogonal. But eigenvectors associated

with repeated eigenvalues are not unique even these are linearly independent. When we

have repeated eigfenvalues, we must consider these eigfenvectors to compute the weighted

projection matrix in Eq. (3-25). This consideration is expressed


L; = Al-P I+A2'P2 "' n'P,, C~ k k i jX(T,~ ZTC) !'33)


Where As = Ay. When we apply this consideration to above example, two weighted

projection matrices, a2P2 and A3P3, muSt be added each other.

2 -1 -1

X2P2 3HP3 = 1-1 2 -1 (3-34)

-1 -1 2










From this fact, we know that this network can not be decomposed. Let us consider

another example of the Laplacian matrix for a detailed explanation.

4 -1 -1 -1 -1

-1 3 -1 0 -1

L = -1-1 2 0 0

-1 0 0 2 -1

-1 -1 0 -1 3


(3-35)


As in Eq. (3-27), we compute the


Above Laplacian matrix L is for a graph in Fig. 3-7.


2 3 4 5
index of vertices


Figure 3-6. Projections on eigenvectors (As qi), For AS, X4, A3, and X2.


projections on eigenvectors, then, find the eigenvector basis vector, which is the 'I;__ -r

contributor to each non-diagonal element in the Laplacian matrix L. Fig. 3-6 shows

projections of L on the eigenvectors such as Xsts, X4q4, X3q3, and 292a. Edge between










node 1 and 2 appeared in the projection on the 5th eigenvector. The projection that has

the N----- 0 differences for nodes, which are elements of the specific edge, is selected as

the eigen projection for the edge. To easily find the proper eigenvector for the specific

edge, we compute the weighted projection matrix of each eigenvector in Eq.(3-25).

Then, compare the values of each weighted projection matrix, which are corresponding

to non-zero and non-diagonal terms in the laplacian matrix. The weighted projection

matrix that has the most negative value are the eigfenvector for the specific edge. This

procedure also be summarized by the pseudo code representation in the algorithm 1.

When the number of edges in the network is E and the number of subgraphs is M~, there



Require: L A. ;,P
Ensure: Trace(L)= 2E, S { Pili E (1, n)}
for eachl edge: e in Gr do
(i, j) = nodes(e)
for for all P in S do
if Pj < Qij for all Q E Pa | Q / P then

else
Skip to the next P
end if
end for
end for


is a maximum of M~ operations necessary to find the subgraph for the edge. Hence, the

entire algorithm runs in time O(M~E), or O(M~2) On a Sparse graphs, when E ~ M~.

This approach also can be represented by other expression. As we mentioned previously,

values for two nodes of the edge must have different signs. For example, the edge between

node 1 and node 2 can be found in the 5th projection. The value for node 1 and node

2 in the 5th projection are 4.89 and -1.11, respectively. Table 3-1 shows absolute values

of the difference for edges in each projection, in this table, we know that edges 1-2, 1-3,

1-4, and 1-5 can be covered by the projection, Xsts and edges 2-3, 2-4, and 2-5 are in the

projection, X4q4. Fig. 3-7 Shows the result of the spectral decomposition. This process is










Table 3-1. Absolute values of the difference for edges in each projection.
Edges Azq, X292 X3q3 444 X5q5
1 -2 - 5.902
1 -3 - 5.902
1 -4 - 5.902
1 -5 - 5.902
2 -3 --3.0 4.0782
2 -5 0.858 5.7674
4 -5 --3.0 4.0782


summarized by the pseudo code in the algorithm 2. In summary, for all edges in the entire

Algorithm 2 Select Projection Asqi
Require: L, Asqi
Ensure: Trace(L) = 2E
for all non-zero, non-diagonal terms in L do
(i, j) = nodes(e)
for for all Aq do
if M~ax of e <| Aq(i) | + | Aq(j) | then
M~ax of e e-| Aq(i) | + | Aq(j)|
else
Keep max of e
end if
end for
e is in col..it,1~,pl Aq at max of e
end for


network, we compute and compare a modified eigenvector centrality of each eigenvector,

and find the eigenvector which maximally contributes to each edge. Every edge in the

entire network must have only one contributive eigenvector. For this reason, subgraphs

cannot share edges with other subgraphs. This process is mathematically equivalent to

finding the maximum contributor to edges among weighted projection matrices in Eq.

(3-25). Fig. 3-7 shows an example of the spectral decomposition of a simple network. In

this example, edges 1-2, 1-3, 1-4, and 1-5 are covered by the largest eigenvector(As). The

second largest eigenvector(A4) COVerS edges 3-2-5-4.











Laplacian Matrix

4 -1 -1 -1 -1
-1 3 -1 0 -1
-1 -1 2 O O
-1 0 0 2 -1
-1 -1 0 -1 3


h=
1.3858
3.0
4.4142
5.0


3 4


DECOMPOSITION
USING LAPLACIAN


11=5.0 h=4.414

Figure 3-7. Example of the decomposition using the Laplacian matrix. This graph can he
decomposed to two subgraphs.


3.2.2 Finding Community Structure in Subgraphs

As we mentioned in section 2.3, the property of a coninunity is a very important

concept in this study. We propose a method to find the coninunity structure in the

network, as the second step of our method. In this approach, we present a framework

to identify coninunity modules front networks by merging subgraphs, the result of the

first step in the previous section. We choose subgraphs and combine them to find which

combination is suitable for finding the coninunity structure in the network. The previous

approaches have focused only on the most significant positive and negative eigenvalues










[60]. Their approach is about clustering in unsupervised botton1-up network analysis

method and they found many quasi-cliques type cluster, which can he used to predict

the function of uncharacterized proteins. But it allows overlapping of components in

clustering and finds only quasi-cliques and quasi-bipartite type clusters. In contrast,

we can decompose uniquely the entire complex network into a linear combination set

of subnetworks. We know that the subgraphs are computed front mutually orthogonal

projection matrices that are linearly independent each other. With the process of selection

and sunination of the subgraphs, we can find the coninunity structure in the network.

Another contribution of this method for finding coninunity structure is that we use

modified eigenvector centrality to classify nodes. In the conventional approach, the

well-known benefit function for finding coninunity structure is the modularity measure

Q and we briefly reviewed in section 2.3. The basic concept of this measure is to find

the partition in which the number of edges between groups is significantly lower than

would be expected purely by chance. In this case, we can claim to have found a significant

coninunity structure in the network. To apply this modularity measure, there must he

edges between groups, which are based on node partitioning. In respect to the clustering

of nodes, our decomposition can he thought of as soft clusteringfs, where each datunt is

assigned to multiple clusters with nientership weights that sunt to one. On the contrary,

our decomposition also can he viewed as a rigid clustering in case of edges. Each subgraph

does not share edges with other subnetworks. Because of these fact, we cannot apply the

conventional modularity to find coninunity structures using subgraphs [3]. In suninary, to

obtain subgraphs is based on an edge partitioning, and it is also a hard clustering, there

is no shared- edges between subgraphs. We cannot directly use the modularity measure

to find the coninunity structure. Instead of the modularity measure, we use another

approach hased on the intuitive concept of coninunities. We can examine the number

of edges and the transitivity within groups and look for the combination of subgraphs

which increases the number of edges and the clustering coefficient of each groups.










First, we compute the correlation values between subgraphs based on the similarity of

nodes in each subgraph. Then, we find the best matched subgraph of each subgraph,

based on the maximum correlation value of each subgraph. By the simple clustering

of these pairs of subgraphs, we can find the community structure. As a foundation for

finding community structures, subgraphs can be computed from projection matrices of

corresponding eigfenvectors, as shown in the first step. The diagonals of the Laplacian

matrix are ahr-l- 1- positive integers. They represent the number of connections that

the particular node makes. The eigenvalues of Laplacian matrix are ahr-l- 1- positive

because of the semi-positive definite property of the Laplacian matrix. We do not have

an imbalance problem as in an .Illi Il:ency matrix. To compute the subgraphs, we find the

minimum value in every weighted projection matrix of corresponding edges. These edges

are contained in this subgraph. Any specific edge appears in only one subgraph. There

is no shared edges between subgraphs. Using subgraphs resulted from the spectral

decomposition, we can find the community structure in the entire network. As we

mentioned previously, since there are no edges between subgraphs, we cannot use the

of the modularity in the section 2.3. We use the degree and transitivity to find optimized

community structure in the network. For analytic process, we compute the correlation

matrix of subgraphs. We define the correlation of subgraphs as in Eq. (3-36).

Number of same entities between ,,,1-, and hiub,,,
corrcolosub,) =(3-36)
Number of entities of ,,,1-, and hiub,,,

In this Eq. (3-36), the correlation has maximum value 1, when ,,,1-, = hiub,,,. We also

express the correlation between subgraphs as the inner product of the membership vector

for subgraphs, which represent the connection of nodes in each subgraph as in Eq. (3-37).


corr~ol.,sub,, = '(3-37)










We can express the membership vector of subgraphs as in Eq. (3-38) and (3-39).


1if node i is in at lair ei d~, x

0I if node i is not in atlo,.~/,, 11, x1(38





V, J, (3-39)



sub,,

As we mentioned previously, we use the intuitive concept of community to find community

structure, increasing of the number of degree for edges and the transitivity (the clustering

coefficient) within groups. We have to look for the combination of subgraphs, which

increase the degree of edges and the clustering coefficient of each groups. The membership

vector of subgraphs defined in Eq. (3-38) and (3-39) is not suitable for this purpose,

because it does not contain the importance of nodes, such as centrality. Let us consider

another representative vector of subgraphs. We consider the degree of node, if a node has

a higher degree than others, this node has a bigger weight. We define the degree vector of

each subgraph as in Eq. (3-40).





Vint,l' = degree of each node in ent..it,, ple x = (3-40)









Eq. (3-41) shows the definition of the correlation matrix of subgraphs. In this matrix, the

diagonal terms are ahr-l-w 1 and this correlation also consider the degree of nodes.


corr (x, x) corr (x, y)

corr(y, x) corr(y, y)

corr (z,x) corr (z, y)


corr(y,z)

cor (z, z)


CORR(x, y, z)


(3-41)


Fig. 3-8 shows a correlation matrix of all subgfraphs in a network, which has 34 nodes.

Usingf the correlation matrix, we can find the best matching subgfraph of each subgfraph.

From set of paired subgraphs, we can infer the community structure in the network. Let


Figure 3-8. Correlation matrix of subgraphs in a social network, which has 34 nodes. X
and Y-axis represent the index of subgraphs.


us consider an example of finding the matched subgraphs. We find the matched subgraph

of subgraph a in Fig. 3-9. First, we compute correlation values between subgraph a and

b, and between subgraph a and c, using two types of the vector in Eq. (3-39) and (3-40).

In Fig. 3-10, subgraph 4,1, and 5 refer subgraph a, b, and c in Fig. 3-9, respectively.










Subgraph b


Subgraph a I%





Correlation a-b

SSubgraphe c








Correlation a-c



Correlation = Similarity
between subgraphs

=Inner product of membership
vectors or degree vectors


Figure :3-9. Finding the best matched subgraph of subgraph a. Subgraph a, b and c refer
subgraph 4, 1, and 5 in Fig. :3-10, respectively.


When we use the nientership vector, which does not consider the importance of nodes,

the correlation value between subgraph 4 and subgraph 1 is nmaxiniun, so subgraph 4

and subgraph 1 are the matched pair. On the other hand, if we use the degree vector

that consider the degree of nodes in subgraph, subgraph 5 is the matched subgraph of

subgraph 4. To check the validity of the degree vector, we compute the increment of

the clustering coefficient for each merged subgraph. In Fig. :3-11, subgraph pair 4 and 5

has the nmaxiniun clustering coefficient and it coincides with the nmaxiniun point of the




































Figure :3-10. Comparison of two types of the vector representation of subgraph.
membership vector is Eq. :3-38, degree vector is Eq. :340. X-axis represents
the index of subgraphs.

Table :3-2. C'lll w !inin coefficient of two combined sub graphs.
Combined Subgraphs Clustering Coefficient Averaged Degree
Sub 4 and 1 0.1915 2.3
Sub 4 and 5 0.5:357 2.29


correlation with the degree vector, so we can see the degree vector is more suitable for

finding the coninunity structure. The pseudo code in the algorithm :$ shows the process

of merging of subgraphs to find coninunity modules in a network. By comparing the

averaged degree and the transitivity for two combined subgraphs, we decide which vector

representation of subgraph is more suitable for finding the coninunity structure in the

network. Front the table :3-2, the clustering coefficient of subgraph 4 and 5 is '?i -< v- than

others. Subgraph 4 and 5 are more likely to be part of coninunity module than subgraph

4 and 1. This shows that the degree vector representation of subgraph, which consider








































Figure 3-11. Comparison of correlated values and clusteringf coefficients for each pair of
subgfraph. X-axis represents the index of subgfraphs


Subgraph 4+Subgraph 5


Subgraph 4+Subgraph 1


Figure 3-12. Combine subgraph 4(a) and 1(b), Combine subgraph 4(a) and 5(c)









Algorithm 3 Merging Subgraphs
Require: degree vector of each Subgraph Vub(,i)
dj is degree of node
for i = 1 to number of subgraphs m do
for j = 1 to nodes a do
Vsub(zi) <- dj
end for
end for
using degree vectors compute correlation matrix
for 1 = 1 to number of subgraphs m do
for j = 1 to number of subgraphs m do
CORRM/AT(sub(i), sub(j)) = Vsub(zi) Vsub(zj) /|Vsub(zi) | | Vsub(zj
end for
end for
find the best matched subgraph of each subgraph Bu ftch
for i = 1 to number of subgraphs m do
BU fmates(i) <-Max[CORRM/AT(i, j = 1 tO m)l
end for
Merge subgraphs using Bu fmates


the degree of nodes, is suitable for finding the community structure in networks. Let us

consider this result in sense of the computation complexity and mathematical reason. In

respect of the computation, the complexity of the direct computation of the clustering

coefficient for the pairs of subgraphs is O(n3), on the contrary, our method for computing

of the correlation matrix runs in time O(n2) and this computation can be done by the

simple matrix computation. We show the mathematical reason for using of the degree

vector in stead of the membership vector to represent each subgraph. We can estimate

the clustering coefficient of the vertices of degrees k' and c(k) by the kind of three-point

correlations, which defined as the probability that two neighbors of a vertex of degree k

are also neighbors themselves [61]. This function can be expressed as


c(k) = P(k", k'|k)pk',k" (3-42)


P(k", k'| k) is the conditional probability that vertex of degree k is simultaneously

connected to two vertices of degrees k' and k". And pk',k"/ iS the probability that vertices










k' and k" are connected given that they have a coninon neighbor [61]. Front the above

relations, we estimate the clustering coefficient for networks is closely related with degrees

of vertices. According to this fact, we must consider information of degrees of vertices

to find the combinations of subgraphs that increase the clustering coefficient. Fig. :3-13

shows a process of finding coninunity structure front 6 subgraphs. First, we compute the

correlation matrix, then find matched subgraph of each subgraph. In this case, subgraph

1, 2, :3, 4, 5, and 6 have subgraph :3, 4, 1, 6, :3, 2 as matched subgraphs, respectively.

For these results, we obtain two coninunity structures, the first coninunity modules

is made of subgraphs 1, :3, and 5, the second one is made of subgraphs 2, 4, and 6. To

classify nodes in networks, find out a specific node belong to which coninunity modules,

we compute the degree distribution and the clustering coefficient of each coninunity

module. Then, we classify nodes according to these values. We will show more example

of this process in section 4. Besides the correlation matrix of subgraphs, we also use

the property of transitivity in the network to find the optimized coninunity structures

by merging subgraphs resulted front the spectral decomposition. Fr-on the nature of

coninunity structure, a good coninunity structure has higher transitivity (clustering

coefficient). By computing the clustering coefficient for an addition a subgraph, we can

check the validation of our method for finding the coninunity structure. The overall

clustering coefficient must he increased in this process.

Interestingly, the subgraphs front the spectral decomposition can he considered

as meaningful partition for finding coninunity structures because there are no edges

between subgraphs. It shows that the proposed method gives us the foundation for findings

coninunity structures in a network. We don't ;?i that this method is the best for all of

the networks, but this method has a good performance for finding coninunity structure

in the network, especially, for the noisy networks. As we mentioned in ChI Ilpter 1, there

are limitation in the conventional approaches and no practical way to define subnetworks.

We think the proposed method will be very useful and lit on the analysis of the complex






































iSubgraph 1 Subgraph 3








|Subgraph 2 Subgraph 4









Subgraph 3 Subgraph 1


_~______________________________________


Subgraph 1


Subgraph 2 | Subgraph 3


Subgraphs from the
decomposition


Subgraph 4 Subgraph 5


Subgraph 6


Compute
Correlation
Matrix


Subgraph 6


Subgraph 4


Subgraph 5 Subgraph 3








Subgraph 6 Subgraph 2


Figure 3-13. Simple example of finding Community Structure from 6 subgraphs.


Check
Transitivity










networks. Since we use the consistent method to reconstruct the Laplacian matrix, the

-II- -_ -rh I1 algorithm contains no free parameter. Fig. 3-14 shows the flow of the whole

process to perform the proposed algorithm.












SPECTRAL DECOMPOSITION AND FINDING
COMMUNITY STRUCTURE


SPECTRAL
DECOMPOSITION
(LAPLACIAN MATRIX)


FROM PROJECTION
MATRICES
FIND the subgraph which has
The largest contribution to
Corresponding edge


1 L /Z r
pari pa,?q~,urlqlurr


44~
(c,
_* o
o
a


tf


i;rt
r


Set of Subgraphs









COMPUTE
CORRELATION MATRIX
AND
FIND MATCHING
SUBGRAPHS









Communit~r t y re


# Checking

- Transitivity
- Degree


** *


*

* *


Based on
Degree Centrality
-Clustering Coefficient
-Modified Eigenvector
Centrality


Figure 3-14. Block diagram of the proposed methods which are spectral decomposition
and finding community structure


CLASSIFICATION
OF NODES









CHAPTER 4
APPLICATIONS

The community structure is an important property of networks, which is the

topological property of the networks, i.e., the division of nodes into some groups within

which the network connections are dense, but sparser between the groups. Communities in

social networks might represent real social groupings, maybe by background or interest of

individuals. We introduced an algorithm for finding the community structure in networks

based on spectral analysis. In order to evaluate this algorithm, we will show the action

of the algorithm on several data sets. and we will use the well-known fast algorithm [62]

hased on the modularity for the comparison with our algorithm. First, we apply our

method to two real-world social networks. Then, we test our algorithm using an artificial

network presented by ?-. i.--us! 1a [6:3]. Finally, as a practical application of this study, we

use the community structure in the social network and the proposed method to analyze

personal email networks for the purpose of spam filtering.

4.1 Detecting Communities in Social Networks

4.1.1 Karate Club Data

We test our algorithm against the friendship network data from Zacharv's karate

club study [64]. This is a social network of friendships between :34 members of a karate

club at a US university in the 1970's. This karate club was broken into two clubs because

of a dispute of two key members, the administrator and the principal teacher. A simple

unweighted, unidirectional network is used to represent this data(available at at ?-. i..--s lI1Ss

homepage [65]). Fig. 4-1 shows the social network of friendship in a karate club. The

network has :34 nodes and 78 edges. To compare with another community finding

algorithm, we also run this karate club network though the fast algorithm [:36], [62],

[66], this method is based on the modularity in section 2.3. During the process of this

algorithm, a benefit function known as the modularity (Q) in Eq. (2-5) and (2-6) is

calculated at each step, with greater modularity indicating a better partitioning of the































Figure 4-1. Friendship network from Zachary's karate club study. It shows the natural
split .


network. This algorithm seeks to maximize the modularity through a greedy choice

of community joins. To see the step by step operation of the algorithm, we show a

dendrogram of the community joins in Fig. 4-2. Fig. 4-3 shows how the modularity of the

network evolves with each additional community merging, the modularity increase until its

peaks at 31 joins and Qmax = 0.3807. This dendrogram in Fig. 4-2 provides the order in

which the method joins nodes, it can be seen this algorithm tends to join outlying nodes

before including nodes in the core of the community. From this result, the modularity

is maximized when the network is split into 3 communities even the natural split is 2

communities. It is possible to note that node 10 is the misclassified node, which connect to

large hubs in both communities.

Now, we apply our algorithm to this same network. After the spectral decomposition,

we have to find the optimized community structure from the combinations of subnetworks.

As we mentioned in the previous section 3.2.2, we cannot directly use the modularity

to find the community structure. We use the clustering coefficient to optimize the











Q=0.3718

- -- -- - -- -- -- Max Q -0,3BO7






















Ve~et~cs 1Missclassified

node:1 0

Figure 4-2. Community dendrogram of Zachary's karate club network


combination of subgraphs instead of the modularity. There are 14 subgraphs, which

have real edges. Fr-om these subgraphs, we compute the correlation matrix and find the

matched subgraphs. In this process, we know that there are two community structures in

this network. To validate this process, we compute the clustering coefficient in each step.

The mean value of the clustering coefficient of the first step is 0.5525 and this value is

smaller than the clustering coefficient of the entire network (0.5879). Once we obtained

the final combination, the clustering coefficient increases to 0.6776, and we can classify the

nodes using the degree, clustering coefficient, and the modified eigenvector centrality in

Eq. (4-1).


j=1










Modularity Evolution for Zachary Karate Club Network


0.2-

S0.15-

0.1-

0.05-




-0.05
0 5 10 15 20 25 30 35
Number of Community Joins


Figure 4-3. Modularity of Zachary's karate club network


where, L is the Laplacian matrix. In Fig. 4-4 and Fig. 4-5, we can see each node has

different values in data 1 and data 2, node (vertex) 6 and 7 have '?V v;i values for data 2

(coninunity A) than data 1 (coninunity B). According to these results, we can classify

node 6 and 7 as nodes of coninunity A. If a node has same values for more than two

coninunities, we check the modified eigenvector centrality in Eq. (4-1). In this network,

node 10 has same values, degree and clustering coefficient, but, by checking the eigenvector

centrality, we know node 10 is belong to the coninunity structure B. Fig. 4-6 shows

the difference of the modified eigenvector centrality of node 10. Fig. 4-7 shows two

coninunities in this network. In Fig. 4-8, we know our algorithm can find the same

coninunity structure as the natural split.

4.1.2 The Social Network of Dolphins

As another application for the social network, we apply our method to the social

network of bottlenose dolphins living in Doubtful Sound, New Zealand [67]. This





















o 10-
O





0 8-0 15 2 5 0 3
One fvrie

Fiue4-.Dgeedsrbuino ec omuiy Dt epeet6-muiy ,dt





Fiue4-.Dgeedsrbuinofec onnuiy.Dt represents coninunity B, at



undirected network consists of 62 nodes, connected by 159 edges. Advanced tools for

the analysis and study of social structure in human populations have been developed

over the last half century [68], [19]. Using these resources, the analysis of animal social

networks can provide substantial insights into the social dynamics of animal populations

and possibly ell---- -1 new nianagenient strategies [69]. Animal social networks are

substantially harder to study than human networks because we cannot make interviews

and questions, and network data must he gathered by direct observation of interactions

between individuals. The network we apply was constructed front observations of a

coninunity of 62 bottlenose dolphins (Tursiops .spp.) over a period of seven years from

1994 to 2001 [67] at the coast of Doubtful Sound, New Zealand. It is known as there are

two coninunities and four sub-coninunities in the dolphin network. First, we run the fast














~tdatal
0.9-
Sdata2

0.8-

0.7-

S0.6-

0.5-

0.4-

0.3-

0.2-

0.1-


0 5 10 15 20 25 30 35
index of vertices


Figure 4-5. Clustering coefficient of each community (data 1 represents community B, data
2 represents community A.)


algorithm to predict substantial community structure; Qmax = 0.4955, forming 4 distinct

communities, as shown in Fig. 4-10.

Fr-om Fig. 4-11, we know that eigfenvalues and the number of edges in corresponding

subgraphs do not coincide each other.

When we apply our method to this network, we can classify 20 nodes from 20

nodes in the main community 1, 22 out of 23 nodes in the main community 2. For

sub-communities, we can classify 14 nodes out of 19 nodes. By the same manner in

the previous data, we use the transitivity as a benefit function for finding community

structures. Fig. 4-12 shows predicted communities.

4.2 Detecting Communities in Dynamic Networks

As a benchmark example of the working of our algorithm, we have generated a large

number of random graphs with known community structure, which we then run through





































15 20
index of vertices


Figure 4-6. Modified eigenvector centrality


Community structure A Community structure B

Figure 4-7. Community structures by adding subgraphs in K~arate club












Split: Our algorithm


Split: Fast algorithm


Figure 4-8. Predicted communities by fast algorithm and our method of Zachary's karate
club network.


Figure 4-9. Social network of 62 dolphins.












Modularity Evolution for Dolphin social network


-0.1
O


10 20 30 40 50 60 70
Number of Community Joins


Figure 4-10. 1\odularity of dolphin social network.


30


25-


20 a


15-



10 Ma

5-~


0 10 20 30 40
index of nodes


50 60 70


Figure 4-11. Eigfenvalues and the number of edgfes in corresponding subgfraphs of dolphin
network.



















e. '


Fiur 4-2 eetd4Cm uiymdls ntesca ewr f6 opi ns
th lgr thmt uniyispromne[] ahgahcnit of a = 28vetie
divided~ inofu ruso 2 ac etxhso vrg zegscnetn tt
membe oftesaegoupadz egstommes fohr rus ithz/adz
chosen': ~ suc tha th ttl xece dgeez, +zn = 6 nti cs.A o sicesd
the reutn rpsps ratradgetrcalnest h o mnt-idn
algoithm Fig 4-3so h rifca ewr hihhsz =2


InFi.4-6 w ho hefacio f etie crecl asind oth ou om uite
by; th aloih safnto f oe stefgr hws ohmtosNw a'





mthod and our memthdpror wel cretldntifyinsprorac 3] ahgrp more stha 11 of the12 vertices










for values of xo,,t < 6. Only when xo,,t approaches the value 8, which the number of within

-and between-community edges per vertex is the same, does the algorithm begin to fail.

When we compare our method to ?-. i.--in! lIs's algorithm, our method is slightly worse than

T. i.--!!! lI1Ss method for values of xo,,t < 6, but our algorithm outperform ?- '. l!! lI1's method

for the value of xent > 6.


Figure 4-13. Artificial network. Z,,t = 2 and Zi,z = 14.


4.3 Personal Email Networks

In this work, we build an email network hased on the information which is available

to a user of an email system, specifically, the header of all the email messages in the

user's inbox. Through the program for pre-processing, we can obtain information about

a personal email data. Every email header has an unique id, date information, the email

address of one sender, the list of recipient addresses, referenced message, and in-reply-to














a 6.- 9 r
+/ 9.
;i r~ : ;Tg
r. gE'a ~--: -m
F~T~F~-~-- O .\
~~~r ar9-~9:~-L- ~`C i
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t
at I ,.C ~ C`-
q


Z
(b)


E?
i
ror,
r c '+~ 9 t
--- + a r .~
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J c t.6 9
+ r
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r- c*r
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i ~ r
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i


(a)


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** ~


and Za = 14.r; j

mesae T ee nor aio resord nth s.- q-id,"dt, "rom ,"T"
Reerece" and "I-el-o fils W re ivea emil ewr yfrtcetn
nodes rersetn al the adrse nte"rm n o ilsadcnie h
mesg-i nth s .-d" rfr ec" ad inrpyt" fild as nods.Edesar
ade ewenmsae-d n adessthtapa in h aehae. hn l oe
rersnigone' mi drsssaermvdbcuew r onl itretdith
lins mog nde tat om uncat va he se [0].Fi. -17 shw a xmpeo

subgrap obtaine by thi press





































0 20 40 60 80 100 120
nz =2038



Figure 4-15. Artificial network. Z,,,, = 7 and Z, = 9.


# Neman's method
-9-- our method


O. 1 2 3 4 5 6
number of inter-community edges Zout


7 8 9


Figure 4-16. Fraction of vertices correctly classified as the number ce,,t is varied.















Message ID: MO
Date : 110900
From: FO
To: TO, T1
Cc: CO, C1

















Figure 4-17. Subgraph resulting front an example for message which has 1\1 ID


To find effective methods for spant filtering, we start analysis of network data by

checking centrality measures, which are some of the most fundamental and frequently

used measures of network structure. We obtained empirical data front one of user's entail

box and the entails have been chopped into a period of 108 dei~ These entails contain

2500 messages and converted to a network which has :3755 nodes and 69:30 edges. All

nodes representing the user's own entail addresses are removed, since we are interested

only in the connections among nodes who coninunicate via the user. Fig. 4-19 shows

a personal entails network for test. Then we applied our method to this network and

obtained subnetworks front the entail network by the spectral decomposition Eq. (:37).

Fig. 4-18 shows that the eigfenvalue spectrum follows the power-law. Using eigfenvalues

and corresponding eigfenvectors, we break the entire entail network into the sunination of

mutually subnetworks. Even if the order of matrix is :3755, most links are covered hv a

small number of subnetworks. Fig. 4-20 shows that 911' of links in the network is covered










-The rank ordered eigenvalue spectruna











-20 0.5 1 1.5 2 2.5 3 3.5 4
index n [IoglO]





Figure 4-18. Rank ordered eigfenvalue spectrum of email network


by only 727 subnetworks. Fig. 4-18 shows cumulative number of links in subnetworks.

We can distinguish 4;:' of non-spam and St.l' of spam without error. It is interesting

to note that !I :' of non-spam can be obtained only using 141 subnetworks and St.l' of

spam can be classified only using 10 subnetworks. It also proves the effectiveness of our

decomposition method. If we consider the addresses of recipient in the sent box, the

performance of this method can be easily enhanced.

To test our method more accurately, let us apply our algorithm to another email

network data, which has been published [2]. Fig. 4-21 shows spam e-mail network.

In the paper [2], they used Newman and Girvan's community-finding algorithm [3]

for separating jointed components, whereas we use our algorithm to find community

structure in email networks. After we obtained community structures of email networks,

we assume components in these communities have non-zero clustering coefficient are

non-spam components and write all nodes in these components to the black list. If a

component's clustering coefficient is zero, we consider all nodes to the black list. All nodes

in components that can not build community structures are considered as the gray list.

Fig. 4-22 shows nonspam e-mail network. In Fig. 4-23, we know that the size of subgraphs




























L~ C-
-t (


It I:

,,


,. ~.. jb~ t'. L ) I r It )~liI~:I~I .ZI~~C "3
t I 1~C j
rr ~ :; ~ ,, ..

i (' i r r~, 2, ,t. ..,. ,,
:i. ri i' '' :. I

/r"''r=:i'~\'~,'\Li~,'~\~*~~'fic ........ ....I1....I ......III. ........ ............ ..I...... ..II....
)nl ,I rrrrl
'' "' '


Figure 4-19. Entire personal email network. This graph contains 2500 messages and has
3755 nodes and 6930 edges.


t ~8 ~Ur~'' '
~ .
rr"ll~ r


I"~r*rr"'

















100

90

80

70


40

30

20

727
10

OO SOO 1000 1500 2000 2500 3000 3500 4000
index n




Figure 4-20. Cumulative-- number- of:-- link in subnetworks







.. .'~.



*'~:~'




** s- *






:*. *',* .

****











se w



Figre4-2. pamemil etor




,rrT80










Table 4-1. Algforithm results for three data sets.
Data Black list White list Gray list Total Corrected


Spant 1

Spant 2

Spant :3


0
701
0
277:3
0
12638


286
:372
201
1172
18:3
969


447
107:3
:39:3
:385:3
:38:3
25:31


:36i."'

48.1.'


51.'7' .
56.(1' .


also follows power-law and 211' of subgfraphs can cover more than 911' of edgfes in the

network.


* .


*






*i
*"



to* .


.


*


*. .*
**
*. \\\ .* /


*. L*


Figure 4-22. Non-spant entail network



Table 4-1 shows the results of using our algorithm on data sets, which used in

the paper [2]. Averaging across data, I17.' of the spant is on the black list, 45' of the
















60
16

1.40


70
0 6 08 1 12 60 6 1 5 2 5 3
inde ofnds[o]nrbro ugah %
(a) (b)

Fiue42.()Cmuaienme flnk nsbewrs()Sz fsbrpsi
nosa emi etok

nosa so h ht it codn oteppr[] '.o h osa so h
whie lst,--. ftesa so h lc it rs mby eko u o m nt
-fining lgoithmis ore uitble or he sam ilteing










CHAPTER 5
CONCLUSIONS

We introduced a new method for edge partitioning and finding the coninunity

structure in the complex network hased on the spectral analysis and the property of the

social network. As the first step of the proposed algorithm, we use the Laplacian matrix

for the spectral decomposition instead of the .Il11 Il-ency matrix, because this matrix is a

senli-positive definite matrix and there is no more imbalance problem for obtaining the

subgraphs. In this process, we also consider the repeated eigfenvalues to decompose the

network. For finding coninunity structure in a network, we present a framework to detect

coninunity modules by merging subgraphs obtained front the spectral decomposition.

We merge subgraphs to find which combination is suitable to represent the coninunity

structure in the network.

The fundamental problem with all the conventional spectral partitioning methods

such as, nmin-cut, nmax-cut, and ratio-cut are that cut sizes are simply not the right thing

to optimize because they do not accurately reflect the intuitive concept of network

coninunities. In addition these methods use only of the leading eigenvector of the

Laplacian matrix and ignores all the others, which throws away useful information

contained in those other vectors. On the contrary, we decompose the network into a set

of subgraphs using all eigenvalues and eigenvectors and we know that these subgraphs are

obtained front mutually orthogonal matrices. we also introduce a modified eigenvector

centrality, which can provide additional information about the real links between nodes,

not only the number and quality of connection as in the conventional eigenvector

centrality. This decomposition can he considered as an edge partitioning and each

subgraph do not share edges with other subnetworks. Our spectral decomposition method

gives not only a set of subnetworks, but also the foundation for detecting coninunity

structure in networks. With the process of selection and merging of the subgraphs, we

can find the coninunity structure in the network. We introduced the correlation matrix










of subgraphs as a similar nietrics to find the optimized coninunity structures front the

subgraphs, since we cannot directly use the modularity Q for this purpose. Using this

concept, we can easily find the nature of coninunity structures in a network. This ability

to find the coninunity structure in a network have practical applications.

As applications of our method, we can find two coninunities in K~arate club network

with a perfect rate. Other algorithm such the G-N algorithm and the fast algorithm have

a nlisclassified node. Fr-on the result of the artificial network, we know our method is

more suitable for the noisy data. According to these fact, when we can apply this method

to the noisy data such as biological network. Interestingly, we find our algorithm has

better performance for the scale-free networks, that is, their degree distribution follows

a power law for large k. As we mentioned, the distribution of the size of subgraphs in

a scale-free network also have power-law property and this property results in a few

large subgraphs during the spectral decomposition. With these larger subgraphs, we can

compute the correlation matrix which has dominant discrimination. Due to this fact,

we can obtain more accurate results of finding the coninunity structure in a network.

Fortunately, it is known that most of real-world networks are scale-free network.

Another application of our method, we use the properties of social networks and

spectral decomposition to distinguish span1 and non-spant entails. Since, the only

information necessary for this method is available in the user's entail headers, the

algorithm can he easily intpleniented and combined with other filtering process. This

effective technique can he easily intpleniented based on graph theoretic methods and the

spectral decomposition of networks.

The best content-hased filters achieve approximately 99.',' accuracy, but require

users to provide a training set of span1 and non-spant messages. This algorithm can

automatically generate an accurate training set for learning of more sophisticated

content-hased filters. The overall performance of this method can he enhanced with a

simple book-keeping- considering the addresses of recipients in the sent box to classify.










Our proposed methods are far front perfect and we have open problems

We hope to improve and generalize the proposed method to handle both directed
and weighted graphs.

We hope to analyze the method and fully express in niathentatical terms.

We hope to combine with other approaches and make an automated coninunity
detector.

The basic principle of our method- using the eigen projection for the spectral

decomposition and making use of the result of edge partitioning to find the coninunity

structure- can he incorporated into more sophisticated method that can automatically

detect coninunity structure regardless of types of networks. We hope that ideas and

algorithms presented here will be improved to be more useful for the determination of

functional clusters in ]?r Ily various networks.










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BIOGRAPHICAL SKETCH

U~ngsik K~im was born in Daegu, K~orea on March 16, 1967. He received his B.S.

degree and M.S. degree in Electronics Engineering from K~yungpook National University,

Daegu, K~orea in 1989 and 1992.

Fr-om 1992 to 2000, he was a Senior Researcher in Agency for Defense Development,

where he was involved in Development of Target Detecting Devices for Surface-to-Air-Missile

systems. He received his M.E.E. degree in Electrical and Computer Engineering from

University of Minnesota, Twin-cities, MN in 2003. Since 2003, he conducted research

for towards his Ph.D. of University of Florida under the guidance of Professor P. Oscar

Boykin.





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IwouldliketothankmyadvisorProfessorOscarBoykinandex-advisorProfessorJianboGaofortheirguidanceandtheirinspirationformyscienticresearch.Besidesmyadvisor,Iwouldalsoliketothanktherestofmycommitteemembers:professorsWilliamOgle,LiuqingYangandJohnHarrisfortheirinterestinmystudy.Iexpressmyappreciationtomycolleagues:Jan-MinLee,andIlPark.Especially,IlParkhelpedmerevisepapersandgavemesomeworthwhilesuggestions.Finally,IwouldliketoshareagreatdealofmyachievementwithmywifeJiwon,parents-in-law,andmykids,Min-suk,Hyun-kyung.Withouttheirloveandencouragement,thisresearchcouldnothavebeencompleted. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 12 1.1GraphTheoreticConcept ........................... 13 1.2ClassicationOfNetworks ........................... 16 1.2.1RandomGraph ............................. 16 1.2.2Small-WorldNetworks ......................... 17 1.2.3Scale-FreeNetworks ........................... 17 1.3SocialNetworks ................................. 19 1.4OrganizationofThisStudy .......................... 20 2RELATEDWORKS:CENTRALITIES,SPECTRALPARTITIONINGANDCOMMUNITYSTRUCTURE ............................ 23 2.1Centrality .................................... 23 2.2GraphPartitioning ............................... 25 2.3FindingCommunityStructure ......................... 26 3PROPOSEDMETHOD:SPECTRALDECOMPOSITIONOFCOMPLEXNETWORKSANDFINDINGCOMMUNITYSTRUCTURE .................. 33 3.1Scale-FreeNetworksandSpectralAnalysis .................. 34 3.2ProposedAlgorithm .............................. 36 3.2.1SpectralDecomposition ......................... 37 3.2.2FindingCommunityStructureinSubgraphs ............. 51 4APPLICATIONS .................................. 64 4.1DetectingCommunitiesinSocialNetworks .................. 64 4.1.1KarateClubData ............................ 64 4.1.2TheSocialNetworkofDolphins .................... 67 4.2DetectingCommunitiesinDynamicNetworks ................ 69 4.3PersonalEmailNetworks ............................ 74 5CONCLUSIONS ................................... 83 REFERENCES ....................................... 86 5

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Table page 3-1Absolutevaluesofthedierenceforedgesineachprojection. ........... 50 3-2Clusteringcoecientoftwocombinedsubgraphs. ................. 57 4-1Algorithmresultsforthreedatasets. ........................ 81 7

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Figure page 1-1ExamplesofgraphsandtheiradjacencymatrixandLaplacianmatrix. ..... 15 1-2Degreedistribution(Eq. 1{5 )oftheprotein-proteininteractionnetwork ..... 18 1-3Exampleofsocialnetwork:Acollaborationnetworkofscientists. ........ 20 2-1Simpleexampleofagraphpartitioning. ...................... 25 2-2Bestdivisionintoequal-sizedpartsfoundedbythespectralpartitioning. .... 26 2-3Schematicrepresentationofanetworkwithcommunitystructure. ........ 28 2-4DendrogramofthecommunitystructureofKarateclub. ............. 32 3-1Rank-orderedeigenvaluespectrumwithnoise. ................... 35 3-2Discretespectrumoftheclassicalrandomnetwork. ................ 37 3-3Discretespectrumofaprotein-proteininteractionnetwork(FruitFly). ..... 38 3-4Twomainstepsoftheproposedalgorithms .................... 39 3-5Projectiononthebasisvectorhas3axescomponents ............... 41 3-6Projectionsoneigenvectors(iqi). ......................... 48 3-7ExampleofthedecompositionusingtheLaplacianmatrix. ............ 51 3-8Correlationmatrixofsubgraphsinasocialnetwork,whichhas34nodes. .... 55 3-9Findingthebestmatchedsubgraphofsubgrapha. ................ 56 3-10Comparisonoftwotypesofthevectorrepresentationofsubgraph. ....... 57 3-11Comparisonofcorrelatedvaluesandclusteringcoecientsforeachpairofsubgraph. ............................................. 58 3-12Combinesubgraph4(a)and1(b),Combinesubgraph4(a)and5(c) ....... 58 3-13SimpleexampleofndingCommunityStructurefrom6subgraphs. ....... 61 3-14Blockdiagramoftheproposedmethods ...................... 63 4-1FriendshipnetworkfromZachary'skarateclubstudy. ............... 65 4-2CommunitydendrogramofZachary'skarateclubnetwork ............ 66 4-3ModularityofZachary'skarateclubnetwork ................... 67 4-4Degreedistributionofeachcommunity. ....................... 68 8

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...................... 69 4-6Modiedeigenvectorcentrality ........................... 70 4-7CommunitystructuresbyaddingsubgraphsinKarateclub ............ 70 4-8Predictedcommunitiesbyfastalgorithmandourmethod ............. 71 4-9Socialnetworkof62dolphins. ........................... 71 4-10Modularityofdolphinsocialnetwork. ........................ 72 4-11Eigenvaluesandthenumberofedges. ........................ 72 4-12Detected4Communitymodulesinthesocialnetworkof62dolphins. ...... 73 4-13Articialnetwork.Zout=2andZin=14. ..................... 74 4-14Communitystructuresofanarticialnetwork ................... 75 4-15Articialnetwork.Zout=7andZin=9. ...................... 76 4-16Fractionofverticescorrectlyclassiedasthenumberzoutisvaried. ....... 76 4-17SubgraphresultingfromanexampleformessagewhichhasM0ID ....... 77 4-18Rankorderedeigenvaluespectrumofemailnetwork ................ 78 4-19Entirepersonalemailnetwork ............................ 79 4-20Cumulativenumberoflinksinsubnetworks .................... 80 4-21Spamemailnetwork ................................. 80 4-22Non-spamemailnetwork ............................... 81 4-23Cumulativelinksandsubgraphsize ......................... 82 9

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Manysystemstaketheformofnetworks,setsofnodesorverticesjoinedtogetherinpairsbylinksoredges.Thesenetworkstructurecanbefoundindiverseeldsasengineering,social,economic,andbiologicalsystems.Duetotheomnipresenceofnetworks,manyeortshavebeenmadetouncovertheorganizingprinciplesthatgoverntheformationandtheevolutionofvariouscomplexnetworks.Oneoftheimportantpropertiesofthenetworksisthatofcommunitystructure-nodesareoftenfoundtoclusterintotightly-knitgroupswithahighdensityofwithin-groupedgesandlowerdensityofbetween-groupedges.Thiscommunitystructureofthenetworksperformsanimportantroleinthestudyofnetworks.Weproposedanewmethodfordetectingsuchcommunity,usingthespectraldecomposition,anditovercomesshortcomingsoftheconventionalspectralpartitioningapproachessuchasmin-cut,andmax-cut.Weshowthismethodcanbeapowerfulapproachforndingthecommunitystructureinthenetworks.Weapplythismethodtothecomputergeneratednetworksandreal-worldnetworksandshowtheadvantagesoftheproposedmethod.Weanalyzepersonalemailsintheformofnetworkdataandproposedanewapproachforclassifyingspamandnon-spamemailsbasedongraphtheoreticapproaches.Theproposedalgorithmcandistinguishbetweenunsolicitedcommercialemails,socalledspamandnon-spamemailsusingonlytheinformationintheemailheaders.Weexploitthepropertiesofsocialnetworksandspectraldecompositiontoimplementouralgorithm.Inthisstudy,wemainlyusedthecommunity 10

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Wearefacingtheexplosionofspam-unsolicitedcommercialemail-everydayandhavingaspamwave-morelikeatsunami.RecentstudyhasshownthatthevolumeofjunkmailontheInternetatlargebeganskyrocketingin2006.Arecentinvestigationrevealedthatmorethanhalfpercentofe-mailuserssayspamhasimpairedthethrustofe-mailand25%saythatthevolumeofspamhasreducedtheire-mailuse[ 1 ].Thiscrisishasdemandedproposalsforabroadrangeofpotentialsolutions,suchasthedesignofecientanti-spamtools,andcallsforanti-spamlaws. Foranyanti-spamtool,itisespeciallyimportantthatthealgorithmshouldgeneratevirtuallynofalsenegative,sinceeachnon-spammessagethatclassiedasaspamunderminesthecondenceoftheuser,anddecreasethelikelihoodthatanti-spamlterswillbeuseduniversally.Theeaseofuseoftheanti-spamtoolisalsoanimportantissueandthereisastrongpreferenceforautomatedalgorithms,whichneedlittleornodirectinterventionfromindividualusers[ 2 ]. Weproposeaneectivetechniquewhichcanbeeasilyimplementedbasedongraphtheoreticmethodsandthespectraldecompositionofnetworks.Networksarethemostcommonfeaturelinkingdiversesystemsrangingfromthetechnological,biological,economic,andsocialsystems[ 3 ],[ 4 ],[ 6 ],[ 7 ].Asoneexampleoftechnologicalsystems,theinternetisacomplexnetworkofcomputersandroutersconnectedbyvariouscommunicationlinks.Inasocialnetwork,nodesarehumanbeingsandedgesrepresentvarioussocialrelationshipsbetweenthesenodes[ 8 ],[ 9 ]. Themainideasinthisalgorithmarebasedontheuniquecharacteristicsofsocialnetworksandaneigen-projectionofmatrixwhichrepresentanetwork.Inoursocialactivities,almostallourcontractualdecisionsdependstronglyoninformationprovidedbyournetworksoffriends.Thereliabilityofthedecisionswemade,then,dependsheavilyonthetrustworthinessofoursocialnetworks[ 2 ].Weseemtohavedevelopedinteraction 12

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Inthischapter,werstpresentsomebackgroundaboutthebasicgraphtheoryandbriefclassicationofnetworks.Thenwebrieyintroducethesocialnetworks.Finallyweoutlinethescopeofthisstudy. 10 ].Whenfollowingconditionsaresatised,agraphG=(V;E;f)isdenedasanundirectedgraph. 13

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Weconsideranemailnetworkasanundirectedgraph.Onthecontrary,adirectedgraphGisanorderedpairG=(V;A)subjecttothefollowingconditions: AgraphisusuallywrittenG(V;E),whenV=1nisthesetofnnodes,andEisthesetofedges.Eachedgei;jisassociatedwithanon-negativeweightwijwhichshowstheadjacencyofnodesiandj.Foranunweightedgraph,weassignwij=0foranynon-adjacentpairofnodesandwij=1foranyadjacentpairofnodes.TheadjacencymatrixofthegraphGisthesymmetricnnmatrixA(G) Theadjacencymatrixofagraphisamatrixwithrowsandcolumnslabeledbygraphvertices,witha1or0inposition(vi;vj)accordingtowhetherviandvjareadjacentornot.Forasimplegraphwithnoself-loops,theadjacencymatrixmusthave0'sonthediagonal.Foranundirectedgraph,theadjacencymatrixissymmetric,wij=wji.LetusdenotetheneighborhoodofibyN(i)=fjjhi;ji2Eg.Thedegreeofnodeiisdeg(i),Pj2N(i)wij.TheLaplacianmatrixLisanothersymmetricnnmatrixthatrepresentedbythediagonalDandadjacencyAcomponents.Thediagonalmatrixshowsthenumberofconnectionsofeachvertex.LetusdenethedegreesmatrixasthenndiagonalmatrixDthatsatisesDii=deg(i). 14

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Figure1-1. ExamplesofgraphsandtheiradjacencymatrixandLaplacianmatrix. 11 ].Thisinherenttendencytoclustercanbequantiedbytheclusteringcoecient[ 25 ].Letusfocusonaselectednodeiinanetwork,havingkiedgeswhichconnecttokiothernodes.Ifthenearestneighborsoftheoriginalnodewerepartofaclique,therewouldbeki(ki1)=2edgesbetweenthem.TheratiobetweenthenumberEiofedgesthatactuallyexistbetweenthesekinodesandthetotalnumberki(ki1)=2givesthevalueoftheclusteringcoecientofnodei, 15

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wherenisthetotalnumberofnodesofthegraphG.Itshouldbenotedthattheaveragepathlengthisatopologicalmeasure,whichisofinteresttostructuralanalysisoflargenetworks. 11 ].First,randomgraph,whicharevariantsoftheErd}os-Renyimodel.Second,motivatedbytheclusteringphenomena,aclassofmodels,collectivelycalledsmall-worldmodels,hasbeenproposed.Finally,thediscoveryofthepower-lawdegreedistributionhasledtotheconstructionofvariousscale-freemodelsthatfocusedonthenetworkdynamics.Wewillbrieyrevieweachmodelofnetwork. 11 ].Traditionally,thestudyofcomplexnetworkshasbeentheterritoryofgraphtheory.Whilegraphtheoryinitiallyfocusedonregulargraphs,sincethe1950slargescalenetworkswithnoapparentdesignprincipleshavebeendescribedasrandomgraphs,proposedasthesimplestandmoststraightforwardrealizationofacomplexnetwork.RandomgraphswererststudiedbytheHungarianmathematiciansPaulErd}osandAlfresRenyi.AccordingtotheErd}os-Renyimodel,westartwithNnodesandconnecteverypairofnodeswithprobabilityp,creatingagraphwithapproximatelypN(N1)=2edgesdistributedrandomly.ThereareCn[N(N1)=2]graphswithNnodesandnedges,formingaprobabilityspaceinwhich 16

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25 ].Therstpropertyofsmall-worldisthattheseparationbetweenanytworandomlychosennodesisverysmall.Theseparationischaracterizedbythenotionofpath,whichisdenedastheshortestdistancebetweennodes.Theaveragepathlengthforasocialnetworkislikelytoreectasmalldegreeofseparation.InthedomainoftheInternet[ 12 ],thewebformsakindofasmall-worldnetworkwithseparationfrompagetopagearound19clicks.Mathematically,suchaseparationcanbedescribedbyanaveragepathlengthinEq.( 1{4 ).Thesecondpropertyofsmall-worldnetworksistheirhighdegreeofclustering.Thiscanalsobeseenfromourdailyexperiencewhere,forexample,ourfriendsarelikelytobefriendsofeachotheraswell,orinotherwords,socialnetworkstendtobeclustered.ThispropertyischaracterizedbytheclusteringcoecientinEq.( 1{3 ).Ahighdegreeofclusteringisnotapropertyofarandomgraph. 17

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1{6 ). Thedegreedistributionofnumerousnetworks,suchastheWorldWideWeb[ 12 ],[ 5 ],socialnetworks[ 13 ]andmolecularnetworks,followthepower-law[ 14 ],[ 15 ].InFig. 1-2 ,weshowthatthedegreedistributionoftheprotein-proteininteractionnetworkofthefruity,whichfollowsthepower-lawproperty.Thispropertyindicatesahighdiversity Figure1-2. Degreedistribution(Eq. 1{5 )oftheprotein-proteininteractionnetworkhas4637nodes.Degreeofthisnetworkfollowsthescale-freeproperty. ofnodedegreesandthatthereisnotypicalnodeinthenetworkthatcouldbeusedtocharacterizetherestofthenodes.Theabsenceofatypicaldegreeisthereasonthesenetworksaredescribedas\scale-free"[ 16 ],[ 17 ],[ 18 ] 18

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19 ].Thepatternoffriendshipsbetweenindividuals,businessrelationshipbetweencompanies,andmarriagesbetweenfamiliesareexampleofthesocialnetwork[ 20 ],[ 21 ],[ 22 ].Inrespectofacademicdisciplines,thesocialscienceshavethelongesthistoryofthequantitativestudyofreal-worldnetworks[ 23 ],[ 24 ],[ 25 ].Thepatternofasocialnetworkhelpsdetermineanetwork'susefulnesstoitsindividuals. Asimplenetworkanalysisconsidersocialrelationshipsintermsofnodesandlinks.Nodesaretheindividualactorswithinthenetworks,andlinksaretherelationshipsbetweentheactors.Therearemanykindsoflinksbetweenthenodes. Asthesimplestform,asocialnetworkisamapofalloftherelevantlinksbetweenthenodes.Thenetworkcanalsobeusedtodeterminethesocialcapitalofindividualactors[ 19 ].Thisconceptcanbedisplayedinasocialnetworkdiagram,wherenodesarethepointsandlinksarethelines.Socialnetworkanalysis,whichisrelatedtonetworktheory,hasemergedasakeytechniqueinmodernsociology,anthropology,sociolinguistics,geography,socialpsychology,informationscienceandorganizationalstudies,aswellasapopulartopicofspeculationandstudy[ 19 ].Fig. 1-3 showsthelargestcomponentofthecollaborationnetwork.SocialnetworkAnalystsreasonfromwholetopart;frombehaviortoattitudefromstructuretorelationtoindividual.Theyeitherstudywholenetworks,allofthelinkscontainingspeciedrelations,orpersonalnetworks,thelinksthatspeciedpeoplehave,suchastheirpersonalcommunities.Insummary,socialnetworkscanberepresentedusinganundirectedordirectedgraph.Edgesrepresentsocialrelationshipssuchasfrequencyofconversation,ratingoffriendship,phone-calls,email,andco-authorship[ 26 ],[ 27 ].Thestructuralpropertiesofasocialnetworkalsorepresenthumanrelationships-statusandpositions.Asanextendedformofasocialnetwork,bibliometricsincludesanynetworkgeneratedbyhuman,citationnetwork,andwwwhyperlinks.Theseconceptofthesocialnetworkcanbeextendedtothecyberspaceaswell,andcanbeusedtondsomeimportantfeaturesforthespamlteringtool. 19

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Exampleofsocialnetwork:Acollaborationnetworkofscientists.AfterM.GirvanandM.E.J.Newman(2002)[ 3 ]. 20

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1. Thismethodcandecomposeuniquelytheentirecomplexnetworkintoasetofsubnetworksbasedonthespectralanalysis. 2. Intherank-orderedeigenvaluespectrumofthecomplexnetwork,weclearlyobserveastraightlinepartinadoublelogarithmscaleplotwhichindicatesthatsomepartoftheeigenvaluespectrumfollowsthepowerlaw.Inaddition,thisslopeisrobustundersignicantamountofnoisewhichisrandomadditionanddeletionofedges.Thissuggeststhattheeigenvaluesandeigenvectorsobtainedfromtheproposedmethodmaybealsoconsideredrobust.Therefore,high-noisenetworkdatacanbeanalyzedproperly. 3. Itshowsthatthepersonalemailnetworkfollowsthepower-lawproperty.Thisproposedmethodshowsthat: 4. Weproposeanewmethodforndingcommunities,andwecanndcommunitystructuresinthenetworkusingsubnetworksasthebuildingblocksofthecommunitystructure.Inthisprocess,weutilizetheclusteringpropertyandcorrelationmatrixofthesubgraphstondthecommunitystructure. 5. Weproposedaneectivespamlteringalgorithmbasedonthepropertyofthesocialnetworkandthespectraldecomposition. Therestofthisproposalisorganizedasfollows.InChapter 2 ,wereviewsomemetricsfortheanalysisofthenetworkandbrieyreviewtheconventionalmethodsofthespectralpartitioningandndingcommunitystructure.InChapter 3 ,weintroducetheproposedalgorithm,spectraldecompositionofthecomplexnetworkandstudyhowtondthecommunitystructureinthenetwork.Inthischapter,wewillexplainmaincontributionofourwork,especiallyforsummary4intheaboveparagraph.Asapplications,inChapter 4 ,weapplythealgorithmtothereal-worldnetworksandanarticialnetworkandshowthattheproposedalgorithmcanbeaneectivemethodtondthecommunitystructuresofthe 21

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5 22

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12 ].Thereareseveraltypesofthecentralitymeasures; 1. degreecentrality 2. eigenvectorcentrality 3. betweennesscentrality 4. closenesscentrality Thesimplestcentralitymeasureisdegreecentrality,alsoknownasdegree.Thedegreeofanodeinanetworkisthenumberofedgesattachedtoit.Thedegreekiofanodeiis Where,AijiselementsofadjacencymatrixA. Evenitissimple,thedegreeisoftenahighlyeectivemeasureoftheimportanceorinuenceofanode.Inmanysocialsituations,peoplewithmoreconnectionstendtohavemorepowerthanotherwithlessconnections.Amorecomplicatedversionofthesameideaiseigenvectorcentrality.Contrarytothedegree,eigenvectorcentralityacknowledgesthatnotallconnectionsareequal.Ingeneral,connectionstopeoplewhoarethemselvesinuentialwilllendapersonmoreinuencethanconnectionstolessinuentialpeople.Ifwedenotethecentralityofanodeibyxi,thenwecanrepresentthiseectbymakingxi

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whereisaconstant.Deningthevectorofcentralitiesx=(x1;x2;:::),wecanrewritethisequationinamatrixformas: andhenceweknowthatxisaneigenvectoroftheadjacencymatrixwithaneigenvalue.Theeigenvectorcentralitydenedinthiswayaccordseachnodeacentralitydependsbothonthenumberandthequalityofitsconnections:havingalargernumberofconnectionsstillcountstomeasure,butanodewithasmallernumberofhigh-qualitycontactsmayoutrankonewithalargernumberofmediocrecontacts.Twoothercentralitymeasuresareclosenesscentralityandbetweennesscentrality.Botharebasedontheconceptofnetworkpaths.Apathinanetworkisasequenceofnodstraversedbyfollowingedgesfromonetoanotheracrossnetwork.Ageodesicpathistheshortestpath,intermsofnumberofedgestraversed,betweenaspecicpairofnodes.Theclosenesscentralityofnodeiisthemeangeodesicdistancefromnodeitoeveryothernode.Closenesscentralityislowerforverticesthataremorecentralinthesenseofhavingashorternetworkdistanceonaveragetoothervertices.Someverticesmaynotbereachablefromnodei-twonodescanlieinseparatecomponentsofanetwork,withnoconnectionbetweenthecomponentatall.Thebetweennesscentralityofanodeiisthefractionofgeodesicpathsbetweenotherverticesthatifallson.Thatis,wendtheshortestpathbetweeneverypairofnodes,andaskonwhatfractionofthosepathsnodeilies.Betweennessisacrudemeasureofthecontroliexertsovertheowofinformationbetweenothers.Ifweimagineinformationowingbetweenindividualsinthenetworkandalwaystakingtheshortestpossiblepath,thenbetweennesscentralitymeasuresthefractionofthatinformationthatwillowthroughionitswaytowhereveritisgoing.Inmanysocialcontextsanodewithhighbetweennesswillexertsubstantialinuencebyvirtueofnotbeinginthemiddleofthenetworkbut 24

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2-1 showsasimplegraphpartitioning.Supposeacomputationrequirestheperformanceofsomenumber Figure2-1. Simpleexampleofagraphpartitioning. ofntasks,eachtobecarriedoutbyaseparateprocess,threadrunning,orprogramononeofdierentcomputerprocessors.Typically,thereisadesirednumberoftasksorvolumeofworktobeassignedtoeachoftheprocessors.Iftheprocessorsareidentical,forinstance,andthetasksareofsimilarcomplexity,wemaywishtoassignthesamenumberoftaskstoeachprocessorsoastosharetheworkloadroughlyequally.Itisalsotypicallythecasethattheindividualtasksrequirefortheircompletionresultsgeneratedduringtheperformanceofothertasks,sotasksmustcommunicatewithoneanothertocompletetheoverallcomputation.Thepatternofrequiredcommunicationscanbethoughtofasanetworkwithnverticesrepresentingthetasksandanedgejoininganypairoftasksthatneedtocommunicate,foratotalofmedges.Intheorytheamountofcommunicationbetweendierentpairsoftaskscouldvary,leadingtoaweightednetwork,butwehererestrictourattentiontothesimplestunweightedcase,whichalreadypresentsinteresting 25

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2-2 showsclusteringofthemeshnetworkin[ 28 ].Spectralclusteringuses Figure2-2. Bestdivisionintoequal-sizedpartsfoundedbythespectralpartitioning.AfterAdameicandGlance(2005)[ 29 ]. eigenvectorsoftheLaplacianmatrixandismostconvenientlyappliedto2-wayclusteringproblemusingasingleeigenvector[ 30 ]. 3 ].Therearemanyalgorithmstodetectsuch 26

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31 ],[ 32 ],theidenticationofthecommunitystructureinnetworksremainsanopenproblem.Thespaceofpossiblepartitionsofevenasmallnetworkisverylargeindeed.Severalmethodshavebeenproposedforndingmeaningfulpartitionsinnetworksofreasonablesize.Thesemethodsvaryconsiderablyfromoneanother,notonlyintheirgeneralapproach,butalsoinsensitivityandcomputationaleort[ 33 ].Ingeneral,moreaccuratemethodstendtobeabletoexplorealargeportionofthepartitionspace,andarecomputationallyexpensive[ 34 ].Ontheotherhand,thosemethodswhichexplodeasmallerregionofthepartitionspacetendtobefaster,butasaconsequence,lessaccurate[ 35 ].Fig. 2-3 showsasimpleexampleofcommunitystructure.Ourspectraldecompositionmethodgivesnotonlyasetofsubnetworks,butalsothefoundationfordetectingcommunitystructureinnetworks.Thefundamentalproblemwithallthesespectralpartitioningmethodssuchas,min-cut,max-cut,andratio-cutarethatcutsizesaresimplynottherightthingtooptimizebecausetheydonotaccuratelyreecttheintuitiveconceptofnetworkcommunities. Toquantifyhowgoodaparticularpartitionsis,themodularitymeasureQwasintroduced[ 36 ].Ithasbeenwidelyusedandmanywell-knownalgorithmsforndingcommunitystructureinnetworksarebasedonthemaximizationofthemodularity.Onapredenedsetofcommunitiesfinanetwork,acommunityconnectionmatrixefgisdened,whereeachmembertheproportionoflinksfromcommunityftocommunityg.Thematrixisnormalized,thatis,eachofthemembersofthematrixefg=Lfg=Ltotal,Lfgbeingthenumberoflinksbetweencommunityfandcommunityg,andLtotalisthetotalnumberoflinksinthenetwork[ 36 ].Theproportionoflinksbelongingtocommunityfisdenotedafandissimplythesum,af=Pfefg.ThecomputationofQisasfollows 27

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Schematicrepresentationofanetworkwithcommunitystructure.Inthisnetworktherearefourcommunitiesofdenselyconnectednodes,withamuchlowerdensityofconnectionsbetweenthem. Themodularity,Q,quantiesthedierencebetweentheintra-communitylinksandtheexpectedvalueforthesamecommunitiesinarandomizednetwork.Themodularityisarelativevalue,andwhileitgivesmeasureofhowgoodapartitionofthenetworkis,itcannotshowuswhetherthispartitionisthebestpossiblepartition. Agooddivisionofanetworkintocommunitiesisnotsimplyoneinwhichthenumberofedgesrunningbetweengroups,intra-grouplinks,issmall.Infact,itisoneinwhichthenumberofedgesbetweengroupsissmallerthanexpected,forasamesizerandomizednetwork.Onlyifthenumberofbetween-groupedgesissignicantlylowerthanwouldbeexpectedpurelybychancecanwecorrectlyclaimtohavefoundasignicantcommunity 28

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36 ].TheseconsiderationsgiveusamodiedbenetfunctionQ,notbasedonpurecutsize,denedas Thisbenetfunctionisalsocalledmodularity[ 36 ].Itisafunctionoftheparticulardivisionofthenetworkintogroups,whichindicatesstrongercommunitystructurewhenithaslargervalues.Inprinciple,wecanndgooddivisionsofanetworkintocommunitiesbyoptimizingthemodularityoverpossibledivisions. TherstterminEq.( 2{6 )iseasytocalculate.However,thesecondtermisrelativelynotclearandneedstobepreciselydenedbeforewecancalculatethemodularity.Tondtheexpectednumberofedgeswithincommunity,wechooseanullmodelagainstwhichtocomparethenetwork.Thedenitionofthismodularityneedsacomparisonofthenumberinsomeequivalentrandomizedmodelnetworkinwhichedgesareplacedwithoutregardtocommunitystructureandthenumberofwithin-groupedgesinarealnetwork.Thisisoneoftheadvantagesofthismodularitythatcanmaketheroleofthisnullmodelclearandexplicit.Thisnullmodelmusthavethesamenumberofverticesnastheoriginalnetwork,sothatwecandivideitintothesamegroupsforcomparison.ThereisalotoffreedomtochooserandomizedmodelsinwhichwespecifytheprobabilityPijforanedgetofallbetweeneverypairofverticesi,j.GivenPij,themodularitycanbedenedasfollows.TheactualnumberofedgesfallingbetweenaparticularpairofverticesiandjisAij,andtheexpectednumberisPij.ThustheactualminusexpectednumberofedgesbetweeniandjisAijPijandthemodularityisthesumofthisquantityoverallpairsofverticesbelongingtothesamecommunity.Letusconsidergitobethecommunitytowhichvertex 29

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2mXij[AijPij](gi;gj)(2{7) Where(f;g)=1,iff=gand0otherwiseandmisthenumberofedgesinthenetwork.Thescalefactor1 2misneededforcompatibilitywiththepreviousdenitionatEq.( 2{5 ).Whenweconsideronlyundirectednetworks,weknowthatPij=PjiandQ=0whenallverticesareplacedinasinglegrouptogether.Inthiscase,wesetallgiinEq.( 2{7 ),andndthatPij[AijPij]=0as Therearemanypossiblechoiceofnullmodelandvariousnullmodelshavebeenconsideredpreviouslyintheliterature[ 34 ],[ 36 ],[ 37 ].Thesimplestisthestandardrandomgraph,inwhichedgesappearwithequalprobabilityPij=pbetweenallvertexpairs,butmanyauthorshaveclaimed[ 34 ],[ 36 ],[ 37 ]thatthismodelisnotagoodrepresentationofmostreal-worldnetworks.Theyusuallyconsidertheactualdegreeofthevertexintherealnetworkandassumetheindependenceoftheprobabilitiesforthetwoendsofasingleedge.TheexpecteddegreeofvertexiisgivenbyPiPij=ki.TheexpectednumberofedgesPijbetweenverticesiandjistheproductf(ki)f(kj)ofaseparatefunctionofthetwodegrees,wherethefunctionmustbethesamesincePijissymmetric.Then foralliandhencef(ki)=CkiforsomeconstantC.CombiningwithEq.( 2{8 )resultsin 2m=XijPij=C2Xijkikj=(2mC)2(2{10) andhenceC=1 30

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38 ].Itisalsocloselyrelatedtothecongurationmodel,whichhasbeenstudiedwidelyinthemathematicsandphysicsliterature[ 38 ],[ 39 ].Letusconsideranalgorithmforndingthecommunitystructurebasedonthemodularity.Ifaparticulardivisiongivesnomorewithin-communityedgesthatwouldbeexpectedbyrandomchancewewillgetQ=0.Valuesotherthan0indicatedeviationsfromrandomness,andinpracticevaluesgreaterthan0.3appeartoindicatesignicantcommunitystructure[ 36 ].Thisfactalsosuggestsanalternativeapproachtondingcommunitystructure[ 35 ].IfahighvalueofQrepresentsagoodcommunitydivision,thebestonecanbefoundbysimpleoptimizingQoverallpossibledivision.TheproblemisthatthetrueoptimizationofQisverycostly.Thenumberofwaystodividenverticesintognon-emptygroupsisgivenbytheStirlingnumberofthesecondkindS(g)n,andhencethenumberofdistinctcommunitydivisionisPng=1S(g)n.Thissumisnotknowninaclosedform,butweobservethatforalln>1,sothatthesummustincreaseatleastexponentiallyinn.ToperformanexhaustivesearchofallpossibledivisionsfortheoptimalvalueofQwouldthereforetakeatleastanexponentialamountoftime,andisinpracticeinfeasibleforsystemslargerthantwentyofthirtyvertices[ 35 ].Variousapproximateoptimizationmethodsareavailable:geneticalgorithm,simulatedannealing,andsoforth.Newman[ 35 ]considersaschemebasedonastandardgreedyoptimizationalgorithm.Thisalgorithmfallsinthegeneralcategoryofagglomerativehierarchicalclusteringmethods[ 19 ],[ 40 ].Startingwithastateinwhicheachnodeistheonlymemberofoneofncommunities,theyrepeatedlyjoincommunitiestogetherinpairs,choosingateachstepthejointhatresultsinthegreatestincrease(orsmalldecrease)inQ.Theprogressofthealgorithmcanberepresentedasadendrogram,atreethatshowstheorderofthejoins.Fig. 2-4 showsasimpleexampleofadendrogram.CutsthroughthisdendrogramatdierentlevelsgivedivisionofthenetworkintosmallerorlargernumberofcommunitiesandwecanselectthebestcutbylookingforthemaximalvalueofQ.Sincethejoiningofapairof 31

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DendrogramofthecommunitystructureofKarateclub.Thenumbersatthebottomrepresentthenodesinthenetwork,andthetreeshowstheorderinwhichtheyjointogethertoformcommunities. communitiesbetweenwhichtherearenoedgesatallcanneverresultinanincreaseinQ,weneedonlyconsiderthosepairsbetweenwhichthereareedges,ofwhichtherewillatanytimebeatmostm,wheremisagainthenumberofedgesinthegraph.Whenweconsideranetworkthathasbeenpartitionedinsomearbitraryway,thechangeinQisgivenby whichcanclearlybecalculatedinconstanttime.Thiscanbeinterpretedasameasureofanityofcommunitiesfandg,andcansubsequentlybeusedtondthetwocommunitieswhicharemostalike(highestdQ).Followingajoin,someofthematrixelementsefgmustbeupdatedbyaddingtogethertherowsandcolumnscorrespondingtothejoinedcommunities,whichtakesworstcasetimecomplexityofO(n).ThuseachstepofthealgorithmtakesworstcasetimecomplexityofO(m+n).Thereareamaximumofn1joinoperationsnecessarytoconstructthecompletedendrogramandhencetheentirealgorithmrunsintimeO((m+n)n),orO(n2)onasparsegraph.WecompareourproposedalgorithmforndingcommunitystructurewiththisalgorithminapplicationpartinChapter 4 32

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Thecontributionofourworkistheintroductionofthespectraldecompositionalgorithmthatcanproducegoodseparationofnetworksintheprojectivespaceandamethodforndingcommunitystructureinthenetwork.Thiscanbeagoodmethodforreal-worldnetworkdataanalysis,especiallyforthescale-freenetworks. Fromtheearlydaysofspectralgraphtheory,matrixandlinearalgebrawereusedtoanalyzedthematrixrepresentationofgraphs.Therearemanypublicationsaboutonspectralgraphtheory[ 41 ],[ 42 ],[ 43 ],[ 44 ],[ 45 ],[ 46 ],[ 47 ],[ 48 ].Basedontheseresults,spectralclusteringbecameapopularapproachfordataclustering,whichincludesaclassofclusteringmethodsthatuseeigenvectorsoftheLaplacianofthesymmetricmatrixW=(wij)containingthepairwisesimilaritybetweendataobjectsi;j.Thespectralpartitioningalgorithmshavetwoobviousshortcomings.First,itbasicallydividesnetworksintotwoclusters,itisextendedtomulti-wayclusteringthroughrecursive2-wayclustering,forreal-worldnetworkscancertainlyhavemorethantwoclusters.Second,itmakesuseonlyoftheleadingeigenvectoroftheLaplacianmatrixandignoresalltheothers,whichthrowsawayusefulinformationcontainedinthoseothervectors[ 49 ]. Ourmethodalsocanbeusedtondthecommunitystructureinanetwork.Therearemanyalgorithmstodetectsuchcommunitiesandalmostallofthesemethodsarebasedonaheuristicapproach.Ourspectraldecompositionmethodgivesnotonlyasetofsubnetworks,butalsothefoundationfordetectingcommunitystructureinnetworks.Hereisabriefoutlineofthischapter.Therstsectionbeginswiththebriefexplanationofthespectralanalysisandthepower-lawpropertyofthescale-freenetwork.Then,weintroduceaspectraldecompositionmethodandanalgorithmtondthecommunitystructureinthenetwork,whichutilizestheresultsfromthespectraldecomposition. 33

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3{1 ). Theempiricalresultsdemonstratethatmanylargenetworksarescalefree,thatis,theirdegreedistributionfollowsapowerlaw[ 15 ],[ 12 ].Therank-orderedeigenvaluespectrumofmanyreal-worldnetworkdataalsofollowsthepower-lawdistribution.Thepower-laweigenvaluespectrumthusprovidesanobjectivewaytodeterminethedimensionofthenetwork.Therank-orderedeigenvaluespectrumoftheprotein-proteininteractionnetworkisshowninFig. 3-1 ,whereweclearlyobserveastraightlinepartinadoublelogarithmscaleplotwhichindicatesthatsomepartoftheeigenvaluespectrumfollowsthepowerlaw.Inaddition,thisslopeisrobustundersignicantamountofnoisewhichisrandomadditionanddeletionofedges.Thissuggeststhattheeigenvaluesandeigenvectorsobtainedfromtheproposedmethodmaybealsoconsideredrobust.Therefore,highlynoisynetworkdata,suchasprotein-proteininteractionnetworkobtainedviahigh-throughputexperiments,canbeanalyzedproperly.Fromthisfact,weknowthattheeigenvaluesandeigenvectorsofthenetworktobeanalyzeddonotshowabigdierencewithnoise.Forthisreason,weassumethattheproposedalgorithmcanberobusttonoisydata.Aswementioned,thestructureofanetworkcanbecompletelydescribedbytheassociatedadjacencymatrix.Theadjacencymatrixofanundirectedgraphissymmetricanditselementsareequaltonumberofedgesbetweenthegiven 34

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Rank-orderedeigenvaluespectrumwithnoise.Thislog-logplotshowsthepower-lawpropertyofthePPInetwork.Notethattheslopeofthestraightpartisresilienttorandomnoise.From20%upto60%oftotalinteractionsarereplacedwithrandomadditionordeletion. vertices.Theeigenvaluesofanadjacencymatrixarerelatedtobasictopologicalinvariantsofnetworksuchas,thediameterofanetwork[ 50 ],[ 51 ].Recently,itwasproposedtoconsiderspectraofeigenvaluesoftheadjacencymatrixasangerprintofthenetworks[ 52 ].Thecharacteristicpolynomialdet(AI)oftheadjacencymatrixAofGiscalledthecharacteristicpolynomialofGanddenotedbyPG(x).TheeigenvaluesofAandthespectrumofA,whichconsistsoftheneigenvalues,arecalledtheeigenvaluesandthespectrumofG,respectively:thesenotionsareindependentofvertexlabeling.Clearly,isomorphicgraphshavethesamespectrum.Theadjacencymatrixissymmetricandnon-negativeinthecaseofundirectednetworksandaccordinglyhasrealeigenvaluesj,j=1;:::N,beingsolutionsofdet(AI)=0.Therelationbetweenfeaturesofnetwork 35

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FollowingFarkasetal[ 51 ]wedenescaledvariablesand = (=p wherep= 3{2 )ofstatescanbecomputedanalyticallyfromrandommatrixtheoryandtheresultistheso-calledWigner'ssemicir-cularlaw[ 53 ].Fig. 3-2 showsthedensityforarandomnetworkwhichhassamenumberofinteractionswiththeprotein-proteininteractionnetworkinFig. 3-3 .Fig. 3-3 shows()fortheprotein-proteininteractionnetworkoffruity(Drosophilamelanogaster)[ 54 ]andhasadistinctbehavior,havingtheemergenceofpeaksatspeciceigenvalues.Usingtheseproperties,wemayknowhowtheprevalenceofspecicpeaksinthediscretespectrumofanetworkreectsthenetworks'topologiesandrelatestootherconceptslikethesearchformotifs[ 55 ],[ 56 ].Butthereareproblemsinthisapproach.First,subgraphsarenotgenerallyrepresentedbytheireigenvaluesinthespectrumofthewholenetwork.Second,isospectralgraphs,whicharesamespectraldensity,arenotnecessarilyisomorphic[ 50 ],[ 57 ].Thisspectralmethodisaindirectedapproachandnoteectivetorevealthestructuralpropertyofanetwork. 36

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Discretespectrumoftheclassicalrandomnetwork,whicharesamesizeoftheprotein-proteininteractionnetwork.Thenumberofnodesis4555 arecomputedfromeigenvaluesandeigenvectorsoftheLaplacianmatrix.Thisprocessisbasedontheorthogonalprojectiononthebasisvector,asanadditionalaccomplishment,wealsointroduceamodiedcentralitymeasureusingeigenvectorsoftheLaplacianmatrix.Asthesecondstep,weintroducethemethodforndingthecommunitystructureinthenetworkwithsubgraphsthatweobtainatthespectraldecomposition.Fig. 3-4 showstwomainstepsofourmethod.Weknowthateigenvectorscorrespondingtothoseeigenvaluescontainalotofdistinguishedinformationonsubnetworks.Thereisastudythatusedeigenvaluesandeigenvectorstosuccessfullyndproteincodingsequence[ 58 ]. 37

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Discretespectrumofaprotein-proteininteractionnetwork(FruitFly).Itshowssomespecicpeaks.Thenumberofnodesis4555. ndstructurallysimilargraphs.Twographsaredeemedtobeisomorphicwhentheyhavethesameeigenvaluespectrum.Thismethodisimperfectsincecospectralnon-isomorphicgraphsexist,inotherwords,eveniftwographshavesameLaplacianeigenvaluespectra,thesegraphscannotbeisomorphicgraphs.ForLaplacianspectra,themethodfailslessthan10to15percentofthecases.Thetopologicalpropertiesofgraphscanbeanalyzedusingspectralgraphtheoryandtheeigenvaluesarerelatedtotheconnectivitypatternofthegraph.Specically,thesecondsmallesteigenvalueisameasureofthecompactnessofagraph.Alargesecondeigenvalueindicatesacompactgraph,whereasasmalleigenvalueimpliesanelongatedtopology[ 31 ].Weshowedthattheprevalenceofspecicpeaksinthediscreteadjacencyspectrumofanetworkreectsthenetwork'stopologiesintheprevioussection 3.1 .ThespectralanalysisbasedonLaplacianspectraisamorepopular 38

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Twomainstepsoftheproposedalgorithms.SpectralEdgepartitioningandFindingcommunitystructure. approachthantheanalysisusingtheadjacencyspectrum.Insummary,theconventionaleigenanalysisofthecomplexnetworksusesthediscretespectrumtondindirectfeatures,theemergenceofspectralpeaks,andtorelatethespectralpropertiestothenetworks'topologiesinanindirectmanner.Similartootherapproachesofthespectralmethod,theconventionalspectralpartitioningmethodsonlyuseafeweigenvectorsforclustering.Onthecontrary,intheproposedmethod,wecanobtainthesubnetworksbasedontheprojectiononeverysignicanteigenvectorinthenetwork.Itispossiblethattheentireoriginalcomplexnetworkcanberepresentedbythesummationofsubnetworks.Thereisnomissingorredundantedgeinallsubnetworksandthesesubnetworksareresultsoftheedgepartitioning. Ouralgorithmisbasedontheorthogonalprojectionofnetworkdataonbasisvectors,whicharecomputedfromeigenvectorsoftheLaplacianmatrix.Werecastthespectraldecompositionofnetworkdataasanapproximationoftheentirenetworkbythesummationoflow-rankmatrices,whichhavethesameformatwiththematrixfortheentirenetwork.Iftheadjacencymatrixisusedtorepresenttheentirenetwork,thelow-rankmatrixalsohasthesamestructurewithanadjacencymatrixsuchaszerodiagonalterms. 39

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=QTAQ(3{4) isadiagonalmatrix.Here=diag(1;2;;m),andthecolumnofQarecorrespondingeigenvectorswhichformanorthonormalbasisofRm. (3{5) (3{7) Eq.( 3{7 )and( 3{8 )arethespectraltheoreminmathematicsandtheprincipalaxistheoremingeometryandphysics[ 59 ].Therepresentedmatrixofthenetworkcanbeconsiderasatransformationmatrix,ifwemultiplythismatrixontheleftofabasisvector,theanswerisanothervectorthatistransformedfromit'soriginalposition,suchasAq=q.Itisthenatureofthetransformationthattheeigenvectorsarisefrom. Where,MistherankofA.Eq.( 3{9 )isthegreatfactorizationQQT,writtenintermsof'sandq0s.Eq.( 3{8 )representsthespectraldecompositionandthisdecompositionalsocanperformthedimensionalityreductionifweusem,whichislessthantherankofthematrixA.Principalcomponentanalysis(PCA)fordataanalysisusuallyusestherstfewlargesteigenvaluesandassociatedeigenvectorstoreducethedimensionalityofthesystem.IncaseoftheconventionalPCA,thematrixusedforeigenvectorsiscomputedfromthecovariancematrixofdata.Inthisstudy,weuseanadjacencymatrixoraLaplacianmatrixofnetworkdataforthespectraldecomposition.First,wecomputeeigenvectors 40

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Projectiononthebasisvectorhas3axescomponents,andtheresultisexpressedwithe1,e2,ande3. andeigenvaluesofthematrix,thenconsidereigenvectorasbasisvectors.AsinEq.( 3{10 ),projectionsofthematrixonthebasisvectorareperpendiculartoeachotherbecauseofthepropertyofeigenvector.Wecanexpressthematrixintermsoftheseperpendiculareigenvectors,insteadoftheoriginalmatrixform.InFig. 3-5 ,wecanseetheprojectionofthematrixonabasisvectorlaysonthevectorandthisisajustscaledbasisvector,notchangingitsdirection. EachEigenvaluetellswhethertheprojectedvectorisstretchedorshrunkorreversedorleftunchangedwhenitismultipliedbyA.FromEq.( 3{9 )and( 3{10 ),thespectraltheoremforsymmetricmatricessaysthatAisacombinationofprojectionmatrix.Pi=qiqTiisaprojectionmatrix.Eachprojectionmatrixcanbeconsideredasalow-ranked 41

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3{7 )and( 3{8 )canberepresentedbytheprojectionmatrixPsatisedP2=P.EverysymmetricmatrixA=AThasthefactorizationQVQTwithrealdiagonalVandorthogonalmatrixQ.Theeigenvaluesofarealsymmetricmatrixarerealandeigenvectorsofarealsymmetricmatrix,whentheycorrespondtodierent,arealwaysperpendicular. Togetthevalidformofsubgraphsfromthespectraldecompositionoftheentirenetwork,wehavetoconsiderafundamentalissue: Fortheissueofthisstudy,weconsidertheeigendecompositionofthematrix.Since,theeigenvectorsoftheoriginaladjacencymatrixareorthonormal,andthespacespannedbytheouterproduct,resultsinprojectionmatrix,ofthemselvesarealsoorthogonal[ 59 ].Hence,theweightedprojectionmatricesdenedbytheouterproductofeacheigenvector,multipliedbytheeigenvalue,willtendtohaveexclusivevaluesforelementsintherespectiveposition.Weutilizethisexclusivenessdirectlyandconverttheweightedprojectionmatrixtotheadjacencymatrixofasubnetwork.Thereforethesummationofeveryadjacencymatrixofthesubnetworkbecomestheadjacencymatrixoftheentirenetwork.Nowletusshowasimpleexampleoftheeigendecompositionofthenetwork. ThismatrixAissymmetricandhaszerodiagonalterms.BecauseofsymmetryoftheadjacencymatrixA,ithasonlyrealeigenvaluesandeigenvectorsofAcanbechosen 42

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EigenvaluesofthisAare1=1;2=1;3=2,andprojectionsofAontheeigenvectorsare Q=2666640:71520:39381:15470:01670:81631:15470:69870:42251:1547377775 Projectionsforeacheigenvectorare1q1=[0:7152;0:0167;0:6987] ,2q2=[0:3938;0:8163;0:4225],and3q3=[1:1547;1:1547;1:1547],respectively.Elementsofeachprojectioncorrespondstoe1,e2,ande3inFig. 3-5 .Fromthesevalues,wecanndtheeigenvectorbasisvector,whichisthebiggestcontributortoeachnon-zeroelementintheadjacencymatrixA.Tondthiseigenvector,wecomputetheweightedprojectionmatrixofeacheigenvectorinEq.( 3{13 ). 43

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Since,thenon-diagonalelementsintheadjacencymatrixAarepositive,weexpectconnectednodeshavesamesignedvaluesinprojections.WeknowthatA=1P1+2P2+3P3,butfortheeigendecomposition,eachweightedprojectionmatrixcanbeconvertedtothesimilarformoftheadjacencymatrix,suchaszerodiagonalterms.LetusconsiderP3.Allofelementsofthismatrixhavethesamevalue(0.6667).WhenwecomparethesevalueswiththeedgeinthematrixA,samevaluesareassigntotwodierentstatus,1and0intheoriginaladjacencymatrixA(A13=1,A33=0).Withthisapproach,wecannotdecomposethismatrixintopropersubnetworks.Fromthesefacts,whenweusetheadjacencymatrixfortheeigendecomposition,wechooseeigenvalues,whichcompensateeachothertogetapproximatednulldiagonalterms.Nowonecanpickthoseeigenvectors;onecould,forexample,usejusttwoeigenvectorscorrespondingto1;orfoureigenvectorscorrespondingto1;2,etc. 2[positivej (3{20) *UsingtheLaplacianMatrixInEq.( 3{22 ),weknowapositiveeigenvalueandacorrespondingnegativeeigenvaluedonotalwaysperfectlycompensatedeachother.Duetothisimbalanceproblem,therearesomeerrorsinthereconstructedadjacencymatrix,whenn
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ThecorrespondingDandAvaluesareasfollows:Eachcolumnandrowintheabovematrixcorrespondstothegraph'svertices.Thecorrespondingadjacencymatrixspeciestheseconnectionsexplicitly.TheLaplacianmatrixL(G)ofagraphG,whereG=(N;E)isanundirected,unweightedgraphwithoutgraphloops(i;i)ormultipleedgesfromonenodetoanother,Disthedegreematrix,Nisthevertexset,andEistheedgeset,isanjNjjNjsymmetricwithonerowandcolumnforeachnode.Similartotheadjacencymatrix,theLaplacianmatrixisalsorepresentedbythesummationofweightedprojectionmatrices. Asincaseoftheadjacencymatrix,wecanexpresstheLaplacianmatrixintermsofperpendiculareigenvectorssuchas Wealreadyknowthattheprojectionmatrixonabasisvector,eigenvectorinthisstudy,laysonthevectoranditisajustscaledbasis.FromprojectionsforeacheigenvectorinEq.( 3{27 ),wecanndtheeigenvectorbasisvector,whichisthebiggestcontributortoeachnon-diagonalelementintheLaplacianmatrixL.Atthispoint,weintroducemodiedeigenvectorcentrality.Itissimilartotheeigenvectorcentrality,whichisdenedintheprevioussection 2.1 .Contrarytothedegreecentrality,eigenvectorcentralityacknowledgesthatnotallconnectionsareequal.Ingeneral,connectionstopeoplewhoarethemselvesinuentialwilllendapersonmoreinuencethanconnectionstolessinuentialpeople.Letusdenethemodiedeigenvectorcentrality,whendenotethecentralityofnodeibyxi,thenwecallrepresentthiseectbymakingxiproportional 45

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Deningthevectorofcentralitiesx=(x1;x2;:::).Since,thenon-diagonalelementsintheLaplacianmatrixarenegative,weexpectnodes,whichhaveconnections,havedierentsignsintheprojectioninEq.( 3{25 ).Themodiedeigenvectorcentralitydenedinherenotonlyaccordseachnodeacentralitydependingbothonthenumberandthequalityofitsconnectionsbutalsoprovidesinformationaboutthelinkssuchastwonodesmusthavedierentsignstohaveconnectionseachother.Tounderstandthisconceptandmethod,weconsideragaintheprevioussimpleexample,thistimeusingtheLaplacianmatrix. BecausethismatrixLisasemi-positivedenitematrix,eigenvaluesofLare1=0;2=3;3=3,andprojectionsofLontheeigenvectorsare Q=26666400:80182:314602:40540:462901:60381:8516377775 Projectionsforeacheigenvectorare1q1=[0;0;0],2q2=[0:8018;2:4054;1:6038],and3q3=[2:3146;0:4629;1:8516],respectively.Elementofeachprojectionisalsocorrespondingtoe1,e2,ande3inFig. 3-5 ,respectively.Onthecontrarytotheadjacencymatrix,wemustconsiderthesignofnodestondtheeigenvectorbasisvector,whichisthebiggestcontributortoeachnon-zeroelementinthenetwork.Forexample,node1andnode3can'tbelinkedintheprojection,2q2,becausebothnodeshavepositivevaluesinthisprojection.Tondeigenvectors,whichcancontributetoedgesin 46

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3{13 ). Weexpectnodes,whichhaveconnections,havethemostnegativevaluesintheprojectionmatrix.Inthiscase,therearemultipleordereigenvalues(2=3=3).Weknowthatanyrealsymmetricmatrixarediagonalizableandtherearealwaysenougheigenvectors(equaltoorderofmatrix)andtheseeigenvectorsareorthogonal.Buteigenvectorsassociatedwithrepeatedeigenvaluesarenotuniqueeventhesearelinearlyindependent.Whenwehaverepeatedeigenvalues,wemustconsidertheseeigenvectorstocomputetheweightedprojectionmatrixinEq.( 3{25 ).Thisconsiderationisexpressed Wherei=j.Whenweapplythisconsiderationtoaboveexample,twoweightedprojectionmatrices,2P2and3P3,mustbeaddedeachother. 47

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AboveLaplacianmatrixLisforagraphinFig. 3-7 .AsinEq.( 3{27 ),wecomputethe Figure3-6. Projectionsoneigenvectors(iqi),For5,4,3,and2. projectionsoneigenvectors,then,ndtheeigenvectorbasisvector,whichisthebiggestcontributortoeachnon-diagonalelementintheLaplacianmatrixL.Fig. 3-6 showsprojectionsofLontheeigenvectorssuchas5q5;4q4;3q3;and2q2.Edgebetween 48

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3{25 ).Then,comparethevaluesofeachweightedprojectionmatrix,whicharecorrespondingtonon-zeroandnon-diagonaltermsinthelaplacianmatrix.Theweightedprojectionmatrixthathasthemostnegativevaluearetheeigenvectorforthespecicedge.Thisprocedurealsobesummarizedbythepseudocoderepresentationinthealgorithm1.WhenthenumberofedgesinthenetworkisEandthenumberofsubgraphsisM,there ifPij
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Absolutevaluesofthedierenceforedgesineachprojection. Edges1q12q23q34q45q5 13----5.902 14----5.902 15----5.902 23--3.04.078225-0.858-5.767445--3.04.0782summarizedbythepseudocodeinthealgorithm2.Insummary,foralledgesintheentire ifMaxofe
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ExampleofthedecompositionusingtheLaplacianmatrix.Thisgraphcanbedecomposedtotwosubgraphs. 2.3 ,thepropertyofacommunityisaveryimportantconceptinthisstudy.Weproposeamethodtondthecommunitystructureinthenetwork,asthesecondstepofourmethod.Inthisapproach,wepresentaframeworktoidentifycommunitymodulesfromnetworksbymergingsubgraphs,theresultoftherststepintheprevioussection.Wechoosesubgraphsandcombinethemtondwhichcombinationissuitableforndingthecommunitystructureinthenetwork.Thepreviousapproacheshavefocusedonlyonthemostsignicantpositiveandnegativeeigenvalues 51

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60 ].Theirapproachisaboutclusteringinunsupervisedbottom-upnetworkanalysismethodandtheyfoundmanyquasi-cliquestypecluster,whichcanbeusedtopredictthefunctionofuncharacterizedproteins.Butitallowsoverlappingofcomponentsinclusteringandndsonlyquasi-cliquesandquasi-bipartitetypeclusters.Incontrast,wecandecomposeuniquelytheentirecomplexnetworkintoalinearcombinationsetofsubnetworks.Weknowthatthesubgraphsarecomputedfrommutuallyorthogonalprojectionmatricesthatarelinearlyindependenteachother.Withtheprocessofselectionandsummationofthesubgraphs,wecanndthecommunitystructureinthenetwork.Anothercontributionofthismethodforndingcommunitystructureisthatweusemodiedeigenvectorcentralitytoclassifynodes.Intheconventionalapproach,thewell-knownbenetfunctionforndingcommunitystructureisthemodularitymeasureQandwebrieyreviewedinsection 2.3 .Thebasicconceptofthismeasureistondthepartitioninwhichthenumberofedgesbetweengroupsissignicantlylowerthanwouldbeexpectedpurelybychance.Inthiscase,wecanclaimtohavefoundasignicantcommunitystructureinthenetwork.Toapplythismodularitymeasure,theremustbeedgesbetweengroups,whicharebasedonnodepartitioning.Inrespecttotheclusteringofnodes,ourdecompositioncanbethoughtofassoftclusterings,whereeachdatumisassignedtomultipleclusterswithmembershipweightsthatsumtoone.Onthecontrary,ourdecompositionalsocanbeviewedasarigidclusteringincaseofedges.Eachsubgraphdoesnotshareedgeswithothersubnetworks.Becauseofthesefact,wecannotapplytheconventionalmodularitytondcommunitystructuresusingsubgraphs[ 3 ].Insummary,toobtainsubgraphsisbasedonanedgepartitioning,anditisalsoahardclustering,thereisnoshared-edgesbetweensubgraphs.Wecannotdirectlyusethemodularitymeasuretondthecommunitystructure.Insteadofthemodularitymeasure,weuseanotherapproachbasedontheintuitiveconceptofcommunities.Wecanexaminethenumberofedgesandthetransitivitywithingroupsandlookforthecombinationofsubgraphswhichincreasesthenumberofedgesandtheclusteringcoecientofeachgroups. 52

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2.3 .Weusethedegreeandtransitivitytondoptimizedcommunitystructureinthenetwork.Foranalyticprocess,wecomputethecorrelationmatrixofsubgraphs.WedenethecorrelationofsubgraphsasinEq.( 3{36 ). InthisEq.( 3{36 ),thecorrelationhasmaximumvalue1,whensubx=suby.Wealsoexpressthecorrelationbetweensubgraphsastheinnerproductofthemembershipvectorforsubgraphs,whichrepresenttheconnectionofnodesineachsubgraphasinEq.( 3{37 ). 53

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3{38 )and( 3{39 ). Aswementionedpreviously,weusetheintuitiveconceptofcommunitytondcommunitystructure,increasingofthenumberofdegreeforedgesandthetransitivity(theclusteringcoecient)withingroups.Wehavetolookforthecombinationofsubgraphs,whichincreasethedegreeofedgesandtheclusteringcoecientofeachgroups.ThemembershipvectorofsubgraphsdenedinEq.( 3{38 )and( 3{39 )isnotsuitableforthispurpose,becauseitdoesnotcontaintheimportanceofnodes,suchascentrality.Letusconsideranotherrepresentativevectorofsubgraphs.Weconsiderthedegreeofnode,ifanodehasahigherdegreethanothers,thisnodehasabiggerweight.WedenethedegreevectorofeachsubgraphasinEq.( 3{40 ). 54

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3{41 )showsthedenitionofthecorrelationmatrixofsubgraphs.Inthismatrix,thediagonaltermsarealways1andthiscorrelationalsoconsiderthedegreeofnodes. Fig. 3-8 showsacorrelationmatrixofallsubgraphsinanetwork,whichhas34nodes.Usingthecorrelationmatrix,wecanndthebestmatchingsubgraphofeachsubgraph.Fromsetofpairedsubgraphs,wecaninferthecommunitystructureinthenetwork.Let Figure3-8. Correlationmatrixofsubgraphsinasocialnetwork,whichhas34nodes.XandY-axisrepresenttheindexofsubgraphs. usconsideranexampleofndingthematchedsubgraphs.WendthematchedsubgraphofsubgraphainFig. 3-9 .First,wecomputecorrelationvaluesbetweensubgraphaandb,andbetweensubgraphaandc,usingtwotypesofthevectorinEq.( 3{39 )and( 3{40 ).InFig. 3-10 ,subgraph4,1,and5refersubgrapha,b,andcinFig. 3-9 ,respectively. 55

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Findingthebestmatchedsubgraphofsubgrapha.Subgrapha,bandcrefersubgraph4,1,and5inFig. 3-10 ,respectively. Whenweusethemembershipvector,whichdoesnotconsidertheimportanceofnodes,thecorrelationvaluebetweensubgraph4andsubgraph1ismaximum,sosubgraph4andsubgraph1arethematchedpair.Ontheotherhand,ifweusethedegreevectorthatconsiderthedegreeofnodesinsubgraph,subgraph5isthematchedsubgraphofsubgraph4.Tocheckthevalidityofthedegreevector,wecomputetheincrementoftheclusteringcoecientforeachmergedsubgraph.InFig. 3-11 ,subgraphpair4and5hasthemaximumclusteringcoecientanditcoincideswiththemaximumpointofthe 56

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Comparisonoftwotypesofthevectorrepresentationofsubgraph.MembershipvectorisEq. 3{38 ,degreevectorisEq. 3{40 .X-axisrepresentstheindexofsubgraphs. Table3-2. Clusteringcoecientoftwocombinedsubgraphs. CombinedSubgraphsClusteringCoecientAveragedDegree correlationwiththedegreevector,sowecanseethedegreevectorismoresuitableforndingthecommunitystructure.Thepseudocodeinthealgorithm3showstheprocessofmergingofsubgraphstondcommunitymodulesinanetwork.Bycomparingtheaverageddegreeandthetransitivityfortwocombinedsubgraphs,wedecidewhichvectorrepresentationofsubgraphismoresuitableforndingthecommunitystructureinthenetwork.Fromthetable 3-2 ,theclusteringcoecientofsubgraph4and5isbiggerthanothers.Subgraph4and5aremorelikelytobepartofcommunitymodulethansubgraph4and1.Thisshowsthatthedegreevectorrepresentationofsubgraph,whichconsider 57

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Comparisonofcorrelatedvaluesandclusteringcoecientsforeachpairofsubgraph.X-axisrepresentstheindexofsubgraphs Figure3-12. Combinesubgraph4(a)and1(b),Combinesubgraph4(a)and5(c) 58

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forj=1tonodesndo endfor forj=1tonumberofsubgraphsmdo endfor 61 ].Thisfunctioncanbeexpressedas c(k)=Xk0;k00P(k00;k0jk)pk0;k00(3{42)P(k00;k0jk)istheconditionalprobabilitythatvertexofdegreekissimultaneouslyconnectedtotwoverticesofdegreesk0andk00.Andpk0;k00istheprobabilitythatvertices 59

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61 ].Fromtheaboverelations,weestimatetheclusteringcoecientfornetworksiscloselyrelatedwithdegreesofvertices.Accordingtothisfact,wemustconsiderinformationofdegreesofverticestondthecombinationsofsubgraphsthatincreasetheclusteringcoecient.Fig. 3-13 showsaprocessofndingcommunitystructurefrom6subgraphs.First,wecomputethecorrelationmatrix,thenndmatchedsubgraphofeachsubgraph.Inthiscase,subgraph1,2,3,4,5,and6havesubgraph3,4,1,6,3,2asmatchedsubgraphs,respectively. Fortheseresults,weobtaintwocommunitystructures,therstcommunitymodulesismadeofsubgraphs1,3,and5,thesecondoneismadeofsubgraphs2,4,and6.Toclassifynodesinnetworks,ndoutaspecicnodebelongtowhichcommunitymodules,wecomputethedegreedistributionandtheclusteringcoecientofeachcommunitymodule.Then,weclassifynodesaccordingtothesevalues.Wewillshowmoreexampleofthisprocessinsection 4 .Besidesthecorrelationmatrixofsubgraphs,wealsousethepropertyoftransitivityinthenetworktondtheoptimizedcommunitystructuresbymergingsubgraphsresultedfromthespectraldecomposition.Fromthenatureofcommunitystructure,agoodcommunitystructurehashighertransitivity(clusteringcoecient).Bycomputingtheclusteringcoecientforanadditionasubgraph,wecancheckthevalidationofourmethodforndingthecommunitystructure.Theoverallclusteringcoecientmustbeincreasedinthisprocess. Interestingly,thesubgraphsfromthespectraldecompositioncanbeconsideredasmeaningfulpartitionforndingcommunitystructuresbecausetherearenoedgesbetweensubgraphs.Itshowsthattheproposedmethodgivesusthefoundationforndingcommunitystructuresinanetwork.Wedon'tsaythatthismethodisthebestforallofthenetworks,butthismethodhasagoodperformanceforndingcommunitystructureinthenetwork,especially,forthenoisynetworks.AswementionedinChapter 1 ,therearelimitationintheconventionalapproachesandnopracticalwaytodenesubnetworks.Wethinktheproposedmethodwillbeveryusefulandlitontheanalysisofthecomplex 60

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SimpleexampleofndingCommunityStructurefrom6subgraphs. 61

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3-14 showstheowofthewholeprocesstoperformtheproposedalgorithm. 62

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Blockdiagramoftheproposedmethodswhicharespectraldecompositionandndingcommunitystructure 63

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Thecommunitystructureisanimportantpropertyofnetworks,whichisthetopologicalpropertyofthenetworks,i.e.,thedivisionofnodesintosomegroupswithinwhichthenetworkconnectionsaredense,butsparserbetweenthegroups.Communitiesinsocialnetworksmightrepresentrealsocialgroupings,maybebybackgroundorinterestofindividuals.Weintroducedanalgorithmforndingthecommunitystructureinnetworksbasedonspectralanalysis.Inordertoevaluatethisalgorithm,wewillshowtheactionofthealgorithmonseveraldatasets.andwewillusethewell-knownfastalgorithm[ 62 ]basedonthemodularityforthecomparisonwithouralgorithm.First,weapplyourmethodtotworeal-worldsocialnetworks.Then,wetestouralgorithmusinganarticialnetworkpresentedbyNewman[ 63 ].Finally,asapracticalapplicationofthisstudy,weusethecommunitystructureinthesocialnetworkandtheproposedmethodtoanalyzepersonalemailnetworksforthepurposeofspamltering. 4.1.1KarateClubData 64 ].Thisisasocialnetworkoffriendshipsbetween34membersofakarateclubataUSuniversityinthe1970's.Thiskarateclubwasbrokenintotwoclubsbecauseofadisputeoftwokeymembers,theadministratorandtheprincipalteacher.Asimpleunweighted,unidirectionalnetworkisusedtorepresentthisdata(availableatatNewman'shomepage[ 65 ]).Fig. 4-1 showsthesocialnetworkoffriendshipinakarateclub.Thenetworkhas34nodesand78edges.Tocomparewithanothercommunityndingalgorithm,wealsorunthiskarateclubnetworkthoughthefastalgorithm[ 36 ],[ 62 ],[ 66 ],thismethodisbasedonthemodularityinsection 2.3 .Duringtheprocessofthisalgorithm,abenetfunctionknownasthemodularity(Q)inEq.( 2{5 )and( 2{6 )iscalculatedateachstep,withgreatermodularityindicatingabetterpartitioningofthe 64

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FriendshipnetworkfromZachary'skarateclubstudy.Itshowsthenaturalsplit. network.Thisalgorithmseekstomaximizethemodularitythroughagreedychoiceofcommunityjoins.Toseethestepbystepoperationofthealgorithm,weshowadendrogramofthecommunityjoinsinFig. 4-2 .Fig. 4-3 showshowthemodularityofthenetworkevolveswitheachadditionalcommunitymerging,themodularityincreaseuntilitspeaksat31joinsandQmax=0:3807.ThisdendrograminFig. 4-2 providestheorderinwhichthemethodjoinsnodes,itcanbeseenthisalgorithmtendstojoinoutlyingnodesbeforeincludingnodesinthecoreofthecommunity.Fromthisresult,themodularityismaximizedwhenthenetworkissplitinto3communitieseventhenaturalsplitis2communities.Itispossibletonotethatnode10isthemisclassiednode,whichconnecttolargehubsinbothcommunities. Now,weapplyouralgorithmtothissamenetwork.Afterthespectraldecomposition,wehavetondtheoptimizedcommunitystructurefromthecombinationsofsubnetworks.Aswementionedintheprevioussection 3.2.2 ,wecannotdirectlyusethemodularitytondthecommunitystructure.Weusetheclusteringcoecienttooptimizethe 65

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CommunitydendrogramofZachary'skarateclubnetwork combinationofsubgraphsinsteadofthemodularity.Thereare14subgraphs,whichhaverealedges.Fromthesesubgraphs,wecomputethecorrelationmatrixandndthematchedsubgraphs.Inthisprocess,weknowthattherearetwocommunitystructuresinthisnetwork.Tovalidatethisprocess,wecomputetheclusteringcoecientineachstep.Themeanvalueoftheclusteringcoecientoftherststepis0.5525andthisvalueissmallerthantheclusteringcoecientoftheentirenetwork(0.5879).Onceweobtainedthenalcombination,theclusteringcoecientincreasesto0.6776,andwecanclassifythenodesusingthedegree,clusteringcoecient,andthemodiedeigenvectorcentralityinEq.( 4{1 ). 66

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ModularityofZachary'skarateclubnetwork where,ListheLaplacianmatrix.InFig. 4-4 andFig. 4-5 ,wecanseeeachnodehasdierentvaluesindata1anddata2,node(vertex)6and7havebiggervaluesfordata2(communityA)thandata1(communityB).Accordingtotheseresults,wecanclassifynode6and7asnodesofcommunityA.Ifanodehassamevaluesformorethantwocommunities,wecheckthemodiedeigenvectorcentralityinEq.( 4{1 ).Inthisnetwork,node10hassamevalues,degreeandclusteringcoecient,but,bycheckingtheeigenvectorcentrality,weknownode10isbelongtothecommunitystructureB.Fig. 4-6 showsthedierenceofthemodiedeigenvectorcentralityofnode10.Fig. 4-7 showstwocommunitiesinthisnetwork.InFig. 4-8 ,weknowouralgorithmcanndthesamecommunitystructureasthenaturalsplit. 67 ].This 67

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Degreedistributionofeachcommunity.Data1representscommunityB,data2representscommunityA. undirectednetworkconsistsof62nodes,connectedby159edges.Advancedtoolsfortheanalysisandstudyofsocialstructureinhumanpopulationshavebeendevelopedoverthelasthalfcentury[ 68 ],[ 19 ].Usingtheseresources,theanalysisofanimalsocialnetworkscanprovidesubstantialinsightsintothesocialdynamicsofanimalpopulationsandpossiblysuggestnewmanagementstrategies[ 69 ].Animalsocialnetworksaresubstantiallyhardertostudythanhumannetworksbecausewecannotmakeinterviewsandquestions,andnetworkdatamustbegatheredbydirectobservationofinteractionsbetweenindividuals.Thenetworkweapplywasconstructedfromobservationsofacommunityof62bottlenosedolphins(Tursiopsspp.)overaperiodofsevenyearsfrom1994to2001[ 67 ]atthecoastofDoubtfulSound,NewZealand.Itisknownastherearetwocommunitiesandfoursub-communitiesinthedolphinnetwork.First,werunthefast 68

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Clusteringcoecientofeachcommunity(data1representscommunityB,data2representscommunityA.) algorithmtopredictsubstantialcommunitystructure;Qmax=0:4955,forming4distinctcommunities,asshowninFig. 4-10 FromFig. 4-11 ,weknowthateigenvaluesandthenumberofedgesincorrespondingsubgraphsdonotcoincideeachother. Whenweapplyourmethodtothisnetwork,wecanclassify20nodesfrom20nodesinthemaincommunity1,22outof23nodesinthemaincommunity2.Forsub-communities,wecanclassify14nodesoutof19nodes.Bythesamemannerinthepreviousdata,weusethetransitivityasabenetfunctionforndingcommunitystructures.Fig. 4-12 showspredictedcommunities. 69

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Modiedeigenvectorcentrality Figure4-7. CommunitystructuresbyaddingsubgraphsinKarateclub 70

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PredictedcommunitiesbyfastalgorithmandourmethodofZachary'skarateclubnetwork. Figure4-9. Socialnetworkof62dolphins. 71

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Modularityofdolphinsocialnetwork. Figure4-11. Eigenvaluesandthenumberofedgesincorrespondingsubgraphsofdolphinnetwork. 72

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Detected4Communitymodulesinthesocialnetworkof62dolphins. thealgorithmtoquantifyitsperformance[ 3 ].Eachgraphconsistsofn=128verticesdividedintofourgroupsof32.Eachvertexhasonaveragezinedgesconnectingittomemberofthesamegroupandzoutedgestomembersofothergroups,withzinandzoutchosensuchthatthetotalexpecteddegreezin+zout=16,inthiscase.Aszoutisincreased,theresultinggraphsposegreaterandgreaterchallengestothecommunity-ndingalgorithm.Fig. 4-13 showthearticialnetworkwhichhaszout=2. InFig. 4-16 ,weshowthefractionofverticescorrectlyassignedtothefourcommunitiesbythealgorithmasafunctionofzout.Asthegureshows,bothmethods-Newman'smethodandourmethod-performwell,correctlyidentifyingmorethan90%ofthevertices 73

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Figure4-13. Articialnetwork.Zout=2andZin=14. 74

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Communitystructuresofanarticialnetwork[(a),(b),(c),and(d)]:Zout=2andZin=14. message.Theseinformationarestoredinthe\message-id",\date",\From",\To",\Reference",and\In-reply-to"elds.Weretrieveanemailnetworkbyrstcreatingnodesrepresentingalltheaddressesinthe\From"and\To"eldsandconsiderthemessage-idinthe\message-id",\reference",and\in-reply-to"eldsasnodes.Edgesareaddedbetweenmessage-idsandaddressesthatappearinthesameheader.Then,allnodesrepresentingowner'semailaddressesareremoved,becauseweareonlyinterestedinthelinksamongnodesthatcommunicateviatheuser[ 70 ].Fig. 4-17 showsanexampleofsubgraphobtainedbythisprocess. 75

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Articialnetwork.Zout=7andZin=9. Figure4-16. Fractionofverticescorrectlyclassiedasthenumberzoutisvaried. 76

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SubgraphresultingfromanexampleformessagewhichhasM0ID Tondeectivemethodsforspamltering,westartanalysisofnetworkdatabycheckingcentralitymeasures,whicharesomeofthemostfundamentalandfrequentlyusedmeasuresofnetworkstructure.Weobtainedempiricaldatafromoneofuser'semailboxandtheemailshavebeenchoppedintoaperiodof108days.Theseemailscontain2500messagesandconvertedtoanetworkwhichhas3755nodesand6930edges.Allnodesrepresentingtheuser'sownemailaddressesareremoved,sinceweareinterestedonlyintheconnectionsamongnodeswhocommunicateviatheuser.Fig. 4-19 showsapersonalemailsnetworkfortest.ThenweappliedourmethodtothisnetworkandobtainedsubnetworksfromtheemailnetworkbythespectraldecompositionEq.( 3{7 ).Fig. 4-18 showsthattheeigenvaluespectrumfollowsthepower-law.Usingeigenvaluesandcorrespondingeigenvectors,webreaktheentireemailnetworkintothesummationofmutuallysubnetworks.Eveniftheorderofmatrixis3755,mostlinksarecoveredbyasmallnumberofsubnetworks.Fig. 4-20 showsthat90%oflinksinthenetworkiscovered 77

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Rankorderedeigenvaluespectrumofemailnetwork byonly727subnetworks.Fig. 4-18 showscumulativenumberoflinksinsubnetworks.Wecandistinguish43%ofnon-spamand36%ofspamwithouterror.Itisinterestingtonotethat43%ofnon-spamcanbeobtainedonlyusing141subnetworksand36%ofspamcanbeclassiedonlyusing10subnetworks.Italsoprovestheeectivenessofourdecompositionmethod.Ifweconsidertheaddressesofrecipientinthesentbox,theperformanceofthismethodcanbeeasilyenhanced. Totestourmethodmoreaccurately,letusapplyouralgorithmtoanotheremailnetworkdata,whichhasbeenpublished[ 2 ].Fig. 4-21 showsspame-mailnetwork.Inthepaper[ 2 ],theyusedNewmanandGirvan'scommunity-ndingalgorithm[ 3 ]forseparatingjointedcomponents,whereasweuseouralgorithmtondcommunitystructureinemailnetworks.Afterweobtainedcommunitystructuresofemailnetworks,weassumecomponentsinthesecommunitieshavenon-zeroclusteringcoecientarenon-spamcomponentsandwriteallnodesinthesecomponentstotheblacklist.Ifacomponent'sclusteringcoecientiszero,weconsiderallnodestotheblacklist.Allnodesincomponentsthatcannotbuildcommunitystructuresareconsideredasthegraylist.Fig. 4-22 showsnonspame-mailnetwork.InFig. 4-23 ,weknowthatthesizeofsubgraphs 78

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Entirepersonalemailnetwork.Thisgraphcontains2500messagesandhas3755nodesand6930edges. 79

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Cumulativenumberoflinksinsubnetworks Figure4-21. Spamemailnetwork 80

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Algorithmresultsforthreedatasets. DataBlacklistWhitelistGraylistTotalCorrected Nonspam1016128644736.2% Spam17010372107365.3% Nonspam2019220139348.6% Spam2277301172385371.9% Nonspam3020018338351.7% Spam312680969253156.0% alsofollowspower-lawand20%ofsubgraphscancovermorethan90%ofedgesinthenetwork. Figure4-22. Non-spamemailnetwork Table 4-1 showstheresultsofusingouralgorithmondatasets,whichusedinthepaper[ 2 ].Averagingacrossdata,65%ofthespamisontheblacklist,45%ofthe 81

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(a)Cumulativenumberoflinksinsubnetworks(b)Sizeofsubgraphsinnonspamemailnetwork. nonspamisonthewhitelist.Accordingtothepaper[ 2 ],34%ofthenonspamisonthewhitelist,56%ofthespamisontheblacklist.Presumably,weknowourcommunity-ndingalgorithmismoresuitableforthespamltering. 82

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Weintroducedanewmethodforedgepartitioningandndingthecommunitystructureinthecomplexnetworkbasedonthespectralanalysisandthepropertyofthesocialnetwork.Astherststepoftheproposedalgorithm,weusetheLaplacianmatrixforthespectraldecompositioninsteadoftheadjacencymatrix,becausethismatrixisasemi-positivedenitematrixandthereisnomoreimbalanceproblemforobtainingthesubgraphs.Inthisprocess,wealsoconsidertherepeatedeigenvaluestodecomposethenetwork.Forndingcommunitystructureinanetwork,wepresentaframeworktodetectcommunitymodulesbymergingsubgraphsobtainedfromthespectraldecomposition.Wemergesubgraphstondwhichcombinationissuitabletorepresentthecommunitystructureinthenetwork. Thefundamentalproblemwithalltheconventionalspectralpartitioningmethodssuchas,min-cut,max-cut,andratio-cutarethatcutsizesaresimplynottherightthingtooptimizebecausetheydonotaccuratelyreecttheintuitiveconceptofnetworkcommunities.InadditionthesemethodsuseonlyoftheleadingeigenvectoroftheLaplacianmatrixandignoresalltheothers,whichthrowsawayusefulinformationcontainedinthoseothervectors.Onthecontrary,wedecomposethenetworkintoasetofsubgraphsusingalleigenvaluesandeigenvectorsandweknowthatthesesubgraphsareobtainedfrommutuallyorthogonalmatrices.wealsointroduceamodiedeigenvectorcentrality,whichcanprovideadditionalinformationaboutthereallinksbetweennodes,notonlythenumberandqualityofconnectionasintheconventionaleigenvectorcentrality.Thisdecompositioncanbeconsideredasanedgepartitioningandeachsubgraphdonotshareedgeswithothersubnetworks.Ourspectraldecompositionmethodgivesnotonlyasetofsubnetworks,butalsothefoundationfordetectingcommunitystructureinnetworks.Withtheprocessofselectionandmergingofthesubgraphs,wecanndthecommunitystructureinthenetwork.Weintroducedthecorrelationmatrix 83

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Asapplicationsofourmethod,wecanndtwocommunitiesinKarateclubnetworkwithaperfectrate.OtheralgorithmsuchtheG-Nalgorithmandthefastalgorithmhaveamisclassiednode.Fromtheresultofthearticialnetwork,weknowourmethodismoresuitableforthenoisydata.Accordingtothesefact,whenwecanapplythismethodtothenoisydatasuchasbiologicalnetwork.Interestingly,wendouralgorithmhasbetterperformanceforthescale-freenetworks,thatis,theirdegreedistributionfollowsapowerlawforlargek.Aswementioned,thedistributionofthesizeofsubgraphsinascale-freenetworkalsohavepower-lawpropertyandthispropertyresultsinafewlargesubgraphsduringthespectraldecomposition.Withtheselargersubgraphs,wecancomputethecorrelationmatrixwhichhasdominantdiscrimination.Duetothisfact,wecanobtainmoreaccurateresultsofndingthecommunitystructureinanetwork.Fortunately,itisknownthatmostofreal-worldnetworksarescale-freenetwork. Anotherapplicationofourmethod,weusethepropertiesofsocialnetworksandspectraldecompositiontodistinguishspamandnon-spamemails.Since,theonlyinformationnecessaryforthismethodisavailableintheuser'semailheaders,thealgorithmcanbeeasilyimplementedandcombinedwithotherlteringprocess.Thiseectivetechniquecanbeeasilyimplementedbasedongraphtheoreticmethodsandthespectraldecompositionofnetworks. Thebestcontent-basedltersachieveapproximately99:9%accuracy,butrequireuserstoprovideatrainingsetofspamandnon-spammessages.Thisalgorithmcanautomaticallygenerateanaccuratetrainingsetforlearningofmoresophisticatedcontent-basedlters.Theoverallperformanceofthismethodcanbeenhancedwithasimplebook-keeping-consideringtheaddressesofrecipientsinthesentboxtoclassify. 84

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Thebasicprincipleofourmethod-usingtheeigenprojectionforthespectraldecompositionandmakinguseoftheresultofedgepartitioningtondthecommunitystructure-canbeincorporatedintomoresophisticatedmethodthatcanautomaticallydetectcommunitystructureregardlessoftypesofnetworks.Wehopethatideasandalgorithmspresentedherewillbeimprovedtobemoreusefulforthedeterminationoffunctionalclustersinmanyvariousnetworks. 85

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[1] D.Fallows,\Spam:HowItISHurtingE-MailandDegradingLifeontheInternet,"PewInternetandAmericanLifeProject,Oct.2003. [2] P.O.BoykinandV.P.Roychowdhury,\LeveragingSocialNetworkstoFightSpam,"IEEEComputerSociety,pp61{68,April2005. [3] M.GirvanandM.E.J.Newman,\Communitystructureinsocialandbiologicalnetworks,"PNAS,vol99,pp7821{7826,June2002. [4] P.Holme,M.Huss,andH.Jeong,\Subnetworkhierarchiesofbiochemicalpathways,"Bioinformatics,vol19,pp532{538,2003. [5] A.Broder,F.Maghoul,andP.Raghavan,\Graphstructureintheweb,"ComputerNetworks,vol33,pp309-320,2000. [6] R.GuimeraandL.A.N.Amaral,\Functionalcartographyofcomplexmetabolicnetworks,"Nature,vol438,pp895{900,2005. [7] G.Palla,I.Derenyi,I.J.Farkas,andT.Vicsek,\Uncoveringtheoverlappingcommunitystructureofcomplexnetworksinnatureandsociety,"Nature,vol435,pp814{818,2005. [8] S.N.Dorogovtsev,A.V.Goltsev,J.F.F.Mendes,andA.N.Samukhin,\Spectraofcomplexnetworks,"PhysicalreviewE,vol68,2003. [9] B.JiangandC.Claramunt,\TopologicalAnalysisofUrbanStreetNetworks,"EnvironmentandPlanningB,2002. [10] F.Harry,GraphTheory,Addison-Wesley,1969. [11] R.AlbertandA.Laszlo,\Stasticalmechanicsofcomplenetworks,"Reviewsofmodernphysics,vol74,pp47{97,Jan2002. [12] R.AlbertandA.L.Barabasi,\Statisticalmechanicsofcomplexnetweoks,"Rev.ModernPhys.,vol74,pp47{97,2002. [13] S.Rander,\Howpopularisyourpaper?Anempiricalstudyofthecitationdistribution,"Eur.Phys.J.B.,pp131{134,1998. [14] H.Jeong,B.Tombor,R.Albert,Z.N.Oltvai,andA.L.Barabasi,\Thelarge-scaleorganizationofmetabolicnetworks,"Nature,vol406,pp651{654,2000. [15] R.Albert,\Scale-freenetworksincellbiology,"JournalofCellScience,vol118,pp4947,2005. [16] A.L.BarabasiandR.Albert,\EmergenceofScalinginRandomNetworks,"Science,1999. 86

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R.AlbertandA.L.Barabasi,\Diameteroftheworld-wideweb,"Nature,vol401,pp130{131,1999. [18] S.N.DorogovtsevandJ.F.F.Mendes,\Evolutionofnetworks,"AdvanceinPhysics,vol51,pp1079{1187,2002. [19] J.Scott,SocialNetworkAnalysis:AHandbook,Sage,London,2000. [20] M.S.Mizruchi,TheAmericanCorporateNetwork,1904-1974,Sage,BeverlyHills,CA,1982. [21] J.F.PadgettandC.K.Ansell,\RobustactionandtheriseoftheMedici,1400-1434,"Amer.J.Social.,vol98,pp1259{1319,1993. [22] A.RapoportandW.J.Horvath,\Astudyofalargesociogram,"BehavioralSci.,vol10,pp279{291,1961. [23] L.A.AdamicandB.A.Huberman,\Power-lawdistributionoftheworldwideweb,"Science,vol287,2115a,2000. [24] L.A.N.Amaral,A.Scala,M.Barthelemy,andH.E.Stanley,\Classesofsmall-worldnetworks,"Proc.Natl.AcadSci.USA,vol97,pp11149{11152,2000. [25] D.J.WattsandS.H.Strogatz,\CollectiveDynamicsof'Small-World'Networks,"Nature,vol393,pp440{442,1998. [26] W.Ailello,F.Chung,andL.Lu,\Arandomgraphmodelformassivegraphs,inProceedingsofthe32ndACM,"TheoryofComputing,AssociationofComputingMachinery,pp171{180,2000. [27] H.Ebel,L.I.Mielsch,andS.Bornholdt,\Scale-freetopologyofe-mailnetworks,"Phys.Rev.E.,vol66,035103,2002. [28] M.Bern,D.Eppstein,andJ.Gilbert,\Provablygoodmeshgeneration,"IEEESymposiumontheFoundationsofComputerScience,pp231{241,1990. [29] L.A.AdameicandN.Galance,\ThepoliticalBlogosphereanfThe2004U.S.Election,"2ndAnnualWorkshopontheWebbloggingEcosystem:Aggregation,AnalysisandDynamics,Japan,2005. [30] C.Ding,X.He,H.Zha,M.Gu,andH.Simon,\Amin-maxcutalgorithmforgraphpartitioninganddataclustering,"Proc.IEEEInt'lConf.DataMining,2001. [31] A.Pothen,\Graphpartitioningalgorithmswithapplicationstoscienticcomputing,"ParallelNumericalAlgorithms,1996. [32] B.W.KernighanandS.Lin,\Anecientheuristicprocedureforpartitioninggraphs,"TheBellSyst.Tech.J.,vol49,pp291,1970. 87

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S.Boccaleti,V.Latora,Y.Moreno,M.Chavez,andD.U.Hwang,\Complexnetworks:Structureanddynamics,"Phys.Rep.,vol424,pp175,2006. [34] J.ReichardandS.Bornholdt,\DetectingfuzzycommunitystructuresincomplexnetworkswithAPottsmodel,"Phys.Rev.Lett.,vol93,218701,2004. [35] M.E.J.Newman,\Thestructureandfunctionofcomplexnetworks,"SIAMRev.,vol45,pp167{256,2003. [36] M.E.J.NewmanandM.Girvan,\Findingandevaluatingcommunitystructureinnetworks,"Phys.Rev.E,vol69,026113,2004. [37] C.P.MassenandJ.P.K.Doye,\Identifyingcommunitieswithinenergylandscapes,"Phys.Rev.E,vol71,046101,2005. [38] F.ChungandL.Lu,\Connectedcomponentsinrandomgraphswithgivendegreesequences,"AnnalsofCombinatorics,vol6,pp125{145,2002. [39] T.Luczak,\Sparserandomgraphswithagivendegreesequence,"ProceedingsoftheSymposiumonRandomGraphs,pp165{182,1989. [40] B.Everitt,ClusterAnalysis,JohnWiley,Newyork,1974. [41] M.AdersonnandT.Morely,\EigenvaluesoftheLaplacianofaGraph,"LinearandMultilinearAlgebra,vol18,pp141{145,1985. [42] M.Fiedler,\APropertyofEigenvectorsofNon-negativeSymmetricMatircesanditsApplicationtoGraphTheory,"Czech.Math.J.,vol85,pp619{633,1975. [43] P.Gould,\TheGeographicalInterpretationofEigenvalues,"InstituteofBritishGeographierTransactions,vol42,pp53{85,1967. [44] R.GroneandR.Merris,\TheLaplacianSpectrumofaGraph,"SIAMJ.MatrixAnal.App.,vol2,pp218{238,1990. [45] R.GroneandR.Merris,\TheLaplacianSpectrumofaGraphII,"SIAMJ.MatrixAnal.App.,vol7,pp221{229,1994. [46] B.Mohar,TheLaplacianSpectrumofGraphs,Wiley,1991. [47] B.Parlett,B.Simon,andL.Stringer,\OnEstimatingtheLargestEigenvaluewiththeLanczosAlgorithm,"MathematicsofComputation,vol38,pp153{165,1982. [48] K.Tinkler,\ThePhysicalInterpretaionofEigenfunctionsofDichotomousMatrices,"Inst.Br.Geog.Trans.,vol55,pp17{64,1972. [49] A.Pothen,H.D.Simon,andKang-PuP.Liu,\PartitioningSpareMatriceswithEigenvectorsofGraphs,"NASASystemDivision,RNR-89-009,1989. 88

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D.Lusseau,K.Schneider,O.J.Boisseau,P.Haase,E.Slooten,andS.M.Dawson,\ThebottlenosedolphincommunityofDoubtfulSoundfeaturesalargeproportionoflong-lastingassociations-Cangeographicisolationexplainthisuniquetrait?,"BehavioralEcologyandSociobiology,vol54,pp396{405,2003. [68] S.WassermanandK.Faust,SocialNetworkAnalysis,CambridgeUniversityPress,Cambridge,1994. [69] L.L.AnthonyandD.T.Blumstein,\Integratingbehaviourintowildlifeconservation:themultiplewaysthatbehaviourcanreduceN-e.,"BiologicalConservation,vol95,pp303{315,2000. [70] M.E.J.Newman,S.Forresy,andJ.Balthrop,\E-MailNetworksandtheSpreadofComputerViruses,"Phys.Rev.E,vol66,035101,2002. 90

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UngsikKimwasborninDaegu,KoreaonMarch16,1967.HereceivedhisB.S.degreeandM.S.degreeinElectronicsEngineeringfromKyungpookNationalUniversity,Daegu,Koreain1989and1992.From1992to2000,hewasaSeniorResearcherinAgencyforDefenseDevelopment,wherehewasinvolvedinDevelopmentofTargetDetectingDevicesforSurface-to-Air-Missilesystems.HereceivedhisM.E.E.degreeinElectricalandComputerEngineeringfromUniversityofMinnesota,Twin-cities,MNin2003.Since2003,heconductedresearchfortowardshisPh.D.ofUniversityofFloridaundertheguidanceofProfessorP.OscarBoykin. 91