UFDC Home myUFDC Home  |   Help
<%BANNER%>

# Edge Partitioning and Finding Community Structure Using Spectral Decomposition

## 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.

## Record Information

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

## Record Information

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

Full Text
xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E20101208_AAAAJT INGEST_TIME 2010-12-08T21:01:06Z PACKAGE UFE0017534_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
FILE SIZE 31872 DFID F20101208_AABWDP ORIGIN DEPOSITOR PATH kim_u_Page_49.QC.jpg GLOBAL false PRESERVATION BIT MESSAGE_DIGEST ALGORITHM MD5
19fed9aa205f4eeaf80121fbc33320d4
SHA-1
003389b49e4c7b880b56b6ec0195565deffc40b3
1932 F20101208_AABVYJ kim_u_Page_49.txt
a88657d7df51c50f332bca1e6352ffd0
6714faf5fa4c6f2193dd4f878857a4538c0c1c8e
2254 F20101208_AABVXU kim_u_Page_34.txt
c9ffd3075b1f40142055a9d70515385b
6006dae712b877dc288b16406f01360d906fd5c8
5580 F20101208_AABWED kim_u_Page_58thm.jpg
abb1512c8481abed9bedca0319a58f52
429b0a5263d4caeeb93de9a29f703cf111368633
5953 F20101208_AABWDQ kim_u_Page_50thm.jpg
8b1c4862efa71f9f268927a637a868ec
1508 F20101208_AABVYK kim_u_Page_50.txt
c2ca14f5a2ec18e0f34912f850cb0364
8bab676e1d1ec5b0ded0dc16666fae58a64482bf
1633 F20101208_AABVXV kim_u_Page_35.txt
3c3d9f37b214768bac962a0725d2d943
3bf1fdc7a8fab253ae1ea821023b389049150434
20408 F20101208_AABWEE kim_u_Page_58.QC.jpg
a89d63ef6d02873a864c0d426cab3fb2
24170 F20101208_AABWDR kim_u_Page_50.QC.jpg
8f2900e8bfc287fbdcb133d3e9bff273
9f8e36d8681694a010b9c4a3b2cc0485ae911100
1029 F20101208_AABVYL kim_u_Page_51.txt
0654ba2caff6bf28452cc97f90448cf499efd0f5
2193 F20101208_AABVXW kim_u_Page_36.txt
aef2d65b3fcff46656b71184eb38efb6
69dc5172c3abc6a5132a92ecb45ba452359d5641
6873 F20101208_AABWEF kim_u_Page_59thm.jpg
173f01d61569ed5db61fb04dcc990931c1316cb1
1416 F20101208_AABVZA kim_u_Page_67.txt
5b26d6033503fa88786c75fe76b84ef7
286cc9f863eb7a2af800706f214e4f540e63081e
6022 F20101208_AABWDS kim_u_Page_51thm.jpg
243c5049c10f34e63366ef31696a5d8c
0cc6769e1dd3278ed1ac4c999fdf5989f5eaac15
2436 F20101208_AABVYM kim_u_Page_52.txt
4ac1ab13c89f7c628d2396835c1b0237
bcc57c24268fa13ba12bbfc71de79701b7ec8f6f
1321 F20101208_AABVXX kim_u_Page_37.txt
64b875c91a74ec1524d279bddf563d50
6ee7c34896ee446fd2cbd72f2a3057216e05f945
27552 F20101208_AABWEG kim_u_Page_59.QC.jpg
cf1cfcb13e486c06382bfc5aa1bace8f
9e631f71bc53e41d9287a4d37b0fb7fc3701543e
1488 F20101208_AABVZB kim_u_Page_69.txt
21257d69906f56cf83dc8f763045689a
6b3bf44d690c1db53a2c134fd538ce962445fcee
21814 F20101208_AABWDT kim_u_Page_51.QC.jpg
48d5696935568aa3727690a6f5b6a4d3fe3c904e
2140 F20101208_AABVYN kim_u_Page_53.txt
9fca5f60b05b2cf39e8a8138c0afd7ec
111c1172ec119947da51d64ca4a2e104084e7fd7
1402 F20101208_AABVXY kim_u_Page_38.txt
0864fbccbf92d2df4c2e97228506f15a
c2723b4d8c1b07a4f106252b3852eb4ff13cf090
8380 F20101208_AABWEH kim_u_Page_60thm.jpg
285 F20101208_AABVZC kim_u_Page_70.txt
1a195936de592d7771acaae0389c3925
8443 F20101208_AABWDU kim_u_Page_52thm.jpg
dd33bc8a25a473530ba3dec92be11be3
7314b82fe4fc1e9e9049d9f49010784a6c184f1c
1266 F20101208_AABVYO kim_u_Page_54.txt
0362ed53a3ef91a8b35f82d0d14b930b8d44bffa
1699 F20101208_AABVXZ kim_u_Page_39.txt
bd696b37e9e1d6ebe9fe65a771671863
06dd5749c3b711500bf320aee0236af9eefa61c5
37327 F20101208_AABWEI kim_u_Page_60.QC.jpg
8aeea249d378b118f9f2957cff12ff51
2d425e3e05804a7c3448aab7615c8419723a978c
237 F20101208_AABVZD kim_u_Page_71.txt
8b0e85fe260df453b122387a2c8cc0ce
37871 F20101208_AABWDV kim_u_Page_52.QC.jpg
90eb2dbec58a3031ca5655b472bb3337
9d32eabfdbac564ce68e8893e6ba6bb8e8b6a02f
1125 F20101208_AABVYP kim_u_Page_55.txt
5fc64285cb1c83b3912f0375314645a4
15dc39fbaca5ec8359798902d3fd93f7ca8e39e4
5784 F20101208_AABWEJ kim_u_Page_61thm.jpg
24724ee642562843a31620441017b2fe
65b12edc8127f35e8e6167b3c8feb5ab7abf5c7b
465 F20101208_AABVZE kim_u_Page_72.txt
22e73ec5b33cbbf21df9c2fe61bdfde2
e80d8633e4d1913ca1fac7f4e39d082ab78d6e74
7573 F20101208_AABWDW kim_u_Page_53thm.jpg
9c830a9a3451bd77d3694890574875c8
1228 F20101208_AABVYQ kim_u_Page_56.txt
8a8bf5cd11e4370a0562566b462d27f8
8cd4d4168ce9ab36977aff6527a200fd714d3546
1485 F20101208_AABWEK kim_u_Page_62thm.jpg
855c1b9f8f1c38126565beda0dfe1827
5f98a49da95fcfee2e10192d19ef504ffd167e69
709 F20101208_AABVZF kim_u_Page_73.txt
b8fa42eff317b2c3e77a93fcd7e29310
32849 F20101208_AABWDX kim_u_Page_53.QC.jpg
6f382854cfc3b7d380342b2560878966
f6c0d037aff7e40e1de11c4dda996b0caff4f388
4808 F20101208_AABWEL kim_u_Page_62.QC.jpg
d7806f0c5e383480f593e09bd8d457ac
c5395e4b1c2424e00f711f8625cdc0892e997e65
1010 F20101208_AABVZG kim_u_Page_74.txt
61ffdd6c6f7d437f8091cef919de07a6
18768 F20101208_AABWDY kim_u_Page_54.QC.jpg
735bef74a3a77a8513a1610124c8f86e
e5dcabe573c340a06bc1a0906434d0a2cddbb2a8
1197 F20101208_AABVYR kim_u_Page_57.txt
c678f0c9d241488db0a6a435194d4286
11308c19c22ea54d70c6db4fe8c2879bc4761b9f
17957 F20101208_AABWFA kim_u_Page_70.QC.jpg
a0d6788178e846f31ce4de377bdfbfc24e1b7699
5486 F20101208_AABWEM kim_u_Page_63thm.jpg
e22449d57e089c78f73d47dc9827ac5f
247051db26f96c750957267c5c3fb0b44c446767
1236 F20101208_AABVZH kim_u_Page_75.txt
59253d89564390cdd6b7fe1a1eb9af9d
8ecc835f498c96faf1da443990f345bf78654aa3
6581 F20101208_AABWDZ kim_u_Page_55thm.jpg
8accaa735152efc3bef83f17844a0daf
371 F20101208_AABVYS kim_u_Page_58.txt
b3c4afcaa73ca1b0e4fe9315e0101778
e52b6e4b8e90b0204d435986e82147fa62364ccd
6549 F20101208_AABWFB kim_u_Page_71thm.jpg
69db8b1d942dae2351c01169c1d97c92
18095 F20101208_AABWEN kim_u_Page_63.QC.jpg
39f1cdaf6ccc0560847f13a0587d442f
3a5df2d499756113c14e724183f44031f876a690
458 F20101208_AABVZI kim_u_Page_76.txt
1b1696beb0a35dab8a23f43f7abfa8bb
80fbfb086b0d56721ce051e968ceb8793e277dc7
2440 F20101208_AABVYT kim_u_Page_60.txt
44da2b16e70e7443c8a6ddfc6f46c6a0
afcd7d249ef2088022e4450d874a1fe32495a4d5
24668 F20101208_AABWFC kim_u_Page_71.QC.jpg
f5bc85d81890634cd3a8a6e9ae80049a
6bf992800795f7cbfcfb9bd290662f2ce35ffd96
8079 F20101208_AABWEO kim_u_Page_64thm.jpg
acf8815011249a51f17531b8c81a35db
6df632c5112e8f17bcc2ef8de08a6bd05f0868a4
1670 F20101208_AABVZJ kim_u_Page_78.txt
8a5e1b4ba8a2d9deb52bf34cf83362a92d44b73f
616 F20101208_AABVYU kim_u_Page_61.txt
0cf5b0f5de44247670456baa8a6e46b1
4805 F20101208_AABWFD kim_u_Page_72thm.jpg
00e908100debfc1aa1f7b2c6c3c196c5
e00d435221bb515564289d46fcb2432677af2b3d
34953 F20101208_AABWEP kim_u_Page_64.QC.jpg
4d2563b888a423c74e6cfd5feb497370
a244b33d7f811710bec2a88695e70eec8d701d37
891 F20101208_AABVZK kim_u_Page_79.txt
edb0de8e30b28c8a9d5751e6e2bd734a
8c06d68aff0dcfb83b30943b3632590c3cfaeeb4
247 F20101208_AABVYV kim_u_Page_62.txt
8ae18db997dc0f88a0a9c4da13aedd1d
851b5d2e2f8763bab29106f5b2d1632263b52267
14637 F20101208_AABWFE kim_u_Page_72.QC.jpg
137f0699c8cb6008240c3f9e11f456a7
13c421f419150e4913aa8d163c4226d8728ed584
8049 F20101208_AABWEQ kim_u_Page_65thm.jpg
240067d9877bc3b342ed9867585863574200b907
885 F20101208_AABVYW kim_u_Page_63.txt
410dc45d6de3a3e07025f6ab3e3ebfbb
a461f131c19696c47a24fffa55080397c95f9cae
6287 F20101208_AABWFF kim_u_Page_73thm.jpg
8b9b74d763a6c665147d9e116a6243ac
3ec2afc312352896a3e4f5eabf9ffb6afa9cc536
32408 F20101208_AABWER kim_u_Page_65.QC.jpg
4542de5b9de838363eb5a14c670f5e47a92d91e2
747 F20101208_AABVZL kim_u_Page_80.txt
4ab1d3e20ff18bb7dacbb21c8eb2c0f3
2259 F20101208_AABVYX kim_u_Page_64.txt
a53d9ef777ea09c26fbd0109454c248d
a24edf64b296300073e7b72a4a68913d5ee5cf33
23026 F20101208_AABWFG kim_u_Page_73.QC.jpg
65d54af9091b944b0a7928f0e6a86239
f4b54361b8e807d79faa1435b41dfa36feee570c
5677 F20101208_AABWES kim_u_Page_66thm.jpg
b70120d77bff3a24ed82dd18c436a52b
b91c0fcc2fc00c55671b5972d22d01a6a7468d09
802 F20101208_AABVZM kim_u_Page_81.txt
09ac882c3d7d384b2cf10c605944e652a888f3d8
1368 F20101208_AABVYY kim_u_Page_65.txt
f52921feb097793f0d9d2528da31319b
23680de557389a60de3a79de8329a90b3b1a51a6
29504 F20101208_AABWFH kim_u_Page_74.QC.jpg
cf376ae93b7206298aeee568164186cb
beeb12204a167d9b8f5548f133b9c03e4b6bb51a
21950 F20101208_AABWET kim_u_Page_66.QC.jpg
efcfed11373e75bc46046bfeff0f739379b09174
523 F20101208_AABVZN kim_u_Page_82.txt
40e6b167511aa18dfc7dc5d2bef7832f
cdf817b7637ef6f39b7b4196c56338b9b85c29a3
1180 F20101208_AABVYZ kim_u_Page_66.txt
432b3472bf0a8c31f985cc26b93350bd2ce732d6
6650 F20101208_AABWFI kim_u_Page_75thm.jpg
6e8f1178d2fbc770588646503beac3cc
aab0a263de62d24fd7dc14dd48d1037ec4c07ef3
6902 F20101208_AABWEU kim_u_Page_67thm.jpg
fbe15cbc2f227cb9834c7b986ccc832a
e90a48e308406a3e2bd562dcc146d4be813df49d
2282 F20101208_AABVZO kim_u_Page_83.txt
52ae856f3764bca7760f73415a06b607
6029 F20101208_AABWFJ kim_u_Page_76thm.jpg
1db768d9b3b60a832c52ae03517091aa00724c8d
27027 F20101208_AABWEV kim_u_Page_67.QC.jpg
d075020500acfd4d933f2b7cda7e5cbe
2373 F20101208_AABVZP kim_u_Page_84.txt
8cf355933960e38fd0442da21787571d
bef427730103f66415c516d06e7919ab02a4d606
22806 F20101208_AABWFK kim_u_Page_76.QC.jpg
07db17cd4c4d888082c973046ea10e49
6230b1a91285726f90b345c064aa52e87f2c13cb
6295 F20101208_AABWEW kim_u_Page_68thm.jpg
591a51b678a715986de6f6e56b49d866
1082960a1499776bba47d152102a01dc92cca8f4
876 F20101208_AABVZQ kim_u_Page_85.txt
06b3e938dd239a9dfa74fd11a54a346b
5c8f44110eb07ac4892a565aa1216f8d6762f39f
7270 F20101208_AABWFL kim_u_Page_77thm.jpg
9ebc5689c5409522523780202506035d
6d1c3bc0b8ea94439da5120cee0ba9d9dbf8a1e3
26403 F20101208_AABWEX kim_u_Page_68.QC.jpg
f8379b19ff1778f24a32ecdcb4916b8a
5450e22a19c7d01f63c8c1b5ec388e6c86632f11
2293 F20101208_AABVZR kim_u_Page_87.txt
e5467688713db6e8e29d6fe5cb55aca7
4edc7bc5077c93f3744a82ec0af93baf12663150
35898 F20101208_AABWGA kim_u_Page_84.QC.jpg
4c1be9cbbf6973331999a0e54e631c3e
b20dc4cb113ca47f3238fdb05335ec384019e20e
26798 F20101208_AABWFM kim_u_Page_77.QC.jpg
9d4a2f3b7703bae5b08894ea31fc3d2e
5e6826e41c8e27d8c39f7110e5c6a19898032f31
6816 F20101208_AABWEY kim_u_Page_69thm.jpg
49250807fc21aecaae5175e88de81b31
8d3ba711bb9754ca977b38890caff0ec952e2fa8
14266 F20101208_AABWGB kim_u_Page_85.QC.jpg
962a887598b3a788eaf0f2d761aaa6e8b4eeb15e
6756 F20101208_AABWFN kim_u_Page_78thm.jpg
4f50ae4bfe15215b0440b99f00cf9e65
c2794d59efb7dc20bbc0e3da461eae51e0eff8f7
5376 F20101208_AABWEZ kim_u_Page_70thm.jpg
bfe4a3e3cf45cdacd55f1a47f8a7b250
2390 F20101208_AABVZS kim_u_Page_88.txt
b44b3031436d3664503618761ac4a001
34cb373b6742ba782ec4f086461454372baaf355
7627 F20101208_AABWGC kim_u_Page_86thm.jpg
8ab8231f4a87dcb9a4ae43670a1172ee
20f58abc082c851874981dbfe1dc28a3b677385d
27423 F20101208_AABWFO kim_u_Page_78.QC.jpg
9f81332fff80826f15cbeb3618b7d124
2415 F20101208_AABVZT kim_u_Page_89.txt
7ea9fe4ed42de2780a5d6d1d0faa3172
402200d9f366e306b157905bda3bf03a378e2137
31024 F20101208_AABWGD kim_u_Page_86.QC.jpg
292e835286b3a7f6a067b35ffa7e51ba
c94fbe1ac609a6f564b25a402a69b7ce4193796e
5408 F20101208_AABWFP kim_u_Page_79thm.jpg
7186dd8ca24a03ec99e6331550358f75
8dd369c3df49a561b407705cc583891fe0cd152f
853 F20101208_AABVZU kim_u_Page_90.txt
8034 F20101208_AABWGE kim_u_Page_87thm.jpg
a335c21289dddf0bc8b9727acbb81df4
ecb400f3eb75f8ac0d38465209680b5c24dbe24f
17251 F20101208_AABWFQ kim_u_Page_79.QC.jpg
14cfea9dea5b44d21e66d9ecb1454d33
1e9a5818a186df4fe0a0032b5dc4d293e3436e58
735 F20101208_AABVZV kim_u_Page_91.txt
07997efaf7f1cffef076f130b93c74f4
06ed15ba106ec4b9626c3b9bdcc9e05964412f49
32749 F20101208_AABWGF kim_u_Page_87.QC.jpg
2d566de91dd7c2df19ee961b999236cd
3a6819e59a4138010b613f8589a327890d8b5069
3798 F20101208_AABWFR kim_u_Page_80thm.jpg
fe0cd0b06e4e4bfc5e2843883e1aa7c3
f9eaecd0fc86c9d1a3f0c0e2ce59228af4bba880
11069521 F20101208_AABVZW kim_u.pdf
d78554f8d9982696acd115ef0bfc0ec7
1ca4923473d5240b40f30a61ca6c5807b7949c5b
8029 F20101208_AABWGG kim_u_Page_88thm.jpg
a44e429ac451cb379a2efccf791540d7
369856351817a3bd0d638213a1d88594044df7a8
13499 F20101208_AABWFS kim_u_Page_80.QC.jpg
1e6be366b12fb0d03696501fc63db002
a720313594bfd0189ab482f898ddf60620cfd2bd
26523 F20101208_AABVZX kim_u_Page_69.QC.jpg
245914c5def10ebdc665e3733864c6bc
3c5636739b4b90c0a8f65a408bdb70af3596d8f4
34163 F20101208_AABWGH kim_u_Page_88.QC.jpg
2f6cff4ee0703798fc42f58e19448e57
a6209590331989d51cc24ce98dcb840e21e1425f
5320 F20101208_AABWFT kim_u_Page_81thm.jpg
5dab6291a6e62eec7761e858c401c09da91e64d6
25535 F20101208_AABVZY kim_u_Page_57.QC.jpg
bea3c409c704d08ec7df7a51889cbc6e
889d34075937c96c5d635ba330735957bf3910e6
20244 F20101208_AABWFU kim_u_Page_81.QC.jpg
548cd0a3d7874712f9c30a358582b9ec
698cd73fd452d7cd5485f5266e8bcbdd1ac8b9e9
4968 F20101208_AABVZZ kim_u_Page_54thm.jpg
303cc354a5d43435b4aea7515f036499
92651038a1f83fcf5726a0450e22c7d1a5854c6c
8227 F20101208_AABWGI kim_u_Page_89thm.jpg
047e7deccb867d73e93720603535a945
e894653a757e9663b9c87af3cd2081ba077bf25f
3421 F20101208_AABWFV kim_u_Page_82thm.jpg
541229c3d576f12b1b8a8fd68a3d91b4
5d8476fe485ddbee6643bd8dc555565511404aca
33877 F20101208_AABWGJ kim_u_Page_89.QC.jpg
d93b94440878f03fdc845f6252bd9978
81764bd98627ae53d677de3cce4307d4901e0617
11447 F20101208_AABWFW kim_u_Page_82.QC.jpg
f1e80dce77e43da10e4acb04aac7cce0
91e31ec6d773e624b41886be2930c7e7d656e86f
11669 F20101208_AABWGK kim_u_Page_90.QC.jpg
65c3ea05396efb50b23183d3f3c89f00
ff629bfaf4cd7bf3385dc1401d00af5a84419601
7652 F20101208_AABWFX kim_u_Page_83thm.jpg
92255896d9b84a9fc8749b28376d4a0a
2993 F20101208_AABWGL kim_u_Page_91thm.jpg
34066 F20101208_AABWFY kim_u_Page_83.QC.jpg
d2912ae3ec4dfa2378ecc62a90013d19
07bd678891b2f23851ef8f4976312796b2a7d9d1
8001 F20101208_AABWFZ kim_u_Page_84thm.jpg
fe384af24a25e1c289565d855baab6ff
96436 F20101208_AABVIA kim_u_Page_14.jpg
e3b1522f21bf627bd0a623925f23a632
042186c9b93f0a5c2bc0253cc2981b24d1ef03a6
24143 F20101208_AABVIB kim_u_Page_66.pro
d927067d3a6568618d3e0ecd95fc567e
d650d4a6e311ee5290879fa67ab55b37324b698b
41962 F20101208_AABVIC kim_u_Page_91.jp2
baaf6d34e38f46d86340513365862da2cd554401
5391 F20101208_AABVID kim_u_Page_18thm.jpg
9b2d276f7c0b20671561de990585e61f
85acaeb48147a42a542e0899cb191a9128041405
1712 F20101208_AABVIE kim_u_Page_30.txt
394c52341a75b7607d0b37eb6d27710a
38ea6ebcf345344e139da3bac5ed6b65074dd88a
6706 F20101208_AABVIF kim_u_Page_46thm.jpg
721d71a2150d3f9fd0daabcd08c98519
91024b77307556aa1f28fa4959f3b0549321276a
1813 F20101208_AABVIG kim_u_Page_59.txt
2f5a07f9227b5dea151b54afa7dee4c7
7828 F20101208_AABVIH kim_u_Page_08thm.jpg
05c1394244cc90ec8bb394ecc49c003d
e2fcee4b65f43e032784fcdc749fb0ba9197d8eb
25403 F20101208_AABVHU kim_u_Page_74.pro
38da08e19259e41469768a2ccd94244f
0d8842331010b08cf728687ec9f6a64a46f9565c
22155 F20101208_AABVII kim_u_Page_56.QC.jpg
75a46f2711623d423acd470dfb6fd550
75644 F20101208_AABVHV kim_u_Page_50.jpg
36620dc748ab27de4bbd6f9a133f3b43
671770b5bc59e61193dd4e7da1b199283ff7497c
935725 F20101208_AABVIJ kim_u_Page_67.jp2
e87dcfae7b90531bef2fc6546aa686fc
87169e68a39e416f301374dec3933f9bb8b83392
2249 F20101208_AABVHW kim_u_Page_86.txt
c18087b534fb4f93e20f194455fcc7f0
660412f1330488f211b425c8244a43cc0b555878
89782 F20101208_AABVIK kim_u_Page_69.jpg
04f1e363125caf0d248ab2a4c7460a30
e1e523d8587f640b1f3bc34dacd81caa0b3c3435
31773 F20101208_AABVHX kim_u_Page_05.QC.jpg
ab5728f5a241f0fae85cfdce59a0b187c185512f
25271604 F20101208_AABVJA kim_u_Page_19.tif
da0e00fdd94f9e2a5feedb26004fdcbe
6c44633ece67b947693c9f0678c7735318606760
1338 F20101208_AABVIL kim_u_Page_02.QC.jpg
8749623ee5c915fab25746d561c3052d
fba851b131567b07d49e7a2e6357018d4137742b
F20101208_AABVHY kim_u_Page_68.tif
35bcba58a397c052258233ac5b88160d
1023679 F20101208_AABVJB kim_u_Page_32.jp2
66046931589caaf5f3201df21bec9ed6
44e515732a4bcd135d52d4edc6c9cedb87e4deed
56219 F20101208_AABVIM kim_u_Page_83.pro
51be32f2cf961397a6f9f2ba6d5ff5f8a3bc9834
1331 F20101208_AABVHZ kim_u_Page_68.txt
128aa2ab50dcba2b4e6ea72cb0559cb4
83f5bc23dd6e8cf6ef6e1338bf092dce858d2243
38094 F20101208_AABVIN kim_u_Page_31.QC.jpg
816a82c586962ea369310ff3d6a13e6d
e5aa903b16fceedfba997ab86ef1256803c5d164
91628 F20101208_AABVJC kim_u_Page_45.jpg
de7b7881a2aac8f36e8b9cbb2f02bf62
121818 F20101208_AABVIO kim_u_Page_52.jpg
59f8987a60bca8c44a8c97dae9a8bc2c
5615d465a9a8c44ace20befff3c930761a0f8599
F20101208_AABVJD kim_u_Page_44.tif
0f26fc6aa8cb200bcccf5eb239fe567e
8370ba6fc464b739492097386ebf798da55b8b07
85518 F20101208_AABVIP kim_u_Page_67.jpg
9d9ce7f593e622234ac2cc64d88f54f6
218be5ed98595b0489cbe30b86af729f9095e22f
1438 F20101208_AABVJE kim_u_Page_77.txt
bcf88b41aa761af354f3aba96ec2d39d
6cc7aebfa233cd4359ac69f97f9807edba50f9e0
7838 F20101208_AABVIQ kim_u_Page_27thm.jpg
966b8f457d0ae1873885301def9da23a
550e8aa71497250f55f9b6a3890730691f3fd6a5
34318 F20101208_AABVJF kim_u_Page_09.pro
9d5e85d6d5e7ec612594a608fca77dab
ebff91ce9c143d998da602e8baaeaa8d1dd67495
12259 F20101208_AABVIR kim_u_Page_91.QC.jpg
f603b73fb35108874d5ae4bf3801467c12b8d478
87247 F20101208_AABVJG kim_u_Page_78.jpg
f10bf8be07e37081eb8a44b0e7805f5a
a6f71a38fe67072b3493940e240f78edb255d4c7
30887 F20101208_AABVIS kim_u_Page_37.pro
737976ca5858fb6c95688bbb152baef8
4bb07f050ba10e6e6d1c5e03571c5247bc099ede
690397 F20101208_AABVJH kim_u_Page_51.jp2
ffb1a01e65ee1d5b699e1d1edf90148c
297a4bec96d1e4993a13c07445f77aff8ba84cba
94526 F20101208_AABVIT kim_u_Page_42.jpg
f6f8db93834fcb6e68e28091b6c361f6
57da3a4d61ba6518fba204ef2d8731dd5875d696
846212 F20101208_AABVJI kim_u_Page_56.jp2
760ee51095778e5a7566094d6e351a8b
f9d890691cbe09e8b46b58df9da6fe1fa5612cec
F20101208_AABVIU kim_u_Page_74.tif
5722ea1f3432b9f59c9de1c8e2f9f7b4
8eb1998724baa217cd94e68117b7455c7c27efb0
1051965 F20101208_AABVJJ kim_u_Page_80.jp2
6c3d16b7d8db59fb8874cb3fa92ed8e4
1a34d6e00bfa820618ab43605e0c41ca43ed234e
7410 F20101208_AABVIV kim_u_Page_74thm.jpg
aa6dede76dcf84b8fbe437acf303d96f
1d41a465fd5af971a678ee2ca1fc2ab0fea63dd5
2870 F20101208_AABVJK kim_u_Page_90thm.jpg
05b05434d4b36b3ec8273da5486bc8b9
aa5dc17f310c40ab743309444afc4664fc03593c
7509 F20101208_AABVIW kim_u_Page_24thm.jpg
d99c478a78622b916c6273efd7c9b9a5
109044 F20101208_AABVKA kim_u_Page_12.jpg
fb1074945f8b1e67b62696ff26820ef7
d90c3e474c593af89cd49f37fbe2ff5ea8cf7f4c
70810 F20101208_AABVJL kim_u_Page_66.jpg
32d13c7421997ec053cf338be58f7aec
59dd9709223ae4154d3ffd1ebee4620225162330
27630 F20101208_AABVIX kim_u_Page_41.pro
4b2a77704011b3b51e0801d7e15cb85d
116110 F20101208_AABVKB kim_u_Page_13.jpg
84b5356d23ce9bfa9729616900ac138d
142cdba5db0c4d3fe204a54f622c10fda1f3f67b
48301 F20101208_AABVJM kim_u_Page_85.jp2
85e87a3f1435f106815ae5cd1856e7d8
dc525c6523234396d6d5301fba172a9431f53e09
26601 F20101208_AABVIY kim_u_Page_01.jpg
0838231c6c96a9b13e12c4105699319e
79020 F20101208_AABVKC kim_u_Page_15.jpg
4e8e8fea5fc308ab8d7af47c7c3c00d4
d72b1051193d8888576fa614ea291d3e27ceb607
104510 F20101208_AABVJN UFE0017534_00001.mets FULL
7de3e548d98d9086e99961a5a77963f8
859f37fa5263f6065ab0aeccd68ab25b79b994bc
806618 F20101208_AABVIZ kim_u_Page_66.jp2
61906d81af681c2259384c1c6219fceaa31232a8
104168 F20101208_AABVKD kim_u_Page_16.jpg
6f0c74048483b9c1f46bfc17d089c472
9393b815c2a3d8558e5f09227f8399248ec66ff2
111389 F20101208_AABVKE kim_u_Page_17.jpg
3f2a6165aef9fb1c8f8b8a6da8e7c15a
3726 F20101208_AABVJQ kim_u_Page_02.jpg
efe04fd1e9a3d893e650538220aa984d
afc7ddc0cb975d8821f3936188dabd5ceeaeb85a
64717 F20101208_AABVKF kim_u_Page_18.jpg
8559882993129e8ee8eec29d8a734fac
055dc9db1932aaa48b7ef3c353bece47b0949db8
9727 F20101208_AABVJR kim_u_Page_03.jpg
61e929c72314f918f65beb7d9536c81f
119461 F20101208_AABVKG kim_u_Page_19.jpg
b0f5215ec823ea80409f6ebd12bf5857
e468e3502060ff80b4acabe9ca542f86bc23abaa
41829 F20101208_AABVJS kim_u_Page_04.jpg
4fffffddc1a6ce601e2b070025552bf5
6928147c88e790bf0f18951f8d05eb2bfa445960
76936 F20101208_AABVKH kim_u_Page_20.jpg
b7742c535019e9f251b0c86239a576d1
4a0f994d8977da00d7de3ceb1a402e9db009afce
107347 F20101208_AABVJT kim_u_Page_05.jpg
ac348a248dc91f06f6d66525fb9a5ab6
0db7b8ca3ab0eed72e781cc20f5f40d04b54fb2c
118740 F20101208_AABVKI kim_u_Page_21.jpg
cab35696a0203bc73d9c26efa4cc47be
81c6184bfee0ca5742c0c82f9dfb841309af9182
7502 F20101208_AABVJU kim_u_Page_06.jpg
6077671015254e8c8f4c3dbdc8708d9b
3bd81c4a40557d61d46acefe82b3fb7025efeffe
23134 F20101208_AABVKJ kim_u_Page_22.jpg
7be0f0b3331159aa2c2ee62f7480e5be
dcc7855fef9bd9fc2f1c05cd3722bd35fb85bca9
20210 F20101208_AABVJV kim_u_Page_07.jpg
3955c31b8dda1e64b5777d1035746b54
2dda34fdcb3cb902e259b7a0c6b0963152fde193
79417 F20101208_AABVKK kim_u_Page_23.jpg
e1de874d390dfd43aa8e62f05c00dce38acacfc8
118817 F20101208_AABVJW kim_u_Page_08.jpg
3c1370e1456a6f8d7e1d3bfc3bd9f260
25fb09d9d904418d0c649d5ac3665b12d6de351a
105293 F20101208_AABVKL kim_u_Page_24.jpg
54d2bc9b5c58ef8e0aacf78a9933b48e
2ea883a765cef4bc5c44bdd78674f7e0cc4b14f2
80347 F20101208_AABVJX kim_u_Page_09.jpg
29b9203b96d8379cff673843fd6d141b
da18df361e4bc0c2cbfc735b7b71cc0d28ef80e8
90939 F20101208_AABVLA kim_u_Page_39.jpg
95e9ce833846236364dc014e98fa34e8
a5197fb5783f48e4de8bfeb1926e5efc51121b81
97851 F20101208_AABVKM kim_u_Page_25.jpg
c2ce86deb3c05df92ba2d9c0b56430b9
c6f743164fba7ff90e0121775499f180e2e75344
102292 F20101208_AABVJY kim_u_Page_10.jpg
dd29163d2ccf1e0edefacc7be76e5999c65c4c2d
86841 F20101208_AABVLB kim_u_Page_40.jpg
96cc35db180ffcb83808f59d8f46ed36
c1a0c002fc364ab025105c41d301aced0227ea76
118239 F20101208_AABVKN kim_u_Page_26.jpg
4ec221c0ef865bb0f51023a188161ab8
326c91d514f22a8abc0e791f57a8f23103337884
24469 F20101208_AABVJZ kim_u_Page_11.jpg
68778 F20101208_AABVLC kim_u_Page_41.jpg
c2d13f31a21de35826dfffb9fe30ea26
c86d6d16a44294f05b5b397a63581934de84caf6
111287 F20101208_AABVKO kim_u_Page_27.jpg
7f1b1635ecfaca893fe44220901ca57e501649eb
65676 F20101208_AABVLD kim_u_Page_43.jpg
e25a88c02e30d9f08a491660c1df7506fc219646
80108 F20101208_AABVKP kim_u_Page_28.jpg
29b4230ef64d09a97a7efe71cd4499d5c2032acf
111524 F20101208_AABVKQ kim_u_Page_29.jpg
ede78080940581392aabaa63b435ea56
c20bd9b2cded338ee46074ed5282b98e6ffb317b
92036 F20101208_AABVLE kim_u_Page_44.jpg
44f06dd5c811bc355d9ab9b69255c08d
14a381cc3ce0250788b47c8e9b8ae3080a1f38b9
85632 F20101208_AABVKR kim_u_Page_30.jpg
58b7a043310e6f4efdcda41cdec46677
2ac7dd67ca936c9759316b3ae82aec04bac1b59f
82848 F20101208_AABVLF kim_u_Page_46.jpg
e1ab973511560412a1e2d122ed285862
25e5a7e975b2c42322ea2468fd5b168df8e466a4
122756 F20101208_AABVKS kim_u_Page_31.jpg
98005c540d4d081cbe9d074fde79d79d
1618c097d81a0b83bb6b6dc7f2bc1cc064fc4396
68412 F20101208_AABVLG kim_u_Page_47.jpg
1f1355c0e8221f4361533a6acf3b05bc
ea09379a3041622a56e7bb1dd41f275583408c58
61527 F20101208_AABVLH kim_u_Page_48.jpg
bfc5a9f3e3dfb21e36e21ce0a3ed38b5
d9940a05f5afa517f7e97bd447407f19d6f3dd5a
96728 F20101208_AABVKT kim_u_Page_32.jpg
a2b1ce759169c48c892d64620ba93e5e
bdda791667c3fc91f795865524bee8d59dd3a995
104400 F20101208_AABVLI kim_u_Page_49.jpg
a9324c9bafd2932faafbf4ea065305ae
a51a9e27dda9e812f92b3a604a91e618efe72fd5
110667 F20101208_AABVKU kim_u_Page_33.jpg
f97c9b79cf2eb04a8723daa763bce4bd
0e6fedf9fc73116eb03be3121d69454a19b23461
62643 F20101208_AABVLJ kim_u_Page_51.jpg
65df69afef7121f92969259615a92210
f601bd1671046599800d35ebab799ab5482fb0c2
109484 F20101208_AABVKV kim_u_Page_34.jpg
1b80eb9716acf24267687c051fd29146
545246ab30b98276bdcd123846c351e4cf19ae15
104044 F20101208_AABVLK kim_u_Page_53.jpg
7b7a6362003ec281c745c79063afd4f3
c01aaafcd34ceacd3b2c8a4fd4a2929fa9973cee
75942 F20101208_AABVKW kim_u_Page_35.jpg
570a9089759cfb33898ff4e55c6b9ac5
1f6d100a9da20036aae26263de5ca7860e84ff33
46129 F20101208_AABVMA kim_u_Page_72.jpg
cc7a224d54ba92fa79c9b05e52e9acc2
c578444419c8c20f1c1395810a247f900f8d134d
56740 F20101208_AABVLL kim_u_Page_54.jpg
85bb045b144e295992ef8d776322e977
4d109452e9a8a9e4db0a4691821a50f8b37555d2
103670 F20101208_AABVKX kim_u_Page_36.jpg
b24309317abc96cf7993aefb969a78778cf19529
69554 F20101208_AABVMB kim_u_Page_73.jpg
e494862f1503fea120603b4ab1aa6f21
fbe510800442594f6fffb7038aff4a0bcbc40c3e
84979 F20101208_AABVLM kim_u_Page_55.jpg
a83ca6cc24ddb1fa900b953c8953d73c
d7bd753d39b2777764fe63a8fd30093c20236251
81160 F20101208_AABVKY kim_u_Page_37.jpg
40120ce7b29524a80c17a0c431ea4ed4
6ab1ccba7465b8821ae3acc9b9bc2167ebb2fe0e
100577 F20101208_AABVMC kim_u_Page_74.jpg
cb32511675d8f69bdc8e3b6f5aea1cbf
31d03b41d5f9a4ddc5f5e49b761f7501fd363d02
68487 F20101208_AABVLN kim_u_Page_56.jpg
14c9ec83cdeefc69545e175b0dc2041e
6845668e7ea14c2fe3064627366daebd2976782f
83674 F20101208_AABVKZ kim_u_Page_38.jpg
eeddd125cb86b8cd29b67fdeddcac0a5
ca212e90b591427e5e0a90cc52946c02fdd9a1c1
81461 F20101208_AABVMD kim_u_Page_75.jpg
77889c5563ee933d2935fcd9b325a2b3
00c14c4437d56689d812b3dbe9bc0f0f36096043
88869 F20101208_AABVLO kim_u_Page_57.jpg
fffa2dba6de92dfb7f4548806bd8b730
0eb1a577b1f45d57fc26a3b77079dbc2654ef261
73192 F20101208_AABVME kim_u_Page_76.jpg
3de4397b672e0790e0e73c9146b2e290
292b4827277083780413e1c77a8aef7146beaf18
64706 F20101208_AABVLP kim_u_Page_58.jpg
0ebeee7e82acafd05859b386d62b5679
92173 F20101208_AABVLQ kim_u_Page_59.jpg
82444 F20101208_AABVMF kim_u_Page_77.jpg
a1c6da28f657552ea75ff130662d4c67
52534b8505919d0c378150fabb967900a8783938
119664 F20101208_AABVLR kim_u_Page_60.jpg
e45fb63312c493a48df3d74ae7fbdb71b51c03de
57052 F20101208_AABVMG kim_u_Page_79.jpg
50e9141321703ec47af482acd15f44fc
48839ea525c4823d7d8decfe045a1f794b3c4f16
60394 F20101208_AABVLS kim_u_Page_61.jpg
5e5fdf2836848bd15dc10e2d7d1b5579
47147 F20101208_AABVMH kim_u_Page_80.jpg
efaa6d8a9b720058f3ee7e6f0f102160
72f4dcef99dc01aa5e69af151f06884c65fedf57
14021 F20101208_AABVLT kim_u_Page_62.jpg
65501 F20101208_AABVMI kim_u_Page_81.jpg
d9c3e46ebf5ee876dfb6fbe67c48ca96
863494ea031bd10006b05467f421bc8051ef4421
53112 F20101208_AABVLU kim_u_Page_63.jpg
f11d658bc9b45f6000e8c0ce1901fd22
36147 F20101208_AABVMJ kim_u_Page_82.jpg
ba6ba93bdfcdc6277c5311e8a0f92535
48a1a805b785cd9f07c7be8d94db8f9bf947f97d
111731 F20101208_AABVLV kim_u_Page_64.jpg
459bd12829e6ee1870262b24f48101c8
2b792dd83ecb2bc516eaf091a3237c2fec20e780
108366 F20101208_AABVMK kim_u_Page_83.jpg
6ab3bc6a13186dca9323ee1da72476d6
c3d66f09db6762c7219e93e69a28de007b956b06
104105 F20101208_AABVLW kim_u_Page_65.jpg
45e89ba4af7dedc438f5fd3ea77de8f4
ce9f445d75de5336237d2d7a93cb9ec61ea86d57
113576 F20101208_AABVML kim_u_Page_84.jpg
6d27fb10ac7fae5f580bb64681383ceb
5df326812b033118cc240243e4b6f265bf392ab6
86858 F20101208_AABVLX kim_u_Page_68.jpg
672c7e7435f05e1d87847e2f2140bea6
c2f3f9e419dd21e63c6d9c2e32bf659cd8d57648
1051924 F20101208_AABVNA kim_u_Page_08.jp2
36ef6a37f276ecae1a4293cc6b80e0ab
55edab46ffc12943582a4fb2913011bae2ac812e
46252 F20101208_AABVMM kim_u_Page_85.jpg
6756832ce77fb3677e2cf8c33e0cd0de
779b078bbcdb9431aed53d8a410751b96c62e410
52633 F20101208_AABVLY kim_u_Page_70.jpg
6be12e84fd0d2cf1e1d7f6258e1f13e0
b09232f0a29f1751db488dbffbcb03935eee546f
1051983 F20101208_AABVNB kim_u_Page_09.jp2
f59dd9dfd04cee1a43b1aabaa050676d
612e7a849ab4487935f3b0278ea57deb3c1ba1f1
105662 F20101208_AABVMN kim_u_Page_86.jpg
03065c494ba9b264dbc1f80a7490af25
81168 F20101208_AABVLZ kim_u_Page_71.jpg
d6ee4bcdc72bbfa53affa35f0261cb08
07e75af24685e928b2ed6fcac9802b963ef97503
111426 F20101208_AABVNC kim_u_Page_10.jp2
9e884a0eb102eb1363d558e7bf785a4a
1dfd6e95a21e315ba9e2417b874285474c5f276e
109728 F20101208_AABVMO kim_u_Page_87.jpg
83c19ea2e5dc4a463118e97ae0579d77
5c6d74c030c7f0b96267e7c0ea319bbc0ac578c8
28409 F20101208_AABVND kim_u_Page_11.jp2
117072 F20101208_AABVMP kim_u_Page_88.jpg
8dc064dfaf321d4bf2a501eeffc4b3db
586f6098614dcb21d14698b9d6ba422309a07a12
1051970 F20101208_AABVNE kim_u_Page_12.jp2
945e8a7ceb6ef344833f9d76ffeec84d
2cc334b74b5f5f2ba0ac629fcfb6646c6788706b
115269 F20101208_AABVMQ kim_u_Page_89.jpg
e690fc36754f94f20db35f736283b1f6
da385f5f2620ef586e56af629d371eafdd4c814d
1051913 F20101208_AABVNF kim_u_Page_13.jp2
d61eb6f74fdf220277b4456b35ab9972
e39bf931b8cec3ba84a9e64f6434e69544ae492e
43162 F20101208_AABVMR kim_u_Page_90.jpg
e8267b26bf2883da7008d32032ab73f2
89a3438eb1a5e6898e73acc4b118f8c65010f12c
39537 F20101208_AABVMS kim_u_Page_91.jpg
168409f4ae6b5c2cddcd3d94c9a5d7b0
26183ffb5be9143abbb0141a83c0c6f4228cc608
101912 F20101208_AABVNG kim_u_Page_14.jp2
79a0b463bda2abd10367dafe8ebde6e2
31a55472c9d32ab9bda6d39b4b7656c7d0589e14
24242 F20101208_AABVMT kim_u_Page_01.jp2
2dacf88eed3ecfef5dea7087b9fee7ca
a0db46ce84b29be36c258daf5343e0e692f3def8
912633 F20101208_AABVNH kim_u_Page_15.jp2
ca876b108d9589e806c0fa75fa1e9a101e95af2f
4873 F20101208_AABVMU kim_u_Page_02.jp2
20ffc823198d43838167f9db6b36a7a4
b5ae97b0ca3da6ba407b2b62b38ac335052aac96
1051985 F20101208_AABVNI kim_u_Page_16.jp2
09d303c5ef53c56e504f5e3cf5c3a681
9d40f4e8eb1c8845dc5d638da76a41c7783c0b59
10506 F20101208_AABVMV kim_u_Page_03.jp2
9d6aa5bbe3683da8aa72698419ac63e8
40f0c955e385f76706f404ffac7611e1f955314e
1051954 F20101208_AABVNJ kim_u_Page_17.jp2
58ff072ed4cc6e13126a8d1ea4a848df
b655b0477ddba3e547e164b938e3463e6558762c
43784 F20101208_AABVMW kim_u_Page_04.jp2
e370630a9b5da59138369fcd198c8180
36c08655d38fc87ba7a11ef44ec15b363341bc5d
702674 F20101208_AABVNK kim_u_Page_18.jp2
a0ee8fdac58b93f1c59b6780c657739b
4413d141f6926c675893be591a0c2e1e0bc964e0
1051977 F20101208_AABVMX kim_u_Page_05.jp2
2c07b789c36d8c82939e831d8abd14b7
e6384b9c9ee1468be07503fc2bfeca0dae40b505
876561 F20101208_AABVOA kim_u_Page_35.jp2
6db9a09ab29b002a453dcb7583c68d0d
1051986 F20101208_AABVNL kim_u_Page_19.jp2
f12fb5b1ca5df3b336395418d955ea1c
63c8dcf71ac7f62823a6fa1c28dcbfe7fa32d84f
63362 F20101208_AABVMY kim_u_Page_06.jp2
752110a643a40859f43c89775fb8a877
3ab32bdd3037ae51268c91db296a055abd6d711c
1051956 F20101208_AABVOB kim_u_Page_36.jp2
a8273aaf4f9b6e78c159a7d375f2b2d8
783576ac214b59e273bbf4c3e9cd10561cd7a6e1
808334 F20101208_AABVNM kim_u_Page_20.jp2
eae14ee8090d85722fab33330f843be3
250d0d54cf7e138e86f6f8c4c23a271810fe5004
267503 F20101208_AABVMZ kim_u_Page_07.jp2
d301852d79657a16bb929c8148d82959
5e5173a8b4733350d969f32c0f25be4cbb449791
925492 F20101208_AABVOC kim_u_Page_37.jp2
7d0ef96c1c0999a38442064d7dfe4ec4
F20101208_AABVNN kim_u_Page_21.jp2
a77a3f4bbbf79360ee59f3cc3b7ee7c3
aef97ba08e2f8fa404d36ecb150ac04225e8044a
927178 F20101208_AABVOD kim_u_Page_38.jp2
a3c9cb8f53000aebff3c2cc5df7ba1c3
0932ef0b4b1bfc7ff6cf5e26154d8fb8851c61bb
238230 F20101208_AABVNO kim_u_Page_22.jp2
769928073ebbe601ea88aca8c4959651
24ae586d41f1e360852a67355b882984dce0c4bf
968576 F20101208_AABVOE kim_u_Page_39.jp2
f8d835d4bbabff7b156a52cbccbc81b7
bd639804ae1a332d28754a7711b7b31576344e27
855482 F20101208_AABVNP kim_u_Page_23.jp2
46216619aba8ff10d8d8cb5681a69d4f
942375 F20101208_AABVOF kim_u_Page_40.jp2
0ede9f5d64ded799f4d8a69a0a11d73a
2f7044cd53d9ab495e8506c367351756735a6ecc
117103 F20101208_AABVNQ kim_u_Page_24.jp2
0f565f4c0c2b450f58eac992bf4aaa36
db869e8fbc2a56488c4b7047d88d54ceb0b8f505
763681 F20101208_AABVOG kim_u_Page_41.jp2
ac965507164532f08b3968057fc05ccd
1051663 F20101208_AABVNR kim_u_Page_25.jp2
a6f0d5cd69becd4b020811bd4f76a9b1
08aaa46931e17abe840d694d952e592b464dbf38
1051975 F20101208_AABVNS kim_u_Page_26.jp2
af56985d6e266d890dec2c6fc7e8c94e
f9e95ceca68c842bcf831474f093512e45195c72
1021572 F20101208_AABVOH kim_u_Page_42.jp2
db34547e4696e88f3b0af0721741dd07
1051926 F20101208_AABVNT kim_u_Page_27.jp2
dddc0a9a6f31ef6e04384917a13a6f02
690738 F20101208_AABVOI kim_u_Page_43.jp2
b42aa8f70ed19fdbab0f5418ee9f4662
e603b0c4ce7ce660bfe70bacc4693beb45a0625f
1051972 F20101208_AABVNU kim_u_Page_28.jp2
4dd00e216dc8cfc2c76d9a8ea17acab9
376099ddd14ba4d2789854949cf3ec20f7ba6f57
1035174 F20101208_AABVOJ kim_u_Page_44.jp2
faa251e7f6d326efbdffa361d5bbaa6c
1051936 F20101208_AABVNV kim_u_Page_29.jp2
ed55617d1b27a5f8a16c1a7b8a3146a9
9887cac9173db9e1913ca3086e4547bdccd4a7e4
1040186 F20101208_AABVOK kim_u_Page_45.jp2
f8169f68671cb362e5a9c01390ec8613
5108495582634ab15d1c1cb21e36db5e0ca78b70
909365 F20101208_AABVNW kim_u_Page_30.jp2
6cc36a69e7ea3a115e3f6e6b37ff3d51
1eea40d39c54473ab0ff96a9bc5e58870d6ff413
727915 F20101208_AABVPA kim_u_Page_63.jp2
68845aed39bdf96b3c23229c1b82515e
2347aa1f6e9a764f8fe7ce55b24ddfff0864c192
908624 F20101208_AABVOL kim_u_Page_46.jp2
7402689bc28eb4b23f61be3a81493124
2c248e73b403e8eb62810bc8dcbe3b9730f22659
1051958 F20101208_AABVNX kim_u_Page_31.jp2
1677883bffeb9eb3800fde2fe87affe7
40bb56576466a33995b58fca2eabfa235954454c
1051967 F20101208_AABVPB kim_u_Page_64.jp2
5a49cf8d5bd2855020b134e2ff21285d
8716c6e91632ca006d9d87881a65d9b4df3f41e5
696493 F20101208_AABVOM kim_u_Page_47.jp2
9da56b9674050d680cb5bec159cb02d0
56fcf49b519fa0da10000f586970c9236b8af7c5
F20101208_AABVNY kim_u_Page_33.jp2
872c8dcb0f5f51a930aa200ef5a1cf49
F20101208_AABVPC kim_u_Page_65.jp2
d5a6ed982db97aa71d66723663d0dcba
c938cb62c306214d53ae002fd67d52c2149610cf
654191 F20101208_AABVON kim_u_Page_48.jp2
f15d3718d62d0b2f6f63aec33a0b848c
dd3d3da91d91cf4e825b912d8954ea813ef8abb0
F20101208_AABVNZ kim_u_Page_34.jp2
a6066a8905d1828c6d04bb0bb966e809
969272 F20101208_AABVPD kim_u_Page_68.jp2
240b8317a9df7341c5a5bee156c3a298
75a1da3b27ae211e332c511a25a0cb70f3f06368
1051974 F20101208_AABVOO kim_u_Page_49.jp2
237d91ec6d0c1d340128a30aa8144429
a781b5d1148fa1142c0e089a9f609482fc54ff2d
F20101208_AABVPE kim_u_Page_69.jp2
a8b664894a35e9a287f88ab6e3b49f95
300e9ea9f32de651d20daac365b220c0eaea553f
789089 F20101208_AABVOP kim_u_Page_50.jp2
b3c9133ae45e5bd33d6639ae2c7a28f8
26293b3655a647eb4560dd269c2c6e9cf87d501e
744543 F20101208_AABVPF kim_u_Page_70.jp2
6d467c5412dd5c21cc64ede5c5aa1cb8
0d5e1eaab4bda41af3f26511a80afe35ccea533b
F20101208_AABVOQ kim_u_Page_52.jp2
1f17a6baca420a2e11f4b50e93c04c4d
b169873bb4f97bbc0358fdfd64f34cd364db65f6
1051964 F20101208_AABVPG kim_u_Page_71.jp2
e95f8bde73f89a05e507b9da98dc3b10
3ae32e723370f8d0490c8563e78fc1f38a34de74
F20101208_AABVOR kim_u_Page_53.jp2
78891d90aabf07738b82073e4f232e2f
505711 F20101208_AABVPH kim_u_Page_72.jp2
27884bdb2becd383de60aa22432a50cb
633597 F20101208_AABVOS kim_u_Page_54.jp2
9283c829f2ffdf5e6d688ffddb360f46
60d98315139eff30cdeef53b0b38f580385353e6
1051976 F20101208_AABVOT kim_u_Page_55.jp2
d4eccc28a4d3f4f72a5463a8df40188b
ca51314d033e8297e2e295d43ff4a338bbd0b140
897162 F20101208_AABVPI kim_u_Page_73.jp2
305b4a8953434e72a837617abd1b105e80de41d0
960436 F20101208_AABVOU kim_u_Page_57.jp2
094b1a03e941cfac8c139aeb2dd4655a
1498211dfc02d791ec68c6b87e37ecb1562c568e
F20101208_AABVPJ kim_u_Page_74.jp2
e7284c5261f49baa99c6e70f36b1653a
f372e6d87c4268e85e1ea32f2662a008acdf185a
795549 F20101208_AABVOV kim_u_Page_58.jp2
c3b0167e833fc0a96c4eece73e3f4d71
F20101208_AABVPK kim_u_Page_75.jp2
297a6c1a898f0740e9c4ef6d05685ae5
81274c942448575da10c9f04bfbd9711bd78dcf0
982545 F20101208_AABVOW kim_u_Page_59.jp2
10edc55dd514209c19f561d8e13b566e
F20101208_AABVPL kim_u_Page_76.jp2
5d2606bcca4268347d0480ff7bc4c660
36f6c5c28b5629ae434ce55a5be74c18b1a6c8a9
1051962 F20101208_AABVOX kim_u_Page_60.jp2
975a134e3e87a00b1f651b3f6f877dd227e41bc6
1053954 F20101208_AABVQA kim_u_Page_03.tif
b04bc109c39b07dba7f848b6db8d991eed00c304
932185 F20101208_AABVPM kim_u_Page_77.jp2
8f33d1a9b51bd0acd78b2dbc329dd3e8
e52b0bf39fd1040f6d4771ed8d3d8944c4bdd339
840419 F20101208_AABVOY kim_u_Page_61.jp2
3274583e662ff2c5d09337c75f5c7c7b
c25c55ddfd1a12c2ef3f848870b18975d437465c
F20101208_AABVQB kim_u_Page_04.tif
08202de0cbe92c1e9c97ca929fc68c60044ba083
994152 F20101208_AABVPN kim_u_Page_78.jp2
b9a7b6477b9dcd337b4576dbe09aed3d0e4aba56
143129 F20101208_AABVOZ kim_u_Page_62.jp2
0fbe7d8dfda3cbd2043959b50c6cf26a
92bed7ede474bc22b28b26e0d3e1275eeb9fc3bb
F20101208_AABVQC kim_u_Page_05.tif
bbfe059c3c8a3bc87a2c3ea92d83b730
844800 F20101208_AABVPO kim_u_Page_79.jp2
00514b50693c15943e714151b5bdf466
b36e3e4b0bdf76698ea99159ef73bc703f25eaba
F20101208_AABVQD kim_u_Page_06.tif
dc719ec9cfb42bc1dbed62244ae04470
93c39498be9251ab0df2e394bd208a3e99b72999
1051984 F20101208_AABVPP kim_u_Page_81.jp2
26a7f23a1b910f54c71cfbbac8a5ccfa
0912ddcc70febe1eb4371a13298c21a0e129e9d7
F20101208_AABVQE kim_u_Page_07.tif
11cef55f23207b8c6442e34e281276b8
fe9e6ef12e752b47d44deb5a14346bbd1401a2d1
F20101208_AABVQF kim_u_Page_08.tif
6abd4ac401445b93d62b2a8ab26d6eb9
9e88f1e64b4a1a84f5d0886e7dfaa741e2c33b5b
398811 F20101208_AABVPQ kim_u_Page_82.jp2
9ff501efb3c488533afd616efaa9e59d
0bdb07349d3105d16f3a83a53c8c56059e0b6997
F20101208_AABVQG kim_u_Page_09.tif
1cb2c4483bc759f824dd42363f4b7cdc
eb0b5c790c75aaa6a4f0e7dbfa079c46e55ddcb9
117406 F20101208_AABVPR kim_u_Page_83.jp2
db803408b17a165f218bcec38b752fe6
d437aa960a145af2319816a52a493b2bf44bd8c4
F20101208_AABVQH kim_u_Page_10.tif
1b7aa2040aeab5f8e2b4de8e2c0bc1551aba9734
124602 F20101208_AABVPS kim_u_Page_84.jp2
b02551d1d8e830a6bbd61533f672dc7f
4c04125b84b85a5f227fbd6907d781e78111c0e5
F20101208_AABVQI kim_u_Page_11.tif
8bc3f90d52d65e325c78252f940075db
115116 F20101208_AABVPT kim_u_Page_86.jp2
e6c4b8e35a333a206f4af45d3b06c5e6
20b1ecd71b7d5d8fe399186bbcbd249ff07471a2
120971 F20101208_AABVPU kim_u_Page_87.jp2
3f097944d52745aba0dce9edb2c4afe8
F20101208_AABVQJ kim_u_Page_12.tif
5865b80b22a86b21fec43e3b600a4a31
049aa3ff2c7a870f4622e0c5dd89993dc497d740
125487 F20101208_AABVPV kim_u_Page_88.jp2
a10c450707fd77508420aff64820a78b
6dc46f48f1a066b6eae5da07e4d2231b237da4c0
F20101208_AABVQK kim_u_Page_13.tif
ae3645720a76dd376997fcac80508005
778868224ae763873886b77707fb316dc60f8937
125377 F20101208_AABVPW kim_u_Page_89.jp2
1bb8e44154fe596e35b365f5d92fbdcb6ec4f72f
F20101208_AABVRA kim_u_Page_30.tif
2c57e52d24dc6265e16ae126ab7cccb6
F20101208_AABVQL kim_u_Page_14.tif
4c626b29c214470afe4fb1f03fe23bdf
da40a1d7a5a4a2df06d35b5b6a73e9fb6bddc0a5
45132 F20101208_AABVPX kim_u_Page_90.jp2
13f4b0da634f25785832176e144aac80
eda964a066a90537e0034a81e2d7e99b98bb69fa
F20101208_AABVRB kim_u_Page_31.tif
61cbbf3acfed4fc9af57d98afe9a0128
611f55666187d49b70a8af87e7094600de2fdb47
F20101208_AABVQM kim_u_Page_15.tif
6d2ef903cf0ceb6ef9671d1f589aa90c
1932078b72db4fed46c5e59787fbe7dd69a395b5
F20101208_AABVPY kim_u_Page_01.tif
926d7ca96a09742c7c38d1a77769e4aae8ec7922
F20101208_AABVRC kim_u_Page_32.tif
76661db397c0107a344a16563ab58b18
3f0b6c3c44e4ecae57fdd2b0bb4694074f9cc826
F20101208_AABVQN kim_u_Page_16.tif
95d9a7bc9810ba387859a435a48cea25
50b0c853b967fefd6ab253e7563df4e4be7e245f
F20101208_AABVPZ kim_u_Page_02.tif
39fd27cc26409cc3a902647a7a820058
75de268d2e38b393f76bfd9ed495417f29a48352
F20101208_AABVRD kim_u_Page_33.tif
37d7e14e69fa96efd5a81fc0c0701cf3
19ac85e9fc1b08dd2d610bcfa2b2e159f7767530
F20101208_AABVQO kim_u_Page_17.tif
b54b9cf1f2f08fc9dee27e8d28b9a7f5
6009a4b5ecea90e38035ca14a7deeebf01b1cb6b
F20101208_AABVRE kim_u_Page_34.tif
c935b4247b5b19a6b95aa09cdf61d030
3e87e101aaf7785096396803c20f09de4a006080
F20101208_AABVQP kim_u_Page_18.tif
69bb0f27419032c7f7b1b9f79bfebc9e
26349467f8de9507bc9742e67bb80d5a159193d8
F20101208_AABVRF kim_u_Page_35.tif
0283d3b21247f6992312aef006228823
1ed6f577ab90ec88e424df06fbe1cd7e0a03fae8
F20101208_AABVQQ kim_u_Page_20.tif
bf5a55a689793b64e9157cfd23b61efb
53efd9f49a51f1a81352904de8280ee67a78ca47
F20101208_AABVRG kim_u_Page_36.tif
e165c225e286a736d698638537687c62
F20101208_AABVQR kim_u_Page_21.tif
cab9456b768efffe65b90721f11de9a44e6f0f23
F20101208_AABVRH kim_u_Page_37.tif
ca532a56d3f84ceffbdcdd24613a55b3
F20101208_AABVQS kim_u_Page_22.tif
098cb813eec23fdbff6d7ab925bbd351
32245e6bc74b6fc103b86b6a7c493f2f187a9305
F20101208_AABVRI kim_u_Page_38.tif
b69b98018f78ebf610ca5a383a5567b3
718f388e2d41ef7f6c96e9b30cb72fb35eb8a83c
F20101208_AABVQT kim_u_Page_23.tif
0f46d44566d1740d4cc385feb9f20f29
fd5e37c3cc42cd15a6f3bd6b13686c7e23cc7d4b
F20101208_AABVRJ kim_u_Page_39.tif
1d8df49d245ef6f73ac3b6519709bd583a306870
F20101208_AABVQU kim_u_Page_24.tif
2c73363c21ed68fa92a094bc71df0618
e150481a45e7fea58620ebfcc720f7f3da655aa4
F20101208_AABVQV kim_u_Page_25.tif
219ecbb2a2f91428f07f4fddf2fa5430
a992d810f228ccde0d0914b09f7f316179f03801
F20101208_AABVRK kim_u_Page_40.tif
95e760b4ae9a0a00d2a03ebc7a87a6b7
F20101208_AABVQW kim_u_Page_26.tif
e0f4348e79de4caaacc069c92e51c1fc
bb0ccf03c7b4f99864868ca8d6673933b1a1a633
F20101208_AABVRL kim_u_Page_41.tif
497519fcde78714522ef291fed8c25dc
F20101208_AABVQX kim_u_Page_27.tif
08b1e44ef6970be767a08a126d93c3e6
be9bcc6f822c4374cff2f3c292897a1b917624ea
F20101208_AABVSA kim_u_Page_57.tif
f05011285630db5e251cff727a8246b9
6a60cf8b10d1f3c49bef951edd5074cfa38bf331
F20101208_AABVRM kim_u_Page_42.tif
d51d5e178ac22450cbb94675800fe0dd
2a662f56fcef10a47143e7b536beaf1835b5155f
F20101208_AABVQY kim_u_Page_28.tif
68be93f39ceb63c18e1d9af12f1cd04a
357a01e259805522a6ec2b03788851e1f54cd3fd
F20101208_AABVSB kim_u_Page_58.tif
3b5cef47408aec1fcce876c6394e9dd1
d4a1514fba5207454ef8de72c196ca1a20af75ea
F20101208_AABVRN kim_u_Page_43.tif
09d024ff6df19c18c02d3a511a3885fe
250cbe33ac61fb9838f214ebfe4646a75cf1d213
F20101208_AABVQZ kim_u_Page_29.tif
b39e9592b3b47f45fcc84b5f79649db6
e58cd71b22a294a9bc18178b7b81ac097068ece7
F20101208_AABVSC kim_u_Page_59.tif
58ffbb32cc5d414766fa57e0ac8f6f19
eb0d3bcb1efbfe5e05c5650cd2d9edd1cf8511cd
F20101208_AABVRO kim_u_Page_45.tif
9a23bfbbcab776d224f8f071eea378f1
5659f58fbabb3266179c010d06e41e9a7f633377
F20101208_AABVSD kim_u_Page_60.tif
01833305950eeb1fc5deb1e067dfdfd9
bd2e523f5f700861793807ec654d1d1abbcdfdd2
F20101208_AABVRP kim_u_Page_46.tif
c11c166fc29561067cd1618edea942e2
bcfe6d8fc9e852a22b9f3143b464ebdcd1b7e95c
F20101208_AABVSE kim_u_Page_61.tif
6437c2ec9e9650d483d2befb373fa3a707cb6e91
F20101208_AABVRQ kim_u_Page_47.tif
a6a3a31c50a2c724fa3ab68643d9d555
199c5850ec86bba1b85fcd6a2f0e597a2aa741af
F20101208_AABVSF kim_u_Page_62.tif
0029a17d309f46b7c6f9e0f86344905f
5463d2088fe04bcb08b8e1f9f775705fa6159a44
F20101208_AABVRR kim_u_Page_48.tif
671a04b07be98456a1919ded56453066
4b879507a411fc5872c4b98fc35dcb1e5fd6d4e8
F20101208_AABVSG kim_u_Page_63.tif
5a77b06a6c120fd0822632c69d463974de5dcd39
F20101208_AABVRS kim_u_Page_49.tif
e61c9f6c6a05b2851960c9ba49b7c044
0becf612c7e89c371f0f8b43508836b052a3a7a1
F20101208_AABVSH kim_u_Page_64.tif
0d7a4e86950c8a07650b87d06f2f8298
24694fbd8740e41c21bca8b2b832b0873c1bc55d
F20101208_AABVRT kim_u_Page_50.tif
cb1c96d44e70710894c1892266fdae2d
283d883fea14d91a409e2a3c9eef3235611ce441
F20101208_AABVSI kim_u_Page_65.tif
5d278c05cc442554d3be485579a5248d
d82b483e179b0d7962fc3471489c1d9b3ac117e4
F20101208_AABVRU kim_u_Page_51.tif
038b7ffc652681a31d9734abe40ed49b
4a353f350e5ce21d2e63e7488dcaa6bce0e0b99e
F20101208_AABVSJ kim_u_Page_66.tif
0099a9ddbaaf54ea76333c947c623ee3
F20101208_AABVRV kim_u_Page_52.tif
ec8d2c77c120df0da0499814fd20b428
928aca2242b361b4c2fbba6fd59279c474e869c0
F20101208_AABVSK kim_u_Page_67.tif
9e41c59528dc928ba3e6e723d87f83e4
04b4d3e2999cc956616ac8be6e4f0c58769a58fb
F20101208_AABVRW kim_u_Page_53.tif
b1c41887afeaa0da807c0070f701f3fb
ded77ba42bb2b8ca62def787795622baa2a06959
F20101208_AABVRX kim_u_Page_54.tif
714a5e8182f0761cfdedc5341a74959a390465db
F20101208_AABVTA kim_u_Page_85.tif
7e0991741fe947a1939bf11974d9e5a48cae1961
F20101208_AABVSL kim_u_Page_69.tif
009f4fc9ce7f689d0d97a25c8728303f
F20101208_AABVRY kim_u_Page_55.tif
b0a989859b93c1e0a5c9f642ded279d1
36523cf65b382264fbbf649d8e2f4ca04a561ae8
F20101208_AABVTB kim_u_Page_86.tif
86cb20e0d8887a8bc551ccf7c57af4ea
081de2ab9a343cc60dd21c52cab294906dc3545a
F20101208_AABVSM kim_u_Page_70.tif
9a9224755bbea60f39ca2d519905ac7a
0421e704b035da877435fd6e70505ba1f41a0486
F20101208_AABVRZ kim_u_Page_56.tif
738ee294a68f5013119d97fd171ee479
2869bf0b80a830e04ab18e8f9cf6dd2721d8c5f3
F20101208_AABVTC kim_u_Page_87.tif
f77fe679a7396c261f7cab7fda5b9d35
445cfeeab1801d01887dac83b610e1ac1a772db4
F20101208_AABVSN kim_u_Page_71.tif
13b53c5128b0a1339b7dc3e52cfa6749
6195971c36c222c07ec7be0b1a19bc3d01dedc31
F20101208_AABVTD kim_u_Page_88.tif
7d180fdb361606b6c0c59fe2b5e6c1d8
0a70a2e9448c6f9fcc734bb93752c451738006cd
F20101208_AABVSO kim_u_Page_72.tif
49f3148235f4fa3af45a07277babf92d
4c4876f4cc491b07936e7854330d67c9b5ed2923
F20101208_AABVTE kim_u_Page_89.tif
bb829f29afb8a2da41fc8ccbcf600be5
5197e8ba71311aaebb8014957c3c061bedb0e58d
F20101208_AABVSP kim_u_Page_73.tif
bbd7d163900518c628583d1e57dee8cd
d8ea777e2a5167f31ed0615d6b18b28f0dea6979
F20101208_AABVTF kim_u_Page_90.tif
a404ebbcf383fc3c13f52902defed69c
199b7a81a25d88298e493e628648f70cb0ceb33a
F20101208_AABVSQ kim_u_Page_75.tif
691fbce61d891d385e41b671107de8a1
26edb0cc39a7227ed6ab773e978d808bfd6dd586
F20101208_AABVTG kim_u_Page_91.tif
fcd5cb1f61b9977c325857873ef12c4d
F20101208_AABVSR kim_u_Page_76.tif
16bb038b606eef19a4a75aaf2187c6cd0e4499d8
8156 F20101208_AABVTH kim_u_Page_01.pro
76d18f3b022ec6d336f987197a5a42c0
260622650d1a0ed54d672cdac4b19f15a46f6c53
F20101208_AABVSS kim_u_Page_77.tif
38680dc624692bf4ee9b61543e9b98b3
912d7ff3fecd15e2894477ff505f4ed864e9d060
722 F20101208_AABVTI kim_u_Page_02.pro
e18d1096e7ae65f277a1155689bd61ff
11feb828abc0f2bb9fbc6c353a578906fd0bf7f8
F20101208_AABVST kim_u_Page_78.tif
12dc703ecfd976c57737b80d20f682a6
8833e6b8780ab35ae2d4779a6f2a6a0d06bd19b5
3311 F20101208_AABVTJ kim_u_Page_03.pro
3325cc09a03481d547d0f65e12262fab
c991149d503596760fd34bb005fd158d6ffc4304
F20101208_AABVSU kim_u_Page_79.tif
9813fc4f15949411429033204fe45254
6b09eded5dd765227c81357e8ddf2d21f88d759e
19267 F20101208_AABVTK kim_u_Page_04.pro
63ea8999ce2d5cfc9e122e7c231f24f9
2366dfc5ce108cab8f80bc40e6791ff3c1c71bf8
F20101208_AABVSV kim_u_Page_80.tif
d436186e0766fa7e81294278faa511d028bb1669
47265 F20101208_AABVTL kim_u_Page_05.pro
f6c5c6b4114abc4c28fda9cff93392b8
783de317579160d633fc7b2785b889034d7fd770
F20101208_AABVSW kim_u_Page_81.tif
52a68557a10df73f428ccf8672517685
3a5e80fa1ae2af3ce4b3749dbe97501b734cd140
62482 F20101208_AABVUA kim_u_Page_21.pro
b90942208d0735fb4343e160fff495c9
2deb91487dabc2d049a038de134c4549752e273d
F20101208_AABVSX kim_u_Page_82.tif
b188d34970d22c7aa4b06936e1859f18
82908eb2992f2cd86fb826a9d576e675486cf80d
10376 F20101208_AABVUB kim_u_Page_22.pro
02475bbceab77de57a8764a19624860b
02d725b458174becbf8a710a15242a879f3e1eb2
1734 F20101208_AABVTM kim_u_Page_06.pro
d6e1c242a084730434e3278121f8fcc3
81696478143f36a8bab8d6f108a09a4b15c2ae3e
F20101208_AABVSY kim_u_Page_83.tif
77a88584437a57690771a4bd12d6715b1c498266
38466 F20101208_AABVUC kim_u_Page_23.pro
db0c45f155f33134770389df13638098
b5149fc25a8fbd14325d1801610b2abcccbecf3e
7077 F20101208_AABVTN kim_u_Page_07.pro
c2399d3f614c4dc690246dd34f278382
4b580bc9d68fc25b93872eeb2a8b63d72edf9af6
F20101208_AABVSZ kim_u_Page_84.tif
18ae4dc1225fcc4b2a771d3452d64912
fb7713a01db1b9eb745284e2c083d35b7ee1f493
57417 F20101208_AABVUD kim_u_Page_24.pro
56f8b2fbd7ae146d734ffa52c37f5cc4
833ae71c145510b08316b9c59d1b8b4554236f65
52559 F20101208_AABVTO kim_u_Page_08.pro
0951c39508e9a7c711e3dd72748f814b
71d12b2c80563458246cedcb06ce15442a4eba71
43509 F20101208_AABVUE kim_u_Page_25.pro
63d4fd133fbdba36e84a2b7a5750849c
a14ac87e5c6eaef152665bb4bf341a284d83ef48
53231 F20101208_AABVTP kim_u_Page_10.pro
13b3e1e8c1a3ed541e1834a34f8de39f
36909026b0681b1d121fea6d0c386105454b7c7e
31249 F20101208_AABVUF kim_u_Page_26.pro
22f460f15b3455b949c31451cd3dd5d0
b245386f1a6da9bb907b4cac03e9af27ba02d517
12054 F20101208_AABVTQ kim_u_Page_11.pro
139a61f35ca123156e9ac01245aeab7d
6835c1d9bfc3645890c17a434fa7cafe909c6848
54935 F20101208_AABVUG kim_u_Page_27.pro
e0840743d1795e0689dd2fa3cffe8c1e
ab5ed4f5fe6af5055917c555f05bc9fcaa65c5cf
55618 F20101208_AABVTR kim_u_Page_12.pro
216d7a9e276d00f21d7cb813e17e415f
b09528b91e5a5059b3e14f3ee1ba0385669f88fc
34055 F20101208_AABWAA kim_u_Page_24.QC.jpg
eb0541dd8a6907ac4a472c1c5eab28a6
d6dc872630fc2db7f77d888d3ea806cdac8c06d3
26726 F20101208_AABVUH kim_u_Page_28.pro
7d4f202bef15af339b7eab412fc8e2ab
22a6010d1805381ffa887e62784f698f927e14df
59100 F20101208_AABVTS kim_u_Page_13.pro
b11fab1c627d7774e579801aa42a6e56
25944 F20101208_AABWAB kim_u_Page_75.QC.jpg
0733b15eb25513ffeb6b6b08163cf68e
64690bd13c26aa77c3c1cebb5979b97b0344e0f4
57408 F20101208_AABVUI kim_u_Page_29.pro
bd7bfbf35e359b4dc7f9088784956175
3a542287d5429e4706706b620416b784d3025ef5
50602 F20101208_AABVTT kim_u_Page_14.pro
8e0417ba7e157c916b1034da26f02a69e80acae7
471 F20101208_AABWAC kim_u_Page_02thm.jpg
6fbd742a4af86009de6d4ef9a4717b73c06aa31d
38525 F20101208_AABVUJ kim_u_Page_30.pro
a1d2977997f47cc01f6d2d9274a1f09a
dacc1fa8a0fd284957025a11cd533ae6f06748dd
37217 F20101208_AABVTU kim_u_Page_15.pro
2f701240da0d06b2c5490c3bbcdbedda
48e4e84de1e3e52e85ce125a95f9baa766fd11b5
f94096b844e28394a647a316a61889fd
47d16b651d9e960be5cb0a32112a29303d2ddcd9
63687 F20101208_AABVUK kim_u_Page_31.pro
aa8a2af6c9df508d662f264bfc0cb033
ee620a7886d081ec201562331dd255d00dac2014
53358 F20101208_AABVTV kim_u_Page_16.pro
2506c9235fd0b082b71f2d6e712ac5fa
fffa097340af5c5c64f55009e14fb7c6e52f3f2f
24017 F20101208_AABWAE kim_u_Page_37.QC.jpg
3d1c01ed1c241f80512250437a3ba5f4
37773 F20101208_AABVUL kim_u_Page_32.pro
1eba56c305654aeee41cf68520dc273f
1fcd6db3b321818411dc82d376faef76b2a797f0
57814 F20101208_AABVTW kim_u_Page_17.pro
536d42c9b7c23df6a994cb6b80ef9b62
1e7cf9868ab4abcac3c4475f9ffff8111ff61d30
8578 F20101208_AABWAF kim_u_Page_31thm.jpg
f158dc5a62748b587db084b654df390c
d86c1d372ed794bf6fff858e324b91df0dcb0afe
55231 F20101208_AABVUM kim_u_Page_33.pro
d5f5a33c5df2e2d8cb22774f1c752fc3
2e94eab65bffeb7ebf2ff2bd074a9f25e73039df
29021 F20101208_AABVTX kim_u_Page_18.pro
2793 F20101208_AABWAG kim_u_Page_03.QC.jpg
c1ebc29d92cf09c97923ccfe17b697ef
eec9e36656c56c1b18ecdb3f883161e819e475f7
48048 F20101208_AABVVA kim_u_Page_49.pro
188ebcf7ae2007672ba3886f774489e8
d064147e03d369033f75ff2ee43896b81af2712e
62612 F20101208_AABVTY kim_u_Page_19.pro
72c2a332c174d47f32f5289d42a856fd
19351 F20101208_AABWAH kim_u_Page_61.QC.jpg
6dd99f9b1ef273315aa28fb2db1fc4b4
13bfc08a3176150345042f2fab475b0b485727e5
35856 F20101208_AABVVB kim_u_Page_50.pro
51482719accf8ee3e018e598d32c68ea
accc3abc9fa2f3d356ba6a46295785462ff10b1b
22798 F20101208_AABVTZ kim_u_Page_20.pro
70511a29f15890f29887ff0de20bb4a6
9e1e3def45fd77e2cbd535d3931621ceac8c3260
8021 F20101208_AABWAI kim_u_Page_11.QC.jpg
b0d3e6dd0a91872259baa7d825b2da7670fc67a4
25391 F20101208_AABVVC kim_u_Page_51.pro
91995f78e2b5b318e0868f48dab0121a
56062 F20101208_AABVUN kim_u_Page_34.pro
783b5a645009301f3ea86a3be39e12f9
79e8f00c0e6752106f4921c65ee658d8669118c8
135140 F20101208_AABWAJ UFE0017534_00001.xml
0190b0edff5f8a219aa5aa7b49aacdb3
62325 F20101208_AABVVD kim_u_Page_52.pro
0fd090b069b6c27cdd4746127f7f269c
b8d1bfddb59ce2b5c72d4c9a237bdaff5e1c7e49
34709 F20101208_AABVUO kim_u_Page_35.pro
906bd3d16863d12880705a02046ac087
1e2e5cf69eb2c59f3504437b00c99b5cd4660263
1846 F20101208_AABWAK kim_u_Page_01thm.jpg
79ac48a2486695c7307b203d5350b9af
f3dcc985ec94b68ac5bbb35fc65cb7753decf41b
51107 F20101208_AABVVE kim_u_Page_53.pro
7b3da90e6aa97223c3974fb26fdbef526d8c61ea
51871 F20101208_AABVUP kim_u_Page_36.pro
d9b09df3e5944a79208b0812f108e971
584852e31047b185bda1f67b89d24677f2e14e21
8094 F20101208_AABWAL kim_u_Page_01.QC.jpg
2c349f8bfb463a536470d1fbc82ce62b
09a0eb1d7dc017cac2443c9e37b26bf79a9d1bd0
26487 F20101208_AABVVF kim_u_Page_54.pro
e974ba7f7c1cf17209936667aa011ed9
8f68a94bd82c8ccc14c6a977ccac16f8d81ea796
32618 F20101208_AABVUQ kim_u_Page_38.pro
34822cc7d9c89f861a321ff00efca088
eeddbf91f8783ba45ed76b689d3a95c9342d1f97
7758 F20101208_AABWBA kim_u_Page_12thm.jpg
389809fdae7a047fe586e6db9a4d5e94
f6e960f1a95c48217a710d5afc152da3840eb8f2
948 F20101208_AABWAM kim_u_Page_03thm.jpg
3337c9bcaf0e09cd098f406f2a9a5da0
c12105d15e5ddb2e1642344f98aeff5397cb3bf4
28567 F20101208_AABVVG kim_u_Page_55.pro
3c8d8c649bc35a97b4da8d70740a4811
78ee14f3b9f002ff8af85042db693b30ff343310
41616 F20101208_AABVUR kim_u_Page_39.pro
57d581cf31a3bd55caff66418fcc3a98
9a548cff1384d99d3c7b28724aca0f66c6d13355
33489 F20101208_AABWBB kim_u_Page_12.QC.jpg
a43ea60b7506dd05fb7501f21f5d6fb9
782fac9dce8ed0d29fd9580c4907aa2fe1609eb8
3028 F20101208_AABWAN kim_u_Page_04thm.jpg
2003840db0acae21c362d98871757960
f6940b22345be7928114fb8c9c18d57f020cf3f2
25044 F20101208_AABVVH kim_u_Page_56.pro
64ba6ed8014b49583df4f10a8bde8658
d9a0a5fef6b6b1a53f5882d2b3a94412e137dc95
42599 F20101208_AABVUS kim_u_Page_40.pro
6d7646e16982426aec26de4509e08911
4e1a1db08416915613142220eba68ff5ab5eb882
8450 F20101208_AABWBC kim_u_Page_13thm.jpg
bfb20a847dfee2afa80dc59dfe4fbdaa
9410515af91c44ff37c4514e24c6bfd0703af972
12976 F20101208_AABWAO kim_u_Page_04.QC.jpg
f70f5454071238e0565238047a7c2668
06ffc3f243356913ff24449065d17577afa45166
28750 F20101208_AABVVI kim_u_Page_57.pro
a172328d55cccbf37327560d2862feb7
489a1b0be18005814645139a63a9ac5300a801c5
46590 F20101208_AABVUT kim_u_Page_42.pro
2ffe8ce46de49b74d06ba8ac83624e78
04b7ca340c64402d5e77fe65a482625b30443007
35874 F20101208_AABWBD kim_u_Page_13.QC.jpg
d4ef6c339b5e1c4d1f45257a6644d98e
6149e631106fb35f2d8970176224aa24ff1846dd
6567 F20101208_AABWAP kim_u_Page_05thm.jpg
093ed4646fa300fb20b9cf549b5e3f005722d9b3
7470 F20101208_AABVVJ kim_u_Page_58.pro
ab15d1a9b9cffc9d76f39f2b7da6ab82
7c4d9781392e537815bd024ed1e6a90929ed2efe
31006 F20101208_AABVUU kim_u_Page_43.pro
36068251d81689b87fb1a6b0424b0c6d
22aae7ed8a1b46b9022c238462022277e8da0661
7193 F20101208_AABWBE kim_u_Page_14thm.jpg
3dfc8705c767fbe3f840469548e4913df2bf3dbc
852 F20101208_AABWAQ kim_u_Page_06thm.jpg
40bac40de42d98929e36dd27a2a21e4f
fdf7cf86d4dd3477439c30bc8351aa8bcaa877ea
44068 F20101208_AABVVK kim_u_Page_59.pro
fb0b91cf664601001148b956d43a1798
b7c73fa580017c48024f8b7313538219ccfdd927
46659 F20101208_AABVUV kim_u_Page_44.pro
ea856db0b24d2b455fbe5026a8830031
30353 F20101208_AABWBF kim_u_Page_14.QC.jpg
b7a90d0e4aa327b77e28de7852c0bdae
ff5a5925acbab90760055840bcf6edd57c36c975
2572 F20101208_AABWAR kim_u_Page_06.QC.jpg
2a370c7da86afe1383b56dde803146f4
4da880f4934cd8d8d463c26a49d0e1a44b879674
62266 F20101208_AABVVL kim_u_Page_60.pro
43627c4b616e179a105bdcc65bfb1ee0
7d09d49a1379828c40789ebe3852d0600dcc5725
48482 F20101208_AABVUW kim_u_Page_45.pro
dfd57752b3c57dc0b42dbeddb9f3fbd3
8de2428f4899829ac4230b8a91a6b14bb0a45d29
7178 F20101208_AABWBG kim_u_Page_15thm.jpg
b1e6fd3b08ccf675950dcbc46dc2b5f91e3c41b4
34431 F20101208_AABVWA kim_u_Page_77.pro
30a280dc95eafda8b199a159cb59bb9d
05083cf4da7252fd656bdd0a72b97a1f4468e832
1844 F20101208_AABWAS kim_u_Page_07thm.jpg
1fc38efd36be0be16daac069f5a5b859
0d3a96cb83ccaeb372711e29ac673b2a1b821d50
11891 F20101208_AABVVM kim_u_Page_61.pro
4e516346faf0632c6c39c1b6cb64a008
a4624c0f1d4e84bb8f5c8fc8653a9e0616328abf
41691 F20101208_AABVUX kim_u_Page_46.pro
27d75ae521925f584258a57e45609124
68f404e5bd08da44403243c8e2a8b3f2fdc225ab
26098 F20101208_AABWBH kim_u_Page_15.QC.jpg
e148dc8762ffed5c5992d6f887627fce
39831 F20101208_AABVWB kim_u_Page_78.pro
3dd59ca5fe2d047c43ed5c9939a03494
5c1b3104ab00da6332b17bcba6857384c137e122
6436 F20101208_AABWAT kim_u_Page_07.QC.jpg
1f8840cba7012dc923f6cf42d001582c
d217c93cfd51cb2b206222646ea8f5474051938e
6031 F20101208_AABVVN kim_u_Page_62.pro
3068a36f391ea5ba39c2a0acc978ce77d2c5aa12
29655 F20101208_AABVUY kim_u_Page_47.pro
aef303de57f894041be684bd253eb453
b6a9b27bc47c5def7a73b3ea22357d2940efe234
7812 F20101208_AABWBI kim_u_Page_16thm.jpg
e1fecd1cfffe795130897795e0729fed1f51c770
13843 F20101208_AABVWC kim_u_Page_79.pro
262621ce4c859aba35bd8da49fac1f71
bc503de56c4e2c289c82a3ec67e44f8eb3dc37ce
36391 F20101208_AABWAU kim_u_Page_08.QC.jpg
ea5875a73eb66fa704ed8fdd7870eb93
18959 F20101208_AABVUZ kim_u_Page_48.pro
faa05029df874a0a3a7762592eb42069
9f22b521c7a5897112946377ce88dceeaca787df
33910 F20101208_AABWBJ kim_u_Page_16.QC.jpg
baa63119c0b636305f0fd4798c65ece6
6547 F20101208_AABVWD kim_u_Page_80.pro
6fc0a18393a758d1f64444bbf296fe32
9c659c017152b284ab51b890b1cdb7ac7d533a4a
5392 F20101208_AABWAV kim_u_Page_09thm.jpg
b20b9afe95a8221a1559767172cb3775
687f1999d7d14807fa2344e2ac521e225ac9379b
17872 F20101208_AABVVO kim_u_Page_63.pro
ba80b3b4f05cfc0f44ff9fd47f8e61a9
086a218bc80c75e0985e95f19643411911a06d51
8293 F20101208_AABWBK kim_u_Page_17thm.jpg
a524200a6b400ebf4e49ac5fd5fe5469
2700d10b521b0d2b1022396237cc82df6db95311
17762 F20101208_AABVWE kim_u_Page_81.pro
dcd3347c6f8536f05381687ed0802564
03a9b912a490e694cd01fe56c41058878fbaba99
25004 F20101208_AABWAW kim_u_Page_09.QC.jpg
b6f1b0fe80838fd8398042f6abd4f3b2
cd6fbe1ccbd93a85fc0e23f67b348a6a56a401d3
55294 F20101208_AABVVP kim_u_Page_64.pro
a79fb0c8209d19450db88240d943983a
16a24d9bbf552b498229df0c91450a2502ac4776
8363 F20101208_AABVWF kim_u_Page_82.pro
5a2704e3fc62231def59a1f0ed6fb60c
fa5b29a85d6a66212b7ba75fef12486173443008
7131 F20101208_AABWAX kim_u_Page_10thm.jpg
819424b93e8faa11501b493d5faa3b95
614c4d498fce93218262782a7e7d9732f5202c4e
33643 F20101208_AABVVQ kim_u_Page_65.pro
9f6589fde065791281b909627e05d660
2886a196b4dd92ac8f481fd4c4c00333f0085c44
35386 F20101208_AABWBL kim_u_Page_17.QC.jpg
d6d6d37cbb719e0649561162a73d7212
2a74921416ac4bacf4122a4b5f98b662a59fc392
60385 F20101208_AABVWG kim_u_Page_84.pro
e519aa2ab3440c257f92586e414bd2bb
dcacd83708da7bc935e6bf333a7ee8240ec162b3
32794 F20101208_AABVVR kim_u_Page_67.pro
a93068f749dd6ca1a574347b69fd79f16ca548c1
35173 F20101208_AABWCA kim_u_Page_26.QC.jpg
7f9e44eae0ea5f65684b13b7bf9799c3
9ce13815540f92f23079bcfddd5c870f469b3955
21066 F20101208_AABWBM kim_u_Page_18.QC.jpg
a025de22758beb455b6124a8d113b1a1
48ce6309fd87c5711131b7c0bcc033c61bd60b83
21244 F20101208_AABVWH kim_u_Page_85.pro
73b533e84bd1b4dec61c13ae6182b0c7
31955 F20101208_AABWAY kim_u_Page_10.QC.jpg
faaf023b0ac4f33b9634cdd6ca10e51f
5132ff029b1c187010c5e50dd424b11e05fb671a
31120 F20101208_AABVVS kim_u_Page_68.pro
458d072caecdc320ece76ebf533dba35
17fdf3bd1577cf090aec46e420a4c55abc2f7545
34569 F20101208_AABWCB kim_u_Page_27.QC.jpg
0cdcfe4fc958481225ea63a4d3b32cd1
ca08c1a6193854e731b005b3088f40470b894071
8345 F20101208_AABWBN kim_u_Page_19thm.jpg
71e5203d888c27d4619963e5f8f5d2fb
54958 F20101208_AABVWI kim_u_Page_86.pro
1ff715bc06e049198e3b37cb10a4d467
2061 F20101208_AABWAZ kim_u_Page_11thm.jpg
11875a7f10f0a5098683c051e2db83e7
250be7641eedefb5abb7bcc6e2e33fa4d966605d
28657 F20101208_AABVVT kim_u_Page_69.pro
7ce0a472e14dd8d60acde4514f661cb9
b34895a2d0ae2d42f99bb612d141a7eaef4e8630
7100 F20101208_AABWCC kim_u_Page_28thm.jpg
edf57be3e8ae2ba71c89eb5438e05b55
2dcdff251fb8e65c01fc72b0e142cc76dee452dc
37928 F20101208_AABWBO kim_u_Page_19.QC.jpg
d466348aae84f65dba3e67b03ac98bed
57131 F20101208_AABVWJ kim_u_Page_87.pro
c8d03d5b717f2804cfe77817e62854c3
b6018aef898a85f3ff2b1b27893c76902a103c4f
5370 F20101208_AABVVU kim_u_Page_70.pro
ace4b5db17c3415c29b8636132228ed4
2a15650436a8737c29a1d516dab5dd0c53efbfba
25971 F20101208_AABWCD kim_u_Page_28.QC.jpg
7842b1ec53ee5ea1b49fd8348c68529f
6819 F20101208_AABWBP kim_u_Page_20thm.jpg
c49827fd144432c74a268b8559ee2d5e
6187a7808a4f0ab5d943f2fa2c57f4cbc107c8cb
59707 F20101208_AABVWK kim_u_Page_88.pro
0512be92113299eb1d0117facde34219
cd7e28eb7486666b04c5fdc61a552f1546189eff
5404 F20101208_AABVVV kim_u_Page_71.pro
f74b7e073ddd138c72d9010d34f42ffff2440c8a
8120 F20101208_AABWCE kim_u_Page_29thm.jpg
aa1570e0c7c3b4f531e5d5a8cde47250
10675326283ae04d3518922ce20986fdc197b8f5
24340 F20101208_AABWBQ kim_u_Page_20.QC.jpg
5f787db6caa6fe703d0fd78088365461
46bec170a39807ee20bc7d0f377841544267e5dc
60235 F20101208_AABVWL kim_u_Page_89.pro
f4c277f5a92f9a1ee2b58fe50896f601
b91b741aeac5d4d6541887bf0be0681668e1b087
9443 F20101208_AABVVW kim_u_Page_72.pro
f13b7e139c2a3d78f32be396d0008860
71629477e4f905a14048c6849946c8fe9f5667fe
35110 F20101208_AABWCF kim_u_Page_29.QC.jpg
bc70b1a16f91ae7886ebb449e3388473c8c7b03f
8151 F20101208_AABWBR kim_u_Page_21thm.jpg
a64f0edd128fee7b8b14fd1e46799bd4
20827 F20101208_AABVWM kim_u_Page_90.pro
f138d0e90756096cc84c03c109c342d4
e510c5f5818d0bc5cd589208fc481111787c4ac3
15694 F20101208_AABVVX kim_u_Page_73.pro
8666723b2a9d31479c8a094de79be21a47dda4e4
7088 F20101208_AABWCG kim_u_Page_30thm.jpg
1905daabe620695dcfaa9c18fa90811a
82b85c55bff81b1dd322e786de25a8f9d74fea62
2351 F20101208_AABVXA kim_u_Page_13.txt
3e0745314f4f6a9b555f1e82a824d0c2
34936 F20101208_AABWBS kim_u_Page_21.QC.jpg
e05343d051c7b109539d9bf0ace79bf4
af169f6219f3fa39379b75ce8af541b67f1ca268
17420 F20101208_AABVWN kim_u_Page_91.pro
e5a3360188f9a9ef0ddc282ee8f5a233
8295c3ec0511e6db1c34d920e9fd0312a591b6ea
20892 F20101208_AABVVY kim_u_Page_75.pro
29e83ef97f2eff7248f8e7f3706205a8
27413 F20101208_AABWCH kim_u_Page_30.QC.jpg
1903ab87d73aa540a3b15b71ec5fe469
2153 F20101208_AABVXB kim_u_Page_14.txt
f90764bc7016751f400c924d62aa3ccd
f5406792ab6514c6670005e1e1aed290c51986c2
2013 F20101208_AABWBT kim_u_Page_22thm.jpg
5708e355ec9ca1c792b6dc2cf28ddfcf
84f411f1e3cec5b488bedb349c6e4b97ac84fe0d
F20101208_AABVWO kim_u_Page_01.txt
8d7167f269a50f63b6c124a78cbf8d7e
9852d0971a0f2b97b45737faa4e5c238d9a22bc1
7848 F20101208_AABVVZ kim_u_Page_76.pro
7776 F20101208_AABWCI kim_u_Page_32thm.jpg
88cbd959f6b0536cb78b27cf1966e49f
5c9821fc94cf022a7aeafb2d8ffeaabc3d8bdca3
1831 F20101208_AABVXC kim_u_Page_15.txt
b46509ab8f8c4ec4cffbf6dbaf254a8f
7995a217428e9307b7218fea1185d7caf06d1217
7733 F20101208_AABWBU kim_u_Page_22.QC.jpg
e587f00108b33fb8cc8511a04dc1ff41
28a82061663538dca9e97b3ccbb42817f85fe95c
31252 F20101208_AABWCJ kim_u_Page_32.QC.jpg
23d2400549f9a1a6117331125af07675
e5cc34313d72fb9688d395a270a160a411b93e40
2212 F20101208_AABVXD kim_u_Page_16.txt
2c6a0ccba37a069364f528082b1d4677
3b8c81ee195e77ab3dd9df1d4282ce6465b75ed7
6317 F20101208_AABWBV kim_u_Page_23thm.jpg
42c095111e5fc564634449383920f76b
1c3ca5e62fe4a87871ba58922f73984e456c8425
80 F20101208_AABVWP kim_u_Page_02.txt
8d9c12a676159b3965c2ae4729464c42
ed761789dccd7c8e87b55cde7e5c1c0a20ea398b
7954 F20101208_AABWCK kim_u_Page_33thm.jpg
98a7562e815e31a5733c2ca428cdd208
09053027e2662dee00987bca4f4b682a74392229
2269 F20101208_AABVXE kim_u_Page_17.txt
a4dcdd382640d244f94479b4fbf0b0bb
24917 F20101208_AABWBW kim_u_Page_23.QC.jpg
0640eec3517d89bd92c4c9510d7f0791
e1c3199cfe9209101c36ea3b4edafaf85e6b17d5
207 F20101208_AABVWQ kim_u_Page_03.txt
e00819c737a1c01da5a7818ac916cb02
889371f81631bb5b7e18d7a7d8e7dae0a9a01e83
34556 F20101208_AABWCL kim_u_Page_33.QC.jpg
46cd83ae21f6e79412ba2b9fc0f8e2e8
4c52d08e4ea8933fe981c981a0601e100625af43
1405 F20101208_AABVXF kim_u_Page_18.txt
6463b11bdf70325158d2e56560c46eae
1d5354561044f70176c8271e67cd539840768f94
7578 F20101208_AABWBX kim_u_Page_25thm.jpg
e687ce68aeb35b8f17dfd7f3a207f360
23e80351e52426faa584ea40d2650623dafe397d
809 F20101208_AABVWR kim_u_Page_04.txt
cc7ec13148343742030732437faba805
0440e1709e992288ef6dccf5e2f5614b2dd322fd
22703 F20101208_AABWDA kim_u_Page_41.QC.jpg
2a5c86ee475a1aff6ccd81feb89af9aa
d0f3dcde5ac938857fab6f83daa7877f3d08ac1b
7951 F20101208_AABWCM kim_u_Page_34thm.jpg
6bfa51d05c6de6a8ee6b6df51607d7c2
297dfbd832f9eb4b8e7ee934b1d44a2317acdcc8
2488 F20101208_AABVXG kim_u_Page_19.txt
dba0943934ff2817583d3c3f5ae5854a39a13068
30422 F20101208_AABWBY kim_u_Page_25.QC.jpg
24876e11411980cec2b00aa4eb21a8ef
2216 F20101208_AABVWS kim_u_Page_05.txt
1bbf44bd565b2d536fb3a5e52dcf1e6a
7076 F20101208_AABWDB kim_u_Page_42thm.jpg
dddeaa896d6d99f38b6c052cedf6206e
fd93b0688a57c019a0d94218069693091f96afc8
34370 F20101208_AABWCN kim_u_Page_34.QC.jpg
cb6b629c6eae8db9206ef61b979eac80
dd147a58f43d8f6b752e5584fd834c34a4e7ab46
946 F20101208_AABVXH kim_u_Page_20.txt
167f0859cf812a9b1dabbf6378960b64
483ddbbc44370aeb4baf03a63e0591075110b4a5
84 F20101208_AABVWT kim_u_Page_06.txt
1109cbb4306792c700f964cb107e8fd1
ed4fc6769ec4f1272e79e0cb83ab56c82b9f2896
29291 F20101208_AABWDC kim_u_Page_42.QC.jpg
b4255f8d5cd852309390cf5a567a36dc45e655ea
5788 F20101208_AABWCO kim_u_Page_35thm.jpg
4eeefe3c9bc9e7515be530b307c3f6e2
2d59043ac3d6ed68033aaa2ee1ebb45e28788815
2547 F20101208_AABVXI kim_u_Page_21.txt
3c7f6cf73ed4ec88fb86dda29fa920e3
f2324470cbf4a1eb7a73254a415fa58158b67e0d
8475 F20101208_AABWBZ kim_u_Page_26thm.jpg
4e3803d66ce7f52451fd04bce6179f15
078c6e74e74d425783fcb2aae0614e6f01a0643f
292 F20101208_AABVWU kim_u_Page_07.txt
8cc7503064e7c12eb4b605ae5faa420a
5673 F20101208_AABWDD kim_u_Page_43thm.jpg
1528a28e48a4048001f417b7b06fdd8f
278bf5b807596c9908ff8ac5c2ea049dc977aa73
23012 F20101208_AABWCP kim_u_Page_35.QC.jpg
e1e715f38a169615689c2e91c931e5b3
14dcde2e82004613166956db0d6a51281733b961
417 F20101208_AABVXJ kim_u_Page_22.txt
fed916bf13514c0f26650c46d68ca615
27bcd12467d2369b77750537fa7ee427e1d11b0d
F20101208_AABVWV kim_u_Page_08.txt
2a2cafff996c8423fe2ae536a16dbd5b
afd6a55436cd158d3104f0c2d7d15832b6fede0a
21395 F20101208_AABWDE kim_u_Page_43.QC.jpg
02ae84b7b9ea157d0a89febd681127fb
7773 F20101208_AABWCQ kim_u_Page_36thm.jpg
fe4508abd472070d15a862d9043e346e
33546740f9d19283cdfe6dd5d3bd7004e98b50bd
1796 F20101208_AABVXK kim_u_Page_23.txt
10499400f5b1ac6d40b697b2934ac548
c61f9fd5a23d203235d06ace98a4cbe3a0c70b73
1526 F20101208_AABVWW kim_u_Page_09.txt
ac3acf0818b167201de2bfc927f50ce85ae6572e
7425 F20101208_AABWDF kim_u_Page_44thm.jpg
2efec5093ce3f042aebd31a612bb0887
d39fccb5ab87ff34eb34359e3fc29448331a0883
32686 F20101208_AABWCR kim_u_Page_36.QC.jpg
f2f9f7b5b89de9b12af8c4f711678c19
47f247f92df5edf8b1d0d256a0d9f48c0477dec0
2377 F20101208_AABVXL kim_u_Page_24.txt
14ab52deaa7447333c1c3eaff027a581
571357b3c1192df27a3913d2cfc733c292606be8
2278 F20101208_AABVWX kim_u_Page_10.txt
185314a0ddfb5514127f2526165619c3
44641e3f3a98de3bd23ec6b8bd5808344d3ce4d2
29197 F20101208_AABWDG kim_u_Page_44.QC.jpg
b3b84f0bddc8c769a28ba2e11a72db22
3a767ae2625d16f50ffa520e2a3d94697d964228
1965 F20101208_AABVYA kim_u_Page_40.txt
91938b33122c2caa3ab8cebd8ed903c8
8dc49721892cac7115df91fe8113bc96ba648a18
6363 F20101208_AABWCS kim_u_Page_37thm.jpg
f9a4466f9717298e5fb695078060f99b
97fd496a3d3515a173f8ddb3056d2f206da410ed
1754 F20101208_AABVXM kim_u_Page_25.txt
eaa8d1468c355705bdae9f022ea95798
51c75c5d8fe8535194644cd717fd8852735a0555
484 F20101208_AABVWY kim_u_Page_11.txt
2237e383834d97626202f6580018b7cc
409c223ed0084182447db33e499e13be8b8c965c
7129 F20101208_AABWDH kim_u_Page_45thm.jpg
22344e1f1119acf83003800598bf3360
55137b052349f6fa01249094e50fe72274fe7c5e
1169 F20101208_AABVYB kim_u_Page_41.txt
3f5dcef6ce1ed3d22de3d76dcb7bde8b
1779582e44141450dbde9bae98fd05d49d793a5b
6603 F20101208_AABWCT kim_u_Page_38thm.jpg
b08e2e69994d665e9397c7f3826f4bd4
47a3b4b29cd29063d797a62aecc83c6a819d7207
1294 F20101208_AABVXN kim_u_Page_26.txt
2f0d34974a11b82fb1476e35797aa1e5
4c17493bd32b749907b2c30bf0839d513cf806d6
2267 F20101208_AABVWZ kim_u_Page_12.txt
80420ab57f83c274e4441092957951fb
4316d8a7bfa850ba7f50cff0b6a9e348ba7900c5
28951 F20101208_AABWDI kim_u_Page_45.QC.jpg
aa33c7e5f13c98a31734f3a464ed667b
cfd8cfa99f86f7df70ebba05e7740d8783f7dab5
2067 F20101208_AABVYC kim_u_Page_42.txt
387161c79ef1531c3c69823e757260f8
047623ddfa753cef0e88c316034eb6b0b45370bd
25713 F20101208_AABWCU kim_u_Page_38.QC.jpg
0cdcdf81ff4b504e57e56b19c9bfa4b7
2168 F20101208_AABVXO kim_u_Page_27.txt
f99f86e81e6afda578cb5f302962d682
cbe1da5cc16353520a5f47db717fd0534ecb8642
27323 F20101208_AABWDJ kim_u_Page_46.QC.jpg
37c01230963ae9a1302257039fb20b70
1575 F20101208_AABVYD kim_u_Page_43.txt
897b86904e9dcb68bc642fa558895107
7180 F20101208_AABWCV kim_u_Page_39thm.jpg
68edcc07f53c21037f2822e0373aa71d
46740bf4501c3871ddeb3d941cbda49e550118cb
1115 F20101208_AABVXP kim_u_Page_28.txt
ee4677d149cd5063ae1c4ba6abe6537a
742caa9b91ac204b789e2fef27f798da84c98028
5660 F20101208_AABWDK kim_u_Page_47thm.jpg
346d395f9ed73c22d24c44e0f5c06e03
35aff665739837b654255fb628438a58a31c91d7
2137 F20101208_AABVYE kim_u_Page_44.txt
d60ac0f977005376c76bd5f290f43c28
14b8dacf9bc6c332c56afd502066f111fc447b88
28756 F20101208_AABWCW kim_u_Page_39.QC.jpg
58037c37ce8946d76c85e61a5c4462f3
9b34ea91d9403f2c4cedbce05a18e7fa33110e2c
21912 F20101208_AABWDL kim_u_Page_47.QC.jpg
0c3bbee905ff0b80637bbb7e15213b2a
65c0800271dab103bef10d3a1a77003cd3b2eefe
2028 F20101208_AABVYF kim_u_Page_45.txt
601f703423e939bbcb36f9cc976709a3de3bfa1c
6696 F20101208_AABWCX kim_u_Page_40thm.jpg
10f800f8b23f1a089944bd7c157eb70c
37abe037efa95466d2572970fc073fb75d1e30ab
2255 F20101208_AABVXQ kim_u_Page_29.txt
ef0a385a9f95a609324d14beb9544725
73a3c407e7421d1e8e4023d3268949e5d8b49bb0
5632 F20101208_AABWDM kim_u_Page_48thm.jpg
e9b0449af5fd1a028d5596b02e51bef4
eaf976383e8c1d9c1bdfa8119fee51fbc85549fc
1886 F20101208_AABVYG kim_u_Page_46.txt
d737d5faba28dfff509d9e8c73f9c7b0
6d79e981dfd7ec7fe1606893b0002c2a9f42f5a1
27138 F20101208_AABWCY kim_u_Page_40.QC.jpg
e95c91380cf3ed2f335298d8fb20c687
2490 F20101208_AABVXR kim_u_Page_31.txt
e729077dc3bbb8b9e187d5a7052e9266
2bf570309762c7695f4824e83f4910641210ec8a
25664 F20101208_AABWEA kim_u_Page_55.QC.jpg
2c6e951ecbc33000ae514f9906c7dfa4
b6a98d1bed88523f0567a9b3e7e1c67912726cfe
20140 F20101208_AABWDN kim_u_Page_48.QC.jpg
51c31c8f13264f0a7437fa757800a3ab6ed957e2
1551 F20101208_AABVYH kim_u_Page_47.txt
b8c750ae9a4906de3f504e103310b293
f19e30aaea6bc5ac736aef6ae80b8d1d15138c68
5769 F20101208_AABWCZ kim_u_Page_41thm.jpg
031961d8fa09c3a6a32f30f45497b252
bdd74c43ca9d75a3178d86f0ec6220634dfb6d81
1555 F20101208_AABVXS kim_u_Page_32.txt
c3ace4b846f3fc23ea677f53ff307112
ce7ced3b96c5dc55934f65b347d83c7d55293129
6333 F20101208_AABWEB kim_u_Page_56thm.jpg
d915a082f4e8ab87092aa5267d421d06
96d3bc9ea4d5729de4797e275fb80a84fd9d5a66
7482 F20101208_AABWDO kim_u_Page_49thm.jpg
4d232675c5e1b89007331acd64feb7df
b2e9ee9142f7963ceb6ddce6dfdbb3c32b6b3b1a
921 F20101208_AABVYI kim_u_Page_48.txt
7ab367b995ac4232a065a3f15768598e
5346dc028115e4d462694f60c3ed04a3459bc565
2240 F20101208_AABVXT kim_u_Page_33.txt
b908994374d849f36b305f64aa487955
c7cf1b34699a7edca52f23c4abe4b452b61353fc
6394 F20101208_AABWEC kim_u_Page_57thm.jpg
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.

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
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
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

a 6.- 9 r
+/ 9.
;i r~ : ;Tg
r. gE'a ~--: -m
F~T~F~-~-- O .\
~~~r ar9-~9:~-L- ~C i
'~r~:+'t~-~h~er~C ~-r.'?8
t
at I ,.C ~ C-
q

Z
(b)

E?
i
ror,
r c '+~ 9 t
--- + a r .~
P +,
%
"' # i
+r-L**
J c t.6 9
+ r
+ -Tt--
I I
r- c*r
r s~ +
+ +~
i ~ r
+ r
~ :fir~ L
O
i

(a)

Y
"i

**)

**~ w

*Y"

** ~

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
mesg-i nth s .-d" rfr ec" ad inrpyt" fild as nods.Edesar
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

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.

REFERENCES

[1] D. Fallows, "Spam: How It IS Hurting E-Mail and Degrading Life on the Internet ,"
Pew Internet and American Life Project, Oct. 200:3.

[2] P. O. Boykin and V. P. Rci-l bowdhury, "Leveraging Social Networks to Fight
Spam," IEEE O'omp~uter S8... .:. It; pp 61-68, April 2005.

[:3] 31. Girvan and 31. E. J. ?-. i..--in lII "Community structure in social and biological
networks," PNAS, vol 99, pp 7821-7826, June 2002.

[4] P. Holme, 31. Huss, and H. .T. ung_ "Subnetwork hierarchies of biochemical
pathir- .va," Bioinformatic~s, vol 19, pp 5:32-538, 200:3.

[5] A. Broder, F. Alaghoul, and P. Raghavan, "Graph structure in the web," C'omp~uter
Networks, vol :33, pp :309-320, 2000.

[6] R. Guimera and L. A. N. Amaral, "Functional cartography of complex metabolic
networks," Nature, vol 4:38, pp 895-900, 2005.

[7] G. Palla, I. Derenyi, I. J. Farkas, and T. Vicsek, "Uncovering the overlapping
community structure of complex networks in nature and society," Nature, vol 4:35,
pp 814-818, 2005.

[8] S. N. Dorogovtsev, A. V. Goltsev, J. F. F. Mendes, and A. N. Samukhin, "Spectra of
complex networks," Ph,;;-.. arl review E, vol 68, 200:3.

[9] B. Jiang and C. Claramunt, "Topological Analysis of Urban Street Networks,"
Environment and Plan t.. .:i( B, 2002.

[10] F. Harry, Grap~h The.. ;; Addison-Wesley, 1969.

[11] R. Albert and A. Laszlo, "Stastical mechanics of complex networks," Reviews of
modern lph;,;i.. vol 74, pp 47-97, Jan 2002.

[12] R. Albert and A. L. Barahasi, "Statistical mechanics of complex netweoks," Rev.
Modern Phys., vol 74, pp 47-97, 2002.

[1:3] S. Rander, "How popular is your paper? An empirical study of the citation
distribution," Eur. Phys. J. B., pp 1:31-134, 1998.

[14] H. Jeong, B. Tombor, R. Albert, Z. N. Oltvai, and A. L. Barahasi, "The large-scale
organization of metabolic networks," Nature, vol 406, pp 651-654, 2000.

[15] R. Albert, "Scale-free networks in cell biology," Jourmal of C'ell Science, vol 118, pp
4947, 2005.

[16] A. L. Barahasi and R. Albert, "Emergence of Scaling in Random Networks,"
Science, 1999.

[17] R. Albert and A. L. Barabasi, "Diameter of the world-wide web," Nature, vol 401,
pp 130-131, 1999.

[18] S. N. Dorogovtsey and J. F. F. Mendes, "Evolution of networks," Advance in
Ph;,:. vol 51, pp 1079-1187, 2002.

[19] J. Scott, Social Network Aith;,i A Handbook, Sage, London, 2000.

[20] M. S. Mizruchi, The American Corp~orate Network, 1904-1974, Sage, Beverly Hills,
CA, 1982.

[21] J. F. Padgett and C. K(. Ansell, "Robust action and the rise of the Medici,
1400-1434," Amer. J. Social., vol 98, pp 1259-1319, 1993.

[22] A. Rapoport and W. J. Horvath, "A study of a large sociogram," Behavioral Sci.,
vol 10, pp 279-291, 1961.

[23] L. A. Adamic and B. A. Huberman, "Power-law distribution of the world wide web,"
Science, vol 287, 2115a, 2000.

[24] L. A. N. Amaral, A. Scala, M. Barthelemy, and H. E. Stanley, "Classes of
small-world networks," Proc. Natl. Acad Sci. USA, vol 97, pp 11149-11152, 2000.

[25] D. J. Watts and S. H. Strogatz, "Collective Dynamics of 'Small-World' Networks,"
Nature, vol 393, pp 440-442, 1998.

[26] W. Ailello, F. Changl5, and L. Lu, "A random graph model for massive graphs, in
Proceedings of the 32nd AC' \! Theory of C'ompluting, Association of Comr 1;,. '9l
Machinery, pp 171-180, 2000.

[27] H. Ebel, L. I. Mielsch, and S. Bornholdt, "Scale-free topology of e-mail networks,"
Phys. Rev. E., vol 66, 035103, 2002.

[28] M. Bern, D. Eppstein, and J. Gilbert, "Provably good mesh generation," IEEE
Symp~osium on the Foundations of Comp~uter Science, pp 231-241, 1990.

[29] L. A. Adameic and N. Galance, "The political Blogosphere anf The 2004 U.S.
Election," -.../ Annual Workshop on the Webblogging Ecosystem:Aggregation,
A,:trle;,.: and D ~ioctriii. Japan, 2005.

[30] C. Ding, X. He, H. Zha, M. Gu, and H. Simon, "A min-max cut algorithm for graph
partitioning and data <1I1-rh 1119 Proc. IEEE Int'1 Conf. Data M~ining, 2001.

[31] A. Pothen, "Graph partitioning algorithms with applications to scientific
computing," Parallel Numerical Algorithms, 1996.

[32] B. W. K~ernighan and S. Lin, "An efficient heuristic procedure for partitioning
graphs," The Bell Syst. Tech. J., vol 49, pp 291, 1970.

[33] S. Boccaleti, V. Latora, Y. Moreno, M. C'!s i.;. ., and D. U. Hwang, "Complex
networks: Structure and dynamics," Phys. Rep?., vol 424, pp 175, 2006.

[34] J. Reichard and S. Bornholdt, "Detecting fuzzy community structures in complex
networks with A Potts model," Phys. Rev. Lett., vol 93, 218701, 2004.

[35] M. E. J. ?-. i.--us! lIs, "The structure and function of complex networks," SIAM~ Rev.,
vol 45, pp 167-256, 2003.

[36] M. E. J. ?-. i.--us! .Is and M. Girvan, "Finding and evaluating community structure in
networks," Phys. Rev. E, vol 69, 026113, 2004.

[37] C. P. Massen and J. P. K(. Dc...; "Identifying communities within energy
landscapes," Phys. Rev. E, vol 71, 046101, 2005.

[38] F. Changl5 and L. Lu, "Connected components in random graphs with given degree
sequences," Annals of Combinatorics, vol 6, pp 125-145, 2002.

[39] T. Luczak, "Sparse random graphs with a given degree sequence," Proceedings of the
Symp~osium on Random Grap~hs, pp 165-182, 1989.

[40] B. Everitt, C'i;,i. r Amderl;: John Wiley, ?-. i.--i-ork, 1974.

[41] M. Adersonn and T. Merely, "Eigenvalues of the Laplacian of a Graph," Linear and
Multilinear Algebra, vol 18, pp 141-145, 1985.

[42] M. Fiedler, "A Property of Eigenvectors of Non-negative Symmetric Matirces and its
Application to Graph Theory," Czech. M~ath. J., vol 85, pp 619-633, 1975.

[43] P. Gould, "The Geographical Interpretation of Eigenvalues," Institute of British
G..y plyl,.: Transactions, vol 42, pp 53-85, 1967.

[44] R. Grone and R. Merris, "The Laplacian Spectrum of a Graph," SIAM~J. Matrix:
Anal. App., vol 2, pp 218-238, 1990.

[45] R. Grone and R. Merris, "The Laplacian Spectrum of a Graph II," SIAM~J. Matrix:
Anal. App., vol 7, pp 221-229, 1994.

[46] B. Mohar, The Lap~lacian Sp~ectrum of Grap~hs, Wiley, 1991.

[47] B. Parlett, B. Simon, and L. Stringer, "On Estimating the Largest Eigenvalue with
the Lanczos Algorithm," M~athematics of Comp~utation, vol 38, pp 153-165, 1982.

[48] K(. Tinkler, "The Physical Interpretaion of Eigenfunctions of Dichotomous Matrices,"
Inst. Br. Geog. Trans., vol 55, pp 17-64, 1972.

[49] A. Pothen, H. D. Simon, and K~ang-Pu P. Liu, "Partitioning Spare Matrices with
Eigenvectors of Graphs," NASA System Division, RNR-89-009, 1989.

[50] D. Cvetkovic, M. Domb, and H. Saches, Spectra of Graphs: Theory and Applications,
John Ambrosius, 1995.

[51] I. J. Farkas, I. Derenyi, A. L. Barabasi, and T. Vicsek, "Spectra of real-world
graphs: Beyond the semicircle law," Phys. Rev. E, vol 64, 2001.

[52] G. Siganos, M. Faloutsos, and C. Faloutsos, "Powe-law and the AS-level internet
topology," IEEE-ACM~~ T. Network, vol 11, pp 514, 2003.

[53] E. Wigner, "Characteristic vector of bordered matrices iwth infinite dimensions,"
Annuals of M~athematics, vol 62, pp 548-564, 1955.

[54] L. Giot, "A Protein Interaction Map of Drosophila melanogaster," Science, vol 302,
pp 1727, Dec. 2003.

[55] C. K~amp and K(. ClIn -I. Is-- is, "Spectral Analysis of Protein-Protein Interactions in
Drosophila melanogaster," arXiv, M .i- 2004.

[56] R. Milo, "Network Motifs: Simple Building Blocks of Complex Networks," Science,
vol 298, pp 824, 2002.

[57] M. A. M. de Aguiar and Y. Bar-Yam, "Spectral analysis and the dynamic response
of complex networks," Ph.;,.: arl Review E, vol 71, 016106, 2005.

[58] J. B. Gao, Y. H. Cao, and J. Hu, "Building innovative representations of DNA
sequences to facilitate gene finding," IEEE Intelligent S;,;l. at- 2004.

[59] G. S. St v ll. Introduction to Linear Algebra, Wellesley Cambridge Press, 1998.

[60] D. Bu, Y. Zhao, and L. Cai, "Topological structure analysis of the protein-protein
interaction network in budding yeast," Nucleic Acids Research, vol 31, pp
2443-2450, 2003.

[61] M. Catanzaro, M. Boguana, and R. Pastor-Satorras "Generation of uncorrelated
random scale-free networks," Phys. Rev. E., vol. 71, pp. 027103, 2005.

[62] M. E. J. ?-. i.--us! lIs, 1 I-I algorithm for detecting community structure in networks,"
Phys. Rev. E, vol 69, 066133, 2004.

[63] M. E. J. ?-. i.--us! lIs, "The Structure and Function of Complex Networks," SIAM~
REV/IEW, vol 45, pp 167-256, 2003.

[64] W. W. Zachary, "An informational flow model for conflict and fission in small
groups," J. Anthropol., vol 33. pp 452-473, 1977.

[65] M. E. J. ?-. i..--slis, I! 1.~ i..ork data," htt~p://www-personal. umich. edu/ mejn/netdata/,
July 2007.

[66] A. Clauset, M.E.J. N. i.--us! .Is, and C. Moore, "Finding community structure in very
large networks," Phys. Rev. E, vol 70, 066117, 2004.

[67] D. Lusseau, K(. Schneider, O. J. Boisseau, P. Haase, E. Slooten, and S. 31. Dawson,
"The bottlenose dolphin community of Doubtful Sound features a large proportion
of long-lasting associations-Can geographic isolation explain this unique trait?,"
Behavioral E, I J..;,i and S .. ..... JI.: I -,I;; vol 54, pp :396-405, 200:3.

[68] S. Wasserman and K(. Faust, Social Network Ame~l;i-.: Cambridge University Press,
Cambridge, 1994.

[69] L. L. Anthony and D. T. Blumstein, lIst.~ gI .1nig behaviour into wildlife
conservation: the multiple v-a--s~ that behaviour can reduce N-e.,"B. .i l
Conservation, vol 95, pp :30:3-315, 2000.

[70] 31. E. J. N -~ i.-s! .Is S. Forresy, and J. Balthrop, "E-Mail Networks and the Spread of
Computer Viruses," Phys. Rev. E, vol 66, 0:35101, 2002.

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.

PAGE 1

1

PAGE 2

2

PAGE 3

3

PAGE 4

PAGE 5

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

PAGE 6

................................ 91 6

PAGE 7

Table page 3-1Absolutevaluesofthedierenceforedgesineachprojection. ........... 50 3-2Clusteringcoecientoftwocombinedsubgraphs. ................. 57 4-1Algorithmresultsforthreedatasets. ........................ 81 7

PAGE 8

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

PAGE 9

...................... 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

PAGE 10

PAGE 11

11

PAGE 12

PAGE 13

PAGE 14

PAGE 15

Figure1-1. ExamplesofgraphsandtheiradjacencymatrixandLaplacianmatrix. 11 ].Thisinherenttendencytoclustercanbequantiedbytheclusteringcoecient[ 25 ].Letusfocusonaselectednodeiinanetwork,havingkiedgeswhichconnecttokiothernodes.Ifthenearestneighborsoftheoriginalnodewerepartofaclique,therewouldbeki(ki1)=2edgesbetweenthem.TheratiobetweenthenumberEiofedgesthatactuallyexistbetweenthesekinodesandthetotalnumberki(ki1)=2givesthevalueoftheclusteringcoecientofnodei, 15

PAGE 16

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

PAGE 17

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

PAGE 18

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

PAGE 19

PAGE 20

Exampleofsocialnetwork:Acollaborationnetworkofscientists.AfterM.GirvanandM.E.J.Newman(2002)[ 3 ]. 20

PAGE 21

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

PAGE 22

5 22

PAGE 23

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

PAGE 24

PAGE 25

PAGE 26

2-2 showsclusteringofthemeshnetworkin[ 28 ].Spectralclusteringuses Figure2-2. Bestdivisionintoequal-sizedpartsfoundedbythespectralpartitioning.AfterAdameicandGlance(2005)[ 29 ]. eigenvectorsoftheLaplacianmatrixandismostconvenientlyappliedto2-wayclusteringproblemusingasingleeigenvector[ 30 ]. 3 ].Therearemanyalgorithmstodetectsuch 26

PAGE 27

PAGE 28

PAGE 29

PAGE 30

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

PAGE 31

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

PAGE 32

PAGE 33

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

PAGE 34

PAGE 35

PAGE 36

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

PAGE 37

Discretespectrumoftheclassicalrandomnetwork,whicharesamesizeoftheprotein-proteininteractionnetwork.Thenumberofnodesis4555 arecomputedfromeigenvaluesandeigenvectorsoftheLaplacianmatrix.Thisprocessisbasedontheorthogonalprojectiononthebasisvector,asanadditionalaccomplishment,wealsointroduceamodiedcentralitymeasureusingeigenvectorsoftheLaplacianmatrix.Asthesecondstep,weintroducethemethodforndingthecommunitystructureinthenetworkwithsubgraphsthatweobtainatthespectraldecomposition.Fig. 3-4 showstwomainstepsofourmethod.Weknowthateigenvectorscorrespondingtothoseeigenvaluescontainalotofdistinguishedinformationonsubnetworks.Thereisastudythatusedeigenvaluesandeigenvectorstosuccessfullyndproteincodingsequence[ 58 ]. 37

PAGE 38

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

PAGE 39

PAGE 40

=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

PAGE 41

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

PAGE 42

PAGE 43

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

PAGE 44

PAGE 45

PAGE 46

PAGE 47

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

PAGE 48

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

PAGE 49

3{25 ).Then,comparethevaluesofeachweightedprojectionmatrix,whicharecorrespondingtonon-zeroandnon-diagonaltermsinthelaplacianmatrix.Theweightedprojectionmatrixthathasthemostnegativevaluearetheeigenvectorforthespecicedge.Thisprocedurealsobesummarizedbythepseudocoderepresentationinthealgorithm1.WhenthenumberofedgesinthenetworkisEandthenumberofsubgraphsisM,there ifPij
PAGE 50

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
PAGE 51

ExampleofthedecompositionusingtheLaplacianmatrix.Thisgraphcanbedecomposedtotwosubgraphs. 2.3 ,thepropertyofacommunityisaveryimportantconceptinthisstudy.Weproposeamethodtondthecommunitystructureinthenetwork,asthesecondstepofourmethod.Inthisapproach,wepresentaframeworktoidentifycommunitymodulesfromnetworksbymergingsubgraphs,theresultoftherststepintheprevioussection.Wechoosesubgraphsandcombinethemtondwhichcombinationissuitableforndingthecommunitystructureinthenetwork.Thepreviousapproacheshavefocusedonlyonthemostsignicantpositiveandnegativeeigenvalues 51

PAGE 52

PAGE 53

2.3 .Weusethedegreeandtransitivitytondoptimizedcommunitystructureinthenetwork.Foranalyticprocess,wecomputethecorrelationmatrixofsubgraphs.WedenethecorrelationofsubgraphsasinEq.( 3{36 ). InthisEq.( 3{36 ),thecorrelationhasmaximumvalue1,whensubx=suby.Wealsoexpressthecorrelationbetweensubgraphsastheinnerproductofthemembershipvectorforsubgraphs,whichrepresenttheconnectionofnodesineachsubgraphasinEq.( 3{37 ). 53

PAGE 54

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

PAGE 55

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

PAGE 56

Findingthebestmatchedsubgraphofsubgrapha.Subgrapha,bandcrefersubgraph4,1,and5inFig. 3-10 ,respectively. Whenweusethemembershipvector,whichdoesnotconsidertheimportanceofnodes,thecorrelationvaluebetweensubgraph4andsubgraph1ismaximum,sosubgraph4andsubgraph1arethematchedpair.Ontheotherhand,ifweusethedegreevectorthatconsiderthedegreeofnodesinsubgraph,subgraph5isthematchedsubgraphofsubgraph4.Tocheckthevalidityofthedegreevector,wecomputetheincrementoftheclusteringcoecientforeachmergedsubgraph.InFig. 3-11 ,subgraphpair4and5hasthemaximumclusteringcoecientanditcoincideswiththemaximumpointofthe 56

PAGE 57

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

PAGE 58

Comparisonofcorrelatedvaluesandclusteringcoecientsforeachpairofsubgraph.X-axisrepresentstheindexofsubgraphs Figure3-12. Combinesubgraph4(a)and1(b),Combinesubgraph4(a)and5(c) 58

PAGE 59

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

PAGE 60

PAGE 61

SimpleexampleofndingCommunityStructurefrom6subgraphs. 61

PAGE 62

3-14 showstheowofthewholeprocesstoperformtheproposedalgorithm. 62

PAGE 63

Blockdiagramoftheproposedmethodswhicharespectraldecompositionandndingcommunitystructure 63

PAGE 64

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

PAGE 65

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

PAGE 66

PAGE 67

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

PAGE 68

Degreedistributionofeachcommunity.Data1representscommunityB,data2representscommunityA. undirectednetworkconsistsof62nodes,connectedby159edges.Advancedtoolsfortheanalysisandstudyofsocialstructureinhumanpopulationshavebeendevelopedoverthelasthalfcentury[ 68 ],[ 19 ].Usingtheseresources,theanalysisofanimalsocialnetworkscanprovidesubstantialinsightsintothesocialdynamicsofanimalpopulationsandpossiblysuggestnewmanagementstrategies[ 69 ].Animalsocialnetworksaresubstantiallyhardertostudythanhumannetworksbecausewecannotmakeinterviewsandquestions,andnetworkdatamustbegatheredbydirectobservationofinteractionsbetweenindividuals.Thenetworkweapplywasconstructedfromobservationsofacommunityof62bottlenosedolphins(Tursiopsspp.)overaperiodofsevenyearsfrom1994to2001[ 67 ]atthecoastofDoubtfulSound,NewZealand.Itisknownastherearetwocommunitiesandfoursub-communitiesinthedolphinnetwork.First,werunthefast 68

PAGE 69

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

PAGE 70

PAGE 71

PredictedcommunitiesbyfastalgorithmandourmethodofZachary'skarateclubnetwork. Figure4-9. Socialnetworkof62dolphins. 71

PAGE 72

Modularityofdolphinsocialnetwork. Figure4-11. Eigenvaluesandthenumberofedgesincorrespondingsubgraphsofdolphinnetwork. 72

PAGE 73

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

PAGE 74

Figure4-13. Articialnetwork.Zout=2andZin=14. 74

PAGE 75

PAGE 76

Articialnetwork.Zout=7andZin=9. Figure4-16. Fractionofverticescorrectlyclassiedasthenumberzoutisvaried. 76

PAGE 77

PAGE 78

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

PAGE 79

Entirepersonalemailnetwork.Thisgraphcontains2500messagesandhas3755nodesand6930edges. 79

PAGE 80

PAGE 81

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

PAGE 82

PAGE 83

PAGE 84

PAGE 85

Thebasicprincipleofourmethod-usingtheeigenprojectionforthespectraldecompositionandmakinguseoftheresultofedgepartitioningtondthecommunitystructure-canbeincorporatedintomoresophisticatedmethodthatcanautomaticallydetectcommunitystructureregardlessoftypesofnetworks.Wehopethatideasandalgorithmspresentedherewillbeimprovedtobemoreusefulforthedeterminationoffunctionalclustersinmanyvariousnetworks. 85

PAGE 86

[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

PAGE 87

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

PAGE 88

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

PAGE 89

D.Cvetkovic,M.Domb,andH.Saches,SpectraofGraphs:TheoryandApplications,JohnAmbrosius,1995. [51] I.J.Farkas,I.Derenyi,A.L.Barabasi,andT.Vicsek,\Spectraofreal-worldgraphs:Beyondthesemicirclelaw,"Phys.Rev.E,vol64,2001. [52] G.Siganos,M.Faloutsos,andC.Faloutsos,\Powe-lawandtheAS-levelinternettopology,"IEEE-ACMT.Network,vol11,pp514,2003. [53] E.Wigner,\Characteristicvectorofborderedmatricesiwthinnitedimensions,"AnnualsofMathematics,vol62,pp548{564,1955. [54] L.Giot,\AProteinInteractionMapofDrosophilamelanogaster,"Science,vol302,pp1727,Dec.2003. [55] C.KampandK.Christensen,\SpectralAnalysisofProtein-ProteinInteractionsinDrosophilamelanogaster,"arXiv,May2004. [56] R.Milo,\NetworkMotifs:SimpleBuildingBlocksofComplexNetworks,"Science,vol298,pp824,2002. [57] M.A.M.deAguiarandY.Bar-Yam,\Spectralanalysisandthedynamicresponseofcomplexnetworks,"PhysicalReviewE,vol71,016106,2005. [58] J.B.Gao,Y.H.Cao,andJ.Hu,\BuildinginnovativerepresentationsofDNAsequencestofaciliatategenending,"IEEEIntelligentSystems,2004. [59] G.S.Strang,IntroductiontoLinearAlgebra,WellesleyCambridgePress,1998. [60] D.Bu,Y.Zhao,andL.Cai,\Topologicalstructureanalysisoftheprotein-proteininteractionnetworkinbuddingyeast,"NucleicAcidsResearch,vol31,pp2443{2450,2003. [61] M.Catanzaro,M.Boguana,andR.Pastor-Satorras\Generationofuncorrelatedrandomscale-freenetworks,"Phys.Rev.E.,vol.71,pp.027103,2005. [62] M.E.J.Newman,\Fastalgorithmfordetectingcommunitystructureinnetworks,"Phys.Rev.E,vol69,066133,2004. [63] M.E.J.Newman,\TheStructureandFunctionofComplexNetworks,"SIAMREVIEW,vol45,pp167{256,2003. [64] W.W.Zachary,\Aninformationalowmodelforconictandssioninsmallgroups,"J.Anthropol.,vol33.pp452{473,1977. [65] M.E.J.Newman,\Networkdata,"http://www-personal.umich.edu/mejn/netdata/,July2007. [66] A.Clauset,M.E.J.Newman,andC.Moore,\Findingcommunitystructureinverylargenetworks,"Phys.Rev.E,vol70,066117,2004. 89

PAGE 90

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

PAGE 91