Network Analysis / Two mode networks

Transcription

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Network Analysis
Two mode
Networks
Vladimir Batagelj
University of Ljubljana
Josh On: They rule 2004
ECPR Summer School, July 19 – August 4, 2007
Faculty of Social Sciences, University of Ljubljana
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V. Batagelj: Network Analysis / Two mode networks
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Outline
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Analysis of two-mode networks . . . . . . . . . . . . . . . . . . . . . . . . . .
1
4
12
Two-mode cores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example: 4-rings in two-mode network . . . . . . . . . . . . . . . . . . . . . .
4
12
15
Directed 4-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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21
Multiplication of networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
25
Example: Kinship relations
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
30
Two-mode network analysis by conversion to one-mode network . . . . . . . . .
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31
34
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Normalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Networks from data tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
EU projects on simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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39
Analysis of ProjInst.net . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Analysis of ProjInst.net . . . . . . . . . . . . . . . . . . . . . . . . . . .
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41
Analysis of Countries.net . . . . . . . . . . . . . . . . . . . . . . . . . .
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ECPR Summer School, Ljubljana, July 19 – August 4, 2007
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Analysis of two-mode networks
A two-mode network or affiliation network is a structure N = (U, V, A, w),
where U and V are disjoint sets of vertices, A is the set of arcs with
the initial vertex in the set U and the terminal vertex in the set V, and
w : A → R is a weight. If no weight is defined we can assume a constant
weight w(u, v) = 1 for all arcs (u, v) ∈ A. The set A can be viewed also
as a relation A ⊆ U × V.
A two-mode network can be formally represented by rectangular matrix
W = [wuv ]U ×V .
w(u, v)
(u, v) ∈ A
wuv =
0
otherwise
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Approaches to two-mode network analysis
For direct analysis of two-mode networks we can use the eigen-vector
approach – a two-mode variant of Kleinberg’s hubs and authorities. The
weight vector (x, y) on U ∪ V is determined by relations y = Wx and
x = WT y.
Two new direct methods will be presented in this lecture: two-mode cores
and 4-rings. In the next lecture we shall also describe the clustering and
blockmodeling in two-mode networks.
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Internet Movie Database http://www.imdb.com/
12th Annual Graph Drawing Contest, 2005. The IMDB network is two-mode and has
1324748 = 428440 + 896308 vertices and 3792390 arcs.
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Two-mode cores
The subset of vertices C ⊆ V is a (p, q)-core in a two-mode network
N = (V1 , V2 ; L), V = V1 ∪ V2 iff
a. in the induced subnetwork K = (C1 , C2 ; L(C)), C1 = C ∩ V1 , C2 =
C ∩ V2 it holds ∀v ∈ C1 : degK (v) ≥ p and ∀v ∈ C2 : degK (v) ≥ q ;
b. C is the maximal subset of V satisfying condition a.
Properties of two-mode cores:
• C(0, 0) = V
• K(p, q) is not always connected
• (p1 ≤ p2 ) ∧ (q1 ≤ q2 ) ⇒ C(p1 , q1 ) ⊆ C(p2 , q2 )
• C = {C(p, q) : p, q ∈ N}. If all nonempty elements of C are different
it is a lattice.
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Algorithm for two-mode cores
To determine a (p, q)-core the procedure similar to the ordinary core
procedure can be used:
repeat
remove from the first set all vertices of degree less than p,
and from the second set all vertices of degree less than q
until no vertex was deleted
It can be implemented to run in O(m) time.
Interesting (p, q)-cores? Table of cores’ characteristics n1 = |C1 (p, q)|,
n2 = |C2 (p, q)| and k – number of components in K(p, q):
• n1 + n2 ≤ selected threshold
• ’border line’ in the (p, q)-table.
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Table (p, q : n1 , n2 ) for Internet Movie Database
1 1590: 1590
1 | 16 39:
2 516: 788
3 | 17 35:
3 212: 1705 18 | 18 32:
4 151: 4330 154 | 19 30:
5 131: 4282 209 | 20 28:
6 115: 3635 223 | 21 26:
7 101: 3224 244 | 22 24:
8
88: 2860 263 | 24 23:
9
77: 3467 393 | 27 22:
10
69: 3150 428 | 29 20:
11
63: 2442 382 | 32 19:
12
56: 2479 454 | 35 18:
13
50: 3330 716 | 36 17:
14
46: 2460 596 | 39 16:
15
42: 2663 739 | 42 15:
2173
2791
2684
2395
2216
1988
1854
34
31
35
34
33
33
29
28
678
995
1080
1063
1087
1087
1153
39
38
52
57
61
65
70
76
|
44
|
46
|
49
|
52
|
56
|
62
|
66
|
72
|
96
| 119
| 141
| 186
| 247
| 1334
|
14:
13:
12:
11:
10:
9:
8:
7:
6:
5:
4:
3:
2:
1:
29
83
29
94
26
95
16
79
34 162
31 177
29 198
22 203
7 114
6 137
8 258
3 186
2 247
1 1334
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(247,2)-core and (27,22)-core
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Give Me Your Soul...
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Girls of the Double D 9
Girls of the A Team, The
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Girls of Cell Block F
Girl with the Heart-Shaped Tattoo, The
Ginger’s Private Party
Ginger’s Greatest Boy/Girl Hits
Ginger Then and Now
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Ginger Lynn: The Movie
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Super Groupie
Sunny After Dark
Summer Break
Sugarpussy Jeans
Suburban Swingers 2
Street Walkers
Strange Sex in Strange Places
Stiff Competition 2
Stiff Competition
Starting Over
Starr
Star, The
Star Cuts 4: Ginger Lynn
Star Cuts 39: Trinity Loren
Star Cuts 37: Buffy Davis
Star 85
Splendor in the Ass
Splash Shots
Spies
Spermbusters
Spellbound
Spectacular Orgasms
Sorority Pink 2: The Initiation
Sophisticated Lady
Sodomania: The Baddest of the Best
Sodomania: Slop Shots 1
Sodomania 18
Sodomania 13
Snatched
Smart Ass, The
Smart Ass Returns, The
Slumber Party
Sloppy Seconds
Slip of the Tongue
Slip Into Ginger and Amber
Slightly Used
Sleeping with Everybody
Slave to Love
Sky Foxes
Skin Games
Sister Dearest
Sindy Does Anal Again
Simply Kia
Simply Blue
Shot From Behind
Sheila’s Deep Desires
Shayla’s Gang
Shaved Pink
Shane’s World
Sexy Secrets N 1
Sexy and 18
Sexual Fantasies
Sextectives
Sexpertease
Sex Toys
Sex Stories
Sex Sluts in the Slammer
Sex Shoot
Sex Plays
Sex Maniacs
Sex Fifth Avenue
Sex Busters
Sex Beat
Sex Appraisals
Sex Academy 2: The Art of Talking Dirty
Sex 3: After Seven
Sensual Exposure
Seeing Red
Seducers, The
Secret of Her Suckcess
Secret Life of Nina Hartley, The
Secret Fantasies 4
Secret Fantasies 3
Screaming Rage
Scarlet Woman, The
Scarlet Bride, The
Savanah Unleased
Satyr
Satin Seduction
Sabotage
Ruthless Affairs
Royals: Ginger Lynn
Roll-x Girls
Rocky Porno Video Show, The
Rising, The
Rise of the Roman Empress 2
Ribald Tales of Canterbury
Return to Sex 5th Avenue
Reincarnation of Don Juan
Red Vibe Diaries 2: Dark Desires
Rear Ended Roommates
Rear Busters
Raunch V
Raunch 6
Rapture
Rambone the Destroyer
Rainwoman 6
Rainwoman 4
Rainbird
Radio-active
Radio K-KUM
Rachel Ryan RR
Queen of Hearts 3
Pussypoppers
Pussyman Takes Hollywood
Pussyman 2: The Prize
Pussyman
Psychic, The
Prom Girls
Project: Ginger
Private Teacher
Private Affairs: Vol 6
Private Affairs: Vol 2
Pretty As You Feel
Precious Peaks
Pouring It On
Possessions
Portrait of Dorian
Pornomania 1
Porn on the 4th of July
Poonies, The
Pleasure Seekers
Pleasure Party
Pleasure Hunt Part II
Plaything 2
Plaything
Playing with a Full Dick
Play It Again... Samantha!
Pink Pussycat, The
Piece of Heaven
Physical II
Photo Flesh
Perverted Passions
Perverted 1
Perks
Perfect Fit
Peggy Sue
Passionate Heiress
Passionate Angels
Passenger 69
Passages 2
Passages 1
Party Doll A Go-Go, Part 1
Party Doll A Go-Go 2
Party Doll
Paradise Lost
Paler Shade of Blue, A
Overtime: Oral Hijinx
Oval Office, The
Outrageous Orgies 5
Outlaw, The
Out of Love
Orgies
Oral Majority 9
Oral Majority 8
Oral Majority 7
Oral Majority 4
Oral Majority 3
Oral Majority 10
Oral Majority
Open Up Traci
Only the Very Best On Video
Only the Best of the 80’s
One Night Stand
On the Loose
On Golden Blonde
Office Girls
Obsession
Nurse Nancy
Nurse Fantasies
Norma Jeane Anal Legend
Nobody’s Looking
No Tell Motel
Nightbreed
Night Temptress
Night Tales
Night Deposit
New Wave Hookers 4
New Wave Hookers 2
New Wave Hookers
Never Say Never
Naughty Thoughts
Naughty 90’s
Nasty Nymphos 3
Nasty Lovers
Naked Truth, The
Naked Ambition
Naked and Nasty
Mystic Pieces
Mystery of the Golden Lotus
Muff ’n’ Jeff
Motel Sex
More Than Friends
Moonstroked
Model Wife
Miscreants
Mirage 2
Mirage
Mind Shadows 2
Mind Shadows
Midslumber’s Night Dream
Midnight Pink
Midnight Hour, The
Megasex
Matter of Size, A
Masque
Mark of Zara
Marilyn Whips Wallstreet
Manbait 2
Manbait
Man Who Loves Women, The
Make My Wife, Please
Make My Night
Make Me Want It
Magic Shower, The
Lust in the Fast Lane
Lust College
Lust Bug, The
Lust at the Top
Luscious Lucy in Love
Lucky Break
Lovin’ USA
Lovers, The
Love Lessons
Love Ghost
Love Bites
Loose Morals
Loose Ends III
Loose Ends II
Loose Ends
Loads of Fun 4
Little Romance
Like a Virgin
Life and Loves of Nikki Charm
Let’s Play Doctor
Legend of Barbi-Q and Little Fawn, The
Laying the Ghost
Latin Lust
Last Temptation, The
Lascivious Ladies of Dr. Lipo, The
Laid Off
Lady in Red, The
KSEX 106.9
Kiss, The
Kiss My Asp
Kinky Vision 2
Kinky Couples
Kinky
Keyhole Video #114: Christy Canyon Special
Kascha and Friends
Just Another Pretty Face
Juicy Sex Scandals
Juicy Lucy
Jezebel
Jaded Love
It’s My Body
Interactive
Insatiable
Inferno
Indecent Itch
Indecent Exposures
Inches for Keisha
In Search of the Golden Bone
Immaculate Erection
Images of Desire
I Want It All
I Touch Myself
I Like to Be Watched
I Dream of Christy
Hunger, The
House On Chasey Lane
House of the Rising Sun
House of Sleeping Beauties 2
House of Sleeping Beauties
House of Blue Dreams
Hottest Ticket
Hothouse Rose Part 1
Hotel Sodom
Hotel Paradise
Hotel Fantasy
Hot Wired
Hot Tight Asses 9
Hot Tight Asses 8
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IMDB cores / Pajek commands
See How to deal with very large networks?
Options/Read-Write/Read-Save vertices labels [Off]
Read/Network [IMDB.net] 1:40
Info/Memory
Net/Partitions/Core/2-Mode Review
Net/Partitions/Core/2-Mode [27 22]
Info/Partition
Operations/Extract from Network/Partition [Yes 1]
Net/Partitions/2-Mode
Net/Transform/Add/Vertices Labels from File [IMDB.nam]
Draw/Draw-Partition
Layers/in y direction
Options/Transform/Rotate 2D [90]
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k-rings
A k-ring is a simple closed chain of length k. Using k-rings we can define a weight
of edges as
wk (e) = # of different k-rings containing the edge e ∈ E
Since for each eadge e of a complete graph Kr ,
r ≥ k ≥ 3 we have wk (e) = (r − 2)!/(r −
k)! the edges belonging to cliques have large
weights. Therefore these weights can be used
to identify the dense parts of a network.
The k-rings can be efficiently determined only
for small values of k – 3, 4, 5. The 3-rings (triangular) weights were implemented in Pajek
in May 2002.
Complete graph K5
On the k-rings we can also base the notion of short cycle connectivity which
provides us with another decomposition of networks.
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4-rings and analysis of two-mode networks
In two-mode network there are no 3-rings. The densest substructures are
complete bipartite subgraphs Kp,q . They contain many 4-rings.
There are
p q
1
= p(p − 1)q(q − 1)
4
2 2
4-rings in Kp,q ; and each of its edges e has
weight
w4 (e) = (p − 1)(q − 1)
The 4-rings weights were implemented in Pajek in August 2005.
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Example: Bibliography from W. Imrich, S. Klavžar: Product graphs:
structure and recognition, JohnWiley & Sons, New York, USA, 2000.
(PDF), (net – two-mode 674×314 network).
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Example: 4-rings in two-mode network
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Example: 1-edge cut for w4
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Example: labeled main part of 1-edge cut for w4
bamu-91
bamu-88
Skrekovski, R.Mulder, H.M.
klsk-96
klmu-98
Klavzar, S.
klmu-99
bamu-83
Bandelt, H.-J.
bamu-94
dam-97
klgu-95
pish-83
klgu-97
gukl-96
Shawe-Taylor, J.
mopi-81
Pisanski, T.
mopi-88
Mohar, B.
mopi-83
mopi-90
Gutman, I.
match-97
haim-99
imkl-99b
auha-91
imkl-93
Hagauer, J.
Schaffer, A.A.
imkl-98
fehe-85
imkl-97
fesc-86
imkl-92
Zerovnik, J.
fesc-92
imkl-99
imze-94
J.
Aurenhammer,Feigenbaum,
F.
imze-96
auha-90
White, A.T.
doim-70
Dorfler, W.
jhag-97
Agnihotri, N.
Kumar, R.
Jha, P.K.
jhsl-92
jhsl-94
hezh-94
doim-72
jhsl-89
jhsl-93
jhag-96
mopi-85
Zhu, X.
Imrich, W.
Hell, P.
Watkins, M.E.
nowa-72a
nowa-72b
Nowitz, L.A.
Slutzki, G.
zh
Zho
hey
Rival, I.
nori-83
nori-88
nora-96
Nowakowski, R
hara-91
Graham, R.L.
Chung, F. R.K.
grpo-71
Saks, M.E.
y
l y
s
chgr-89
s
Ullman, J.D.
Hopcroft, J.E. Alexe, G.
Olaru, E.
Aho, A.V.
olal-98
s
alol-97
ECPR Summer
School, Ljubljana,
July 19 – August 4, 2007
ahho-87
s
&
s
Jacobson, M.S.
jaki-83
Kinch, L.F.
fija-85
Rall, D.F.
brno-96
hara-97
Hartnell, B.L.
hara-95
s
jaki-86
popu-81
Pultr, A.
Poljak, S.
poro-83
lone-80
poro-81
nepu-78
Rodl, V.
nero-78
Nesetril, J.
nero-85
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Directed 4-rings
There are 4 types of directed 4-rings:
cyclic
transitive
genealogical
diamond
In the case of transitive rings Pajek provides a special weight counting on
how many transitive rings the arc is a shortcut.
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Simple line islands in IMDB for w4
We obtained 12465 simple line islands on 56086 vertices. Here is their size
distribution.
Size Freq
Size Freq
Size Freq
Size Freq
-------------------------------------------------------2 5512
20
19
38
4
59
2
3 1978
21
18
39
3
61
1
4 1639
22
15
40
2
64
1
5
968
23
9
42
2
67
1
6
666
24
13
43
3
70
1
7
394
25
12
45
3
73
1
8
257
26
6
46
4
76
1
9
209
27
6
47
5
82
1
10
148
28
5
48
1
86
1
11
118
29
6
49
2
106
1
12
87
30
3
50
2
122
1
13
55
31
6
51
1
135
1
14
62
32
5
52
2
144
1
15
46
33
3
53
1
163
1
16
39
34
1
54
2
269
1
17
27
35
5
55
1
301
1
18
28
36
4
57
1
332
2
19
29
37
7
58
1
673
1
--------------------------------------------------------
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Example: Islands for w4 / Charlie Brown and Adult
Morgan, Jonathan (I)
Kesten, Brad
Brando, Kevin
Robbins, Peter (I)
Shea, Christopher (I)
Altieri, Ann
Schoenberg, Jeremy
Boy, T.T.
Hauer, Brent
Voyeur, Vince
Charlie Brown and Snoopy Show
Reilly, Earl ’Rocky’
Charlie Brown Celebration
Ornstein, Geoffrey
You Don’t Look 40, Charlie Brown
He’s Your Dog, Charlie Brown
Making of ’A Charlie Brown Christmas’
You’re In Love, Charlie Brown
It’s the Great Pumpkin, Charlie Brown
Charlie Brown’s All Stars!
Life Is a Circus, Charlie Brown
Charlie Brown Christmas
Dough, Jon
Sanders, Alex (I)
North, Peter (I)
Michaels, Sean
Race for Your Life, Charlie Brown
Be My Valentine, Charlie Brown
Mendelson, Karen
Stratford, Tracy
Davis, Mark (V)
Horner, Mike
It’s Magic, Charlie Brown
Dryer, Sally
Melendez, Bill
You’re a Good Sport, Charlie Brown
Drake, Steve (I)
It’s a Mystery, Charlie Brown
Boy Named Charlie Brown
It’s an Adventure, Charlie Brown
Byron, Tom
Silvera, Joey
It’s Flashbeagle, Charlie Brown
Play It Again, Charlie Brown
Momberger, Hilary
West, Randy (I)
Is This Goodbye, Charlie Brown?
Charlie Brown Thanksgiving
There’s No Time for Love, Charlie Brown
Jeremy, Ron
You’re Not Elected, Charlie Brown
Snoopy Come Home
It’s the Easter Beagle, Charlie Brown
Wallice, Marc
Savage, Herschel
Thomas, Paul (I)
Shea, Stephen
Pajek
Pajek
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Example: Islands for w4 / Mark Twain and Abid
Sergeant Madden
Honky Tonk
Sawak nus el lail
Hoodlum Saint, The
Roaring Twenties, The
Soltan, Hoda
Malak el zalem, El
Rostom, Hind
Unconquered
Union Pacific
Phelps, Lee (I)
El Dekn, Tewfik
Flavin, James
Big City
Sittat afarit, alFatat el mina
Hareb min el ayyam
Abu Hadid
Elf laila wa laila
Souk el selah
Nashal, El
Maktub alal guebin
Fatawat el Husseinia
Amir el antikam
Abid el gassad
Ghaltet ab
Abu Dahab
Aguazet seif
Hamida
Batal lil nehaya
Namrud, El
Ebn el-hetta
Nassab, El
Zoj el azeb, El
Abid el mal
Cass el azab
Ghazal al-banat
Rasif rakam khamsa
Laab bil nar, El
Iskanderija... lih?
Imlak, El
Matloub zawja fawran
Sarhan, Shukry
Port Said
Riad, Hussein
Saum, Cliff
Wells Fargo
Star Is Born, A
Fatawa, El
Dunn, Ralph
Tarik el saada
Hub fil zalam
San Quentin
You Can’t Take It with You
Vogan, Emmett
Chandler, Eddy
Flowers, Bess
Hamama, Faten
O’Connor, Frank (I)
Hamdi, Imad
Whole Town’s Talking, The
Dust Be My Destiny
Ard el ahlam
Sullivan, Charles (I)
Nancy Drew... Reporter
Shawqi, Farid
Meet John Doe
Holmes, Stuart
Baad al wedah
Massiada, Al
Asrar el naas
Baba Amin
Beyt al Taa
Haked, El
Osta Hassan, El
Ibn al ajar
Ana bint min?
Murra kulshi, El
Mohtal, El
Zalamuni el habaieb
Ashki limin?
Ana zanbi eh?
Castle on the Hudson
Valley of the Giants
Racket Busters
Kid Galahad
Go Getter, The
They Made Me a Criminal
Women in the Wind
Mower, Jack
Man Who Talked Too Much, The
Naughty But Nice
Yankee Doodle Dandy
Kid From Kokomo, The
King of the Underworld
They Drive by Night
Secret Service of the Air
Bad Men of Missouri
Adventures of Mark Twain, The
Knockout
Smashing the Money Ring
El-Meliguy, Mahmoud
Abu Ahmad
Pajek
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Example: Island for w4 / Polizeiruf 110 and Starkes Team
Maranow, Maja
Starkes Team, Ein
Starkes Team - Eins zu Eins, Ein
’Affre Semmeling, Die’
Starkes Team - Kollege Mrder, Ein
Starkes Team - Sicherheitsstufe 1, Ein
Martens, Florian
Starkes Team - Erbarmungslos, Ein
Starkes Team - Das Bombenspiel, Ein
Polizeiruf 110 - Ein Bild von einem Mrder
Polizeiruf 110 - Kopf in der Schlinge
Polizeiruf 110 - Zerstrte Trume
Polizeiruf 110 - Angst um Tessa Blow
Polizeiruf 110 - Rosentod
Starkes Team - Blutsbande, Ein
Polizeiruf 110 - Doktorspiele
Starkes Team - Tdliche Rache, Ein
Polizeiruf 110 - Jugendwahn
Starkes Team - Der Mann, den ich hasse, Ein
Polizeiruf 110 - Heikalte Liebe
Starkes Team - Kindertrume, Ein
Starkes Team - Mrderisches Wiedersehen, Ein
Lansink, Leonard
Starkes Team - Auge um Auge, Ein
Schwarz, Jaecki
Starkes Team - Lug und Trug, Ein
Starkes Team - Der letzte Kampf, Ein
Polizeiruf 110 - Todsicher
Polizeiruf 110 - Der Spieler
Polizeiruf 110 - Mordsfreunde
Starkes Team - Kleine Fische, groe Fische, Ein
Starkes Team - Roter Schnee, Ein
Bademsoy, Tayfun
Starkes Team - Der Verdacht, Ein
Starkes Team - Der Todfeind, Ein
Polizeiruf 110 - Kurschatten
Starkes Team - Mordlust, Ein
Winkler, Ein
Wolfgang
Starkes Team - Das groe Schweigen,
Polizeiruf 110 - Tote erben nicht
Starkes Team - Der schne Tod, Ein
Polizeiruf 110 - Der Pferdemrder
Starkes Team - Trume und Lgen, Ein
Polizeiruf 110 - Henkersmahlzeit
Starkes Team - Bankraub, Ein
Starkes Team - Verraten und verkauft, Ein
Starkes Team - Braunauge, Ein
Starkes Team - Im Visier des Mrders, Ein
Starkes Team - Die Natter, Ein
Lerche, Arnfried
Pajek
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5-rings
In the future we intend to implement in Pajek also weights w5 . Again
there are only 4 types of directed 5-rings.
cyclic
transitive
????
????
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Multiplication of networks
To a simple two-mode network N = (I, J , E, w); where I and J are sets
of vertices, E is a set of edges linking I and J , and w : E → R (or some
other semiring) is a weight; we can assign a network matrix W = [wi,j ]
with elements: wi,j = w(i, j) for (i, j) ∈ E and wi,j = 0 otherwise.
Given a pair of compatible networks NA = (I, K, EA , wA ) and NB =
(K, J , EB , wB ) with corresponding matrices AI×K and BK×J we call a
product of networks NA and NB a network NC = (I, J , EC , wC ), where
EC = {(i, j) : i ∈ I, j ∈ J , ci,j 6= 0} and wC (i, j) = ci,j for (i, j) ∈ EC .
The product matrix C = [ci,j ]I×J = A ∗ B is defined in the standard way
X
ci,j =
ai,k · bk,j
k∈K
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s
s
In the case when I = K = J we are dealing with ordinary one-mode
networks (with square matrices).
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Fast sparse matrix multiplication
The standard matrix multiplication has the complexity O(|I| · |K| · |J |) – it
is too slow to be used for large networks. For sparse large networks we can
multiply faster considering only nonzero elements:
for k in K do
for i in NA (k) do
for j in NB (k) do
if ∃ci,j then ci,j := ci,j + ai,k ∗ bk,j
else new ci,j := ai,k ∗ bk,j
NA (k): neighbors of vertex k in network NA
NB (k): neighbors of vertex k in network NB
In general the multiplication of large sparse networks is a ’dangerous’
operation since the result can ’explode’ – it is not sparse.
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Complexity of fast sparse matrix multiplication
Let A and B be matrices of networks NA = (I, K, EA , wA ) and NB =
(K, J , EB , wB ).
Assume that the body of the loops can be computed in the constant time c.
Then the complexity of product is
X X
X
X
C=
c=c·
degA (k) · degB (k)
k∈K i∈NA (k) j∈NB (k)
k∈K
B
Let ∆A
K = maxk∈K degA (k) and ∆K = maxk∈K degB (k) and consider
the well known equality
X
X
degA (k) =
degA (i) = |EA |
k∈K
i∈I
A
We get C ≤ c · min(|EA | · ∆B
K , |EB | · ∆K ).
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If at least one of the sparse networks NA and NB has small maximal degree
on K then also the resulting product network NC is sparse.
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More detailed complexity analysis
Let dmin (k) = min(degA (k), degB (k)), ∆min = maxk∈K dmin (k),
dmax (k) = max(degA (k), degB (k)), K(d) = {k ∈ K : dmax (k) ≥ d},
d∗ = argmind (|K(d)| ≤ d) and K∗ = K(d∗ ). Then |K∗ | ≤ d∗ and
C =c·
X
degA (k) · degB (k) = c ·
k∈K
=c·(
X
dmin (k) · dmax (k)
k∈K
dmin (k) · dmax (k) +
k∈K∗
≤ c · (∆min ·
X
X
dmin (k) · dmax (k))
k∈K\K∗
X
k∈K∗
∗
dmax (k) + d ·
X
dmin (k))
k∈K\K∗
≤ c · d∗ · (∆min · max(|I|, |J|) + min(|EA |, |EB |))
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s
If for the sparse networks NA and NB the quantities ∆min and d∗ are small
then also the resulting product network NC is sparse.
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Example: Kinship relations
Anthropologists typically use a basic vocabulary of kin types to represent
genealogical relationships. One common version of the vocabulary for
basic relationships:
Kin Type
P
F
M
C
D
S
G
Z
B
E
H
W
English Type
Parent
Father
Mother
Child
Daughter
Son
Sibling
Sister
Brother
Spouse
Husband
Wife
The genealogies are usually described in GEDCOM format. Examples
family, Bouchards.
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Ore graph
m-grandmother
m-grandfather
f-grandfather
mother
sister-in-law
brother
daughter-in-law
father
I
son
f-grandmother
stepmother
wife
son-in-law
sister
daughter
In Ore graph
every person is represented by a
vertex,
marriages, relation
is a spouse of ,
are represented with edges,
and relations
is a mother of
and
is a father of
as arcs pointing from parents to
their children.
grandson
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Calculating kinship relations
Pajek generates three relations when reading genealogy as Ore graph:
F:
is a father of
M:
is a mother of
E:
is a spouse of
Additionally we must generate two binary diagonal matrices, to distinguish
between male and female:
L:
is a male
J:
is a female
/ 1-male, 0-female
/ 1-female, 0-male
F ∩ M = ∅,
L ∪ J ⊆ I,
L∩J=∅
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s
y
s
s
s
&
%
* 6
28
'
$
Derived kinship relations
Other basic relations can be obtained using macros based on identities:
is a parent of
P = F ∪M
is a child of
C = PT
is a son of
S = L∗C
is a daughter of
D = J ∗C
is a husband of
H = L∗E
is a wife of
W = J ∗E
is a sibling of
G = ((F T ∗ F ) ∩ (M T ∗ M )) \ I
is a brother of
B = L∗G
is a sister of
Z = J ∗G
is an uncle of
U = B∗P
is an aunt of
A = Z ∗P
is a semi-sibling of
Ge = (P T ∗ P ) \ I
and using them other relations can be determined
is a grand mother of
is a niece of
M2
Ni
=
=
M ∗P
D∗G
s
s
l y
s
y
s
s
s
&
%
* 6
29
'
$
Relative sizes of kinship relations in genealogies
Kin Type
P-Parent
F-Father
M-Mother
C-Child
D-Daughter
S-Son
G-Sibling
Z-Sister
B-Brother
E-Spouse
H-Husband
W-Wife
U-Uncle
A-Aunt
Ge-Semi-sibling
n
mE = Spouse
mA = Parent
Turks Ragusa
1.000 1.000
0.514 0.532
0.486 0.468
1.000 1.000
0.431 0.384
0.569 0.616
1.250 0.943
1.135 0.746
1.366 1.140
0.205 0.215
0.205 0.215
0.205 0.215
1.920 1.789
1.750 1.143
1.473 1.155
1269
407
1987
5999
2002
9315
Loka
1.000
0.504
0.496
1.000
0.480
0.520
1.019
0.983
1.055
0.208
0.208
0.208
1.200
1.190
1.128
Silba
1.000
0.519
0.481
1.000
0.469
0.531
0.811
0.760
0.861
0.230
0.230
0.230
1.181
1.097
0.932
Royal
1.000
0.540
0.460
1.000
0.427
0.573
0.767
0.707
0.828
0.306
0.306
0.306
0.927
0.798
0.905
47956
14154
68052
6427
2217
9627
3010
1138
3724
s
s
l y
s
y
s
s
s
&
%
* 6
30
'
$
Two-mode network analysis by conversion to
one-mode network
Often we transform a two-mode network N = (U, V, E, w) into an
ordinary (one-mode) network N1 = (U, E1 , w1 ) or/and N2 = (V, E2 , w2 ),
(1)
where E1 and w1 are determined by the matrix W(1) = WWT , wuv =
P
(1)
(1)
T
w
·
w
.
Evidently
w
=
w
. There is an edge (u : v) ∈ E1 in
uv
vu
uz
zv
z∈V
(1)
N1 iff N (u) ∩ N (v) 6= ∅. Its weight is w1 (u, v) = wuv .
The network N2 is determined in a similar way by the matrix W(2) =
WT W.
The networks N1 and N2 are analyzed using standard methods.
s
s
l y
s
y
s
s
s
&
%
* 6
31
'
$
Normalizations
The normalization approach was developed for quick inspection of (1mode) networks obtained from two-mode networks – a kind of network
based data-mining.
In networks obtained from large two-mode networks there are often huge
differences in weights. Therefore it is not possible to compare the vertices
according to the raw data. First we have to normalize the network to make
the weights comparable.
There exist several ways how to do this. Some of them are presented in the
following table. They can be used also on other networks.
s
s
s
s
s
s
In the case of networks without loops we define the diagonal weights for
undirected networks as the sum of out-diagonal elements in the row (or
P
column) wvv = u wvu and for directed networks as some mean value
P
P
1
of the row and column sum, for example wvv = 2 ( u wvu + u wuv ).
Usually we assume that the network does not contain any isolated vertex.
&
%
y l y
* 6
32
'
$
. . . Normalizations
Geouv
=
wuv
√
wuu wvv
Inputuv
=
wuv
wvv
Minuv
=
MinDiruv
=
wuv
min(wuu , wvv )
wuv
wuu ≤ wvv
wuu
0
otherwise
GeoDeguv
=
wuv
p
degu degv
Outputuv
=
wuv
wuu
Maxuv
=
MaxDiruv
=
wuv
max(wuu , wvv )
wuv
wuu ≤ wvv
wvv
0
otherwise
After a selected normalization the important parts of network are obtained
by line-cuts or islands approaches.
Slovenian journals and magazins.
Reuters Terror News: GeoDeg, MaxDir, MinDir.
s
s
l y
s
y
s
s
s
&
%
* 6
33
'
$
GeoDeg normalization of Reuters terror news network
s
s
l y
s
y
s
s
s
&
%
* 6
34
'
$
Networks from data tables
A data table T is a set of records T = {Tk : k ∈ K}, where K is the set of
keys. A record has the form Tk = (k, q1 (k), q2 (k), . . . , qr (k)) where qi (k)
is the value of the property (attribute) qi for the key k.
s
s
l y
s
y
s
s
s
&
%
* 6
35
'
$
. . . Networks from data tables
Suppose that the property q has the range Q. If Q is finite (it can always be
transformed in such set by partitioning the set Q and recoding the values)
we can assign to the property q a two-mode network K × q = (K, Q, E, w)
where (k, v) ∈ E iff q(k) = v, and w(k, v) = 1.
Also, for properties qi and qj we can define a two-mode network qi × qj =
(Qi , Qj , E, w) where (u, v) ∈ E iff ∃k ∈ K : (qi (k) = u ∧ qj (k) = v), and
w(u, v) = card({k ∈ K : (qi (k) = u ∧ qj (k) = v)}).
It holds [qi × qj ]T = qj × qi and qi × qj = [K × qi ]T ∗ [K × qj ] =
[qi × K] ∗ [K × qj ].
We can join a pair of properties qi and qj also with respect to the third
property qs : we get a two-mode network [qi ×qj ]/qs = [qi ×qs ]∗[qs ×qj ].
s
s
l y
s
y
s
s
s
&
%
* 6
36
'
$
EU projects on simulation
For the meeting The Age of Simulation at Ars Electronica in Linz, January
2006 a dataset of EU projects on simulation was collected by FAS research,
Vienna and stored in the form of Excel table (SimPro.csv).
The rows are the projects participants (idents) and colomns correspond to
different their properties. Three two-mode networks were produced from
this table using Jürgen Pfeffer’s Text2Pajek program:
• project.net – P = [idents × projects]
• country.net – C = [idents × countries]
• institution.net – U = [idents × institutions]
|idents| = 8869, |projects| = 933,
|countries| = 60.
|institutions| = 3438,
s
s
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s
y
s
s
s
&
%
* 6
37
'
$
EU projects – network multiplication
Since all three networks have the common set (idents) we can derive from
them using network multiplication several interesting networks:
• ProjInst.net – W = [projects × institutions] = PT ∗ U
• Countries.net – S = [countries × countries] = CT ∗ C
• Institutions.net – Q = [institutions × institutions] = WT ∗W
• ...
s
s
l y
s
y
s
s
s
&
%
* 6
38
'
$
EU projects – deleted projects
Some projects (27) from original data set have to be deleted – in the final
data set SimPro they were marked as deleted. When producing two-mode
networks we could first physicaly delete them. Instead of this, we used
another approach: we produced the cluster CD of deleted idents and from
it a two-mode matrix D (idents × idents). Matrix D is a ’diagonal’ matrix
with value 1 for idents not belonging to CD and 0 otherwise. Using matrix
D we can determine the network ProjInst.net by W = PT ∗ D ∗ U
– the deleted idents don’t contribute to the network.
s
s
l y
s
y
s
s
s
&
%
* 6
39
'
$
Analysis of ProjInst.net
For identifying important parts of ProjInst.net we first computed the
4-rings weights and in the obtained network we determined the line islands
Net/Count/4-rings/Undirected
Net/Partitions/Islands/Line Weights[Simple [2,200]
We obtain 101 islands. We extracted 18 islands of the size at least 5. There
are two most important islands: aviation companies and car companies.
In labels we used a new option \n.
For analysis of two-mode networks we can use also (p, q)−cores.
s
s
l y
s
y
s
s
s
&
%
* 6
40
'
$
Analysis of ProjInst.net
ARMINES
PSI FUR PRODUKTE UND
SYS.E DER INFORMATIONSTECH.
BICC GENERAL CABLE
TQT SRL
ESI SOFTWARE SA
CHALMERS TEKNISKA HOEGSKOLA
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T3.5/99
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HPSE-CT-2002-00108
THE AARHUS SCHOOL OF BUSINESS
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METALLURGICAL RESEARCH
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IST-2000-28177
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JOR3980200
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CENTRE DE ROBOTIQUE
FRIMEKO INT. AB
CRE GROUP LTD.
MJM GROUP, A.S.
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TPS TERMISKA PROCESSER AB
PROLEXIA
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FERNSEHEN BETEILIGUNGS & CO
INGENIORHOJSKOLEN HELSINGOR TEKNIKUM
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Pajek
s
s
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s
y
s
s
s
&
%
* 6
41
'
$
Analysis of Countries.net
To obtain picture in which the
stronger lines cover weaker lines we
have to sort them
Kazakhstan
Afghanistan
Japan
Azerbaijan
Georgia
Morocco
Liechtenstein
Ecuador
India
Uzbekistan
Armenia
Net/Transform/Sort
Iceland
Belarus
Tunisia
Jordan
Canada
Lebanon
France
lines/Line values/Ascending
Algeria
United Kingdom
Italia
The Netherlands
Russian F.
Finland
Moldavia
China
Turkey
Greece
Germany
Portugal
Spain
Switzerland
Turkmenistan
Cyprus
Sweden
Denmark
Thailand
Israel
Austria
Estonia
Slovakia
USA
Belgium
Poland
Croatia
Luxembourg
Latvia
Malta
Ukraine
Norway
Slovenia
Ireland
Serbia-Montenegro
Macedonia
Hungary
Romania
Bulgaria Czech R.
Lithuania
Albania
For dense (sub)networks we get
better visualization by using matrix
display. In this case we also recoded
values (2,10,50).
To determine
clusters we used Ward’s clustering
procedure with dissimilarity measure
d5 (corrected Euclidean distance).
Pajek
The permutation determined by hierarchy can often be improved by
changing the positions of clusters. We get a typical center-periphery
structure.
s
s
l y
s
y
s
s
s
&
%
* 6
42
'
$
Analysis of Countries.net
Pajek - shadow [0.00,4.00]
Ecuador
Thailand
Armenia
Turkmenist
Uzbekistan
Moldavia
Japan
Kazakhstan
Azerbaijan
India
Macedonia
Albania
Liechtenst
Serbia-Mon
Iceland
Canada
Estonia
China
Belarus
Georgia
Tunisia
Lebanon
Jordan
Algeria
Malta
Morocco
Afghanista
Luxembourg
Croatia
Latvia
Lithuania
Cyprus
Turkey
Bulgaria
Ukraine
Slovenia
Romania
Slovakia
USA
Portugal
Denmark
Poland
Finland
Switzerlan
Austria
Czech R.
Ireland
Norway
Hungary
Israel
Russian F.
Sweden
Greece
Belgium
Spain
The Nether
France
United Kin
Germany
Italia
Ecuador
Thailand
Armenia
Turkmenist
Uzbekistan
Moldavia
Japan
Kazakhstan
Azerbaijan
India
Macedonia
Albania
Liechtenst
Serbia-Mon
Iceland
Canada
Estonia
China
Belarus
Georgia
Afghanista
Morocco
Malta
Tunisia
Lebanon
Jordan
Algeria
Croatia
Latvia
Lithuania
Luxembourg
Cyprus
Turkey
Bulgaria
Ukraine
Slovenia
Romania
Slovakia
USA
Russian F.
Israel
Hungary
Ireland
Czech R.
Norway
Poland
Finland
Portugal
Denmark
Switzerlan
Austria
Sweden
Greece
Belgium
Spain
The Nether
Italia
France
United Kin
Germany
Pajek - Ward [0.00,4785.14]
Ecuador
Thailand
Armenia
Turkmenist
Uzbekistan
Moldavia
Japan
Kazakhstan
Azerbaijan
India
Macedonia
Albania
Liechtenst
Serbia-Mon
Iceland
Canada
Estonia
China
Belarus
Georgia
Afghanista
Morocco
Malta
Tunisia
Lebanon
Jordan
Algeria
Croatia
Latvia
Lithuania
Luxembourg
Cyprus
Turkey
Bulgaria
Ukraine
Slovenia
Romania
Slovakia
USA
Russian F.
Israel
Hungary
Ireland
Czech R.
Norway
Poland
Finland
Portugal
Denmark
Switzerlan
Austria
Sweden
Greece
Belgium
Spain
The Nether
Italia
France
United Kin
Germany
s
s
l y
s
y
s
s
s
&
%
* 6
43
'
$
Analysis of Institutions.net
U.A.S.ZITTAU/GOERLITZ
U.THE AEGEAN
MIT-MANAGEMENT INTELLIGENTER TECH.N
U.PARIS-DAUPHINE
OTTO VON GUERICKE MAGDEBURG U.
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To identify the most important institutions we first computed pS -cores vector and use it to determine the corresponding vertex islands. We got
essentially one large island. Again
the corresponding subnetwork is very
dense. We prepared also a matrix display.
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%
* 6

Network Analysis / Two mode networks

Transcription

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