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Analysis and visulization of large networks with Pajek
Vladimir Batagelj University of Ljubljana
Vienna, St. Stephen’s Cathedral
Methodenforum der Fakult¨ t fur Sozialwissenschaften a ¨ Universit¨ t Wien, 21-22. 6. 2007 a &
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Outline
Networks . . . . . . . . . . . . . . . . . . Complexity of algorithms . . . . . . . . . Pajek . . . . . . . . . . . . . . . . . . Approaches to large networks . . . . . . . Statistics . . . . . . . . . . . . . . . . . . Clusters, clusterings, partitions, hierarchies Representations of properties . . . . . . . . Example: Snyder and Kick World Trade . . Clustering . . . . . . . . . . . . . . . . . Contraction of cluster . . . . . . . . . . . Subgraph . . . . . . . . . . . . . . . . . . Important vertices in network . . . . . . . Dense groups . . . . . . . . . . . . . . . . Connectivity . . . . . . . . . . . . . . . . Cuts . . . . . . . . . . . . . . . . . . . . Citation weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6 7 11 12 19 20 30 35 38 42 44 52 60 64 69 70 75
k-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Islands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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84 91 96 101 105 107 116
Bipartite cores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Directed 4-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pattern searching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84 91 96
Multiplication of networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Networks from data tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 EU projects on simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 What else? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
http://vlado.fmf.uni-lj.si/pub/networks/doc/tut/Vienna07.pdf
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Networks
A network is based on two sets – set of vertices (nodes), that represent the selected units, and set of lines (links), that represent ties between units. They determine a graph. A line can be directed – an arc, or undirected – an edge. Additional data about vertices or lines can be known – their properties (attributes). For example: name/label, type, value, . . .
Alexandra Schuler/ Marion Laging-Glaser: Analyse von Snoopy Comics
Network = Graph + Data
The data can be measured or computed. &
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Networks / Formally
A network N = (V, L, P, W) consists of: • a graph G = (V, L), where V is the set of vertices and L = E ∪ A is the set of lines; A is the set of arcs and E is the set of edges. n = |V|, m = |L| • P vertex value functions / properties: p : V → A • W line value functions / weights: w : L → B
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12 "f12" 13 "f13" 14 "f14" V. Batagelj: Analysis and visulization of large networks with Pajek 15 "f15" 16 "f16" 17 "f17" 18 "f18" 19 "f19" 20 "f20" *Edges 1 2 2 1 3 10 inter.net inter.net sex.clu 1 4 4 1 5 5 *Vertices 20 *vertices 20 1 6 5 1 "m01" 1 1 7 9 2 "m02" 1 1 8 7 3 "m03" 1 1 9 4 4 "m04" 1 1 10 3 5 "m05" 1 1 11 3 6 "f06" 2 1 12 7 7 "f07" 2 1 13 3 8 "f08" 2 1 14 2 9 "f09" 2 1 15 5 10 "f10" 2 1 16 1 11 "f11" 2 1 17 4 12 "f12" 2 1 18 1 13 "f13" 2 2 3 5 14 "f14" 2 2 4 1 15 "f15" 2 2 5 3 16 "f16" 2 2 6 1 17 "f17" 2 2 7 4 18 "f18" 2 2 8 2 19 "f19" 2 2 9 6 20 "f20" 2 2 10 2 *Edges 2 11 5 1 2 2 2 12 4 1 3 10 2 13 3 1 4 4 2 14 2 1 5 5 2 15 2 1 6 5 2 ... 6 16 1 7 9 2 17 3 1 8 7 2 18 1 1 9 4 2 19 1 1 notes: Important 10 3 0 is not3allowed as vertex number. 4 8 1 11 3 3 5 9 1 3 6 5 lines should12 ended with CR LF. be 7 1 13 3 3 7 11 1 14 2 3 8 7 1 15 5 3 9 8 1 16 1 3 Universit¨ t Wien, 21-22. 6. 2007 10 8a Methodenforum der Fakult¨ t17 r Sozialwissenschaften, u 1 a f¨ 4 3 11 14 1 18 1 3 12 17 2 3 5 3 13 9
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Example: Wolfe Monkey Data
age.vec
*vertices 20 15 10 10 8 7 15 5 11 8 9 16 10 14 5 7 11 7 5 15 4
rank.per
*vertices 20 1 2 3 4 5 10 11 6 12 9 7 8 18 19 20 13 14 15 16 17
Pajek doesn’t support Unix text files –
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Size of network
The size of a network/graph is expressed by two numbers: number of vertices n = |V| and number of lines m = |L|. In a simple undirected graph (no parallel edges, no loops) m ≤ 1 n(n − 1); and in 2 2 a simple directed graph (no parallel arcs) m ≤ n . The quotient γ =
m mmax
is a density of graph.
Small networks (some tens of vertices) – can be represented by a picture and analyzed by many algorithms (UCINET, NetMiner). Also middle size networks (some hundreds of vertices) can still be represented by a picture (!?), but some analytical procedures can’t be used. Till 1990 most networks were small – they were collected by researchers using surveys, observations, archival records, . . . The advances in IT allowed to create networks from the data already available in the computer(s). Large networks became reality. Large networks are too big to be displayed in details; special algorithms are needed for their analysis (Pajek ).
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Large Networks
Large network – several thousands or millions of vertices. Can be stored in computer’s memory – otherwise huge network. Usually sparse m < n2 ; typical: m = O(n) or m = O(n log n) . <
network ODLIS dictionary Citations SOM Molecula 1ATN Comput. geometry English words 2-8 Internet traceroutes Franklin genealogy World-Wide-Web Internet Movie DB Wikipedia US patents SI internet size 61K 168K 74K 140K 520K 1.7M 12M 3.6M 113.6M 53.8M 82M 38M n = |V | 2909 4470 5020 7343 52652 124651 203909 325729 1324748 659388 3774768 5547916 m = |L| 18419 12731 5128 11898 89038 207214 195650 1497135 3792390 16582425 16522438 62259968 source ODLIS online Garfield’s collection Brookhaven PDB BiBTEX bibliographies Knuth’s English words Internet Mapping Project Roperld.com gedcoms Notre Dame Networks IMDB Wikimedia Nber Najdi Si
Pajek datasets. &
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Complexity of algorithms
From some thousands to some (tens) millions of units (vertices). Let us look to time complexities of some typical algorithms:
T(n) LinAlg LogAlg SqrtAlg SqrAlg CubAlg O(n) O(n log n) √ O(n n) O(n2 ) O(n3 ) 1.000 0.00 s 0.00 s 0.01 s 0.07 s 0.10 s 10.000 0.015 s 0.06 s 0.32 s 7.50 s 1.67 m 100.000 0.17 s 0.98 s 10.0 s 12.5 m 1.16 d 1.000.000 2.22 s 14.4 s 5.27 m 20.8 h 3.17 y 10.000.000 22.2 s 2.8 m 2.78 h 86.8 d 3.17 ky
For the interactive use on large graphs already quadratic algorithms, O(n2 ), are too slow.
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Pajek
The main goals in the design of Pajek are: • to support abstraction by (recursive) decomposition of a large network into several smaller networks that can be treated further using more sophisticated methods; • to provide the user with some powerful visualization tools; • to implement a selection of efficient subquadratic algorithms for analysis of large networks. With Pajek we can: find clusters (components, neighbourhoods of ‘important’ vertices, cores, etc.) in a network, extract vertices that belong to the same clusters and show them separately, possibly with the parts of the context (detailed local view), shrink vertices in clusters and show relations among clusters (global view).
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Pajek’s data types
In Pajek analysis and visualization are performed using 6 data types:
• network (graph), • partition (nominal or ordinal properties of vertices), • vector (numerical properties of vertices), • cluster (subset of vertices), • permutation (reordering of vertices, ordinal properties), and • hierarchy (general tree structure on vertices). Pajek supports also multi-relational, temporal and two-mode networks.
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. . . Pajek’s data types
The power of Pajek is based on several transformations that support different transitions among these data structures. Also the menu structure of the main Pajek’s window is based on them. Pajek’s main window uses a ‘calculator’ paradigm with list-accumulator for each data type. The operations are performed on the currently active (selected) data and are also returning the results through accumulators. The procedures are available through the main window menus. Frequently used sequences of operations can be defined as macros. This allows also the adaptations of Pajek to groups of users from different areas (social networks, chemistry, genealogy, computer science, mathematics. . . ) for specific tasks. Pajek supports also repetitive operations on series of networks. &
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ESNA Pajek
An introduction to social network analysis with Pajek is available in the book ESNA (de Nooy, Mrvar, Batagelj 2005). Pajek– program for analysis and visualization of large networks is freely available, for noncommercial use, at its web site.
http://vlado.fmf.uni-lj.si/pub/networks/pajek/ & Methodenforum der Fakult¨ t f¨ r Sozialwissenschaften, Universit¨ t Wien, 21-22. 6. 2007 a u a S G S
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Approaches to large networks
In analysis of a large network (several thousands or millions of vertices, the network can be stored in computer memory) we can’t display it in its totality; also there are only few algorithms available. To analyze a large network we can use statistical approach or we can use the described decomposition approach – identify smaller (sub) networks that can be analyzed further using more sophisticated methods.
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Statistics
Input data • numeric → vector • ordinal → permutation • nominal → clustering (partition) Computed properties global: number of vertices, edges/arcs, components; maximum core number, . . . local: degrees, cores, indices (betweeness, hubs, authorities, . . . ) inspections: partition, vector, values of lines, . . . Associations between computed (structural) data and input (measured) data. &
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Degrees
degree of vertex v, deg(v) = number of lines with v as end-vertex; indegree of vertex v, indeg(v) = number of lines with v as terminal vertex (end-vertex is both initial and terminal); outdegree of vertex v, outdeg(v) = number of lines with v as initial vertex.
n = 12, m = 23, indeg(e) = 3, outdeg(e) = 5, deg(e) = 6 indeg(v) =
v∈V v∈V
outdeg(v) = |A| + 2|E|,
v∈V
deg(v) = 2|L| − |E0 |
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. . . Statistics
The global computed properties are reported by Pajek’s commands or can be seen using the Info option. In repetitive commands they are stored in vectors. The local properties are computed by Pajek’s commands and stored in vectors or partitions. To get information about their distribution use the Info option. As an example, let us look at The Edinburgh Associative Thesaurus network. The EAT is a network of word association as collected from subjects (students). The weight on the arcs is the count of word associations.
File/Network/Read eatRS.net Info/Network/General
It has 23219 vertices and 325624 arcs (564 loops); number of lines with value=1 is 227481. &
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. . . Statistics
To identify the vertices with the largest degree:
Net/Partitions/Degree/All Partition/Make vector Info/Vector +10
The largest degrees have the vertices:
1 2 3 4 5 6 7 8 9 10 vertex 12720 12459 8878 18122 13793 13181 23136 15080 13948 22973 deg 1108 1074 878 875 803 799 732 723 720 716 label ME MAN GOOD SEX NO MONEY YES PEOPLE NOTHING WORK
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Statistics / Pajek and R
Pajek (0.89 and higher) supports interaction with statistical program R and the use of other external programs as tools (menu Tools). In Pajek we determine the degrees of vertices and submit them to R
info/network/general Net/Partitions/Degree/All Partition/Make Vector Tools/Program R/Send to R/Current Vector
In R we determine their distribution and plot it
summary(v2) t <- tabulate(v2) c <- t[t>0] i <- (1:length(t))[t>0] plot(i,c,log=’xy’,main=’degree distribution’, xlab=’deg’,ylab=’freq’)
The obtained picture can be saved with File/Save as in selected format A (PDF or PS for LTEX; Windows metafile format for inclusion in Word). Attention! The vertices of degree 0 are not considered by tabulate. & Methodenforum der Fakult¨ t f¨ r Sozialwissenschaften, Universit¨ t Wien, 21-22. 6. 2007 a u a S G S
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EAT all-degree distribution
EAT all−degree distribution
5000
q q q q q q q q q qq qqq q qqqq qqq qqqq q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q qqq q q q q q q q q q qq q qq q q qq q q qq qq q q qq qq qq q q q q q q qq q q q qqq q q q qq q q qq q q q qqq q qq q q q q qq qq q q q qq qq qq q qq q q qq q q qq q q q q qqqq q q q qqqq q qqq qq q qqqqqqqqq qqqqqq qq qq qqqq q q qq q q q q
q
freq
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50
500
q qqqqqqqq qqqq q q q qqqqqqqqq qqq q qqqqqq q q q q qqqq q q q q qq q q
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Degree distribution
Random graph degree distribution, n=100000, degav=30
qq q q q
US Patents degree distribution
1 e+06
q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q qq qq qq q q q q qq q q q qq q q q q q q qq qq qq q q qq qq qqq q q q qq q qq qq q qq qqq qq qq qqq q q qq q q qq qq q qq q q q qq q q qq qqqqq qq q q qqqq q qq q q q qq q q qq q qqq q q qq q q q qqqq q qqqq qqq qq q qqqq qq qq qqqq qq qq qqqq q qq q q
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q q q
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q q q q q q q qqq qqqq q q
2000
q q q q q q qqqq q q
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qqq qqq q q qqq q qqq qqq q q qqq qqqqqq q q qq q qq q q q q q
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Real-life networks are usually not random in the Erd˝ s/Renyi sense. The o analysis of their distributions gave a new view about their structure – Watts (Small worlds), Barab´ si (nd/networks, Linked). a &
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Clusters, clusterings, partitions, hierarchies
A nonempty subset C ⊆ V is called a cluster (group). A nonempty set of clusters C = {Ci } forms a clustering. Clustering C = {Ci } is a partition iff ∪C =
i
Ci = V
in i = j ⇒ Ci ∩ Cj = ∅
Clustering C = {Ci } is a hierarchy iff Ci ∩ Cj ∈ {∅, Ci , Cj } Hierarchy C = {Ci } is complete, iff ∪C = V; and is basic if for all v ∈ ∪C also {v} ∈ C. &
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Representations of properties
Properties of vertices P and lines W can be measured in different scales: numerical, ordinal and nominal. They can be input as data or computed from the network. In Pajek numerical properties of vertices are represented by vectors, nominal properties by partitions or as labels of vertices. Numerical property can be displayed as size (width and height) of vertex (figure), as its coordinate; and a nominal property as color or shape of the figure, or as a vertex label (content, size and color). We can assign in Pajek numerical values to links. They can be displayed as value, thickness or grey level. Nominal vales can be assigned as label, color or line pattern (see Pajek manual, section 4.3). &
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Some comments
While the technical graph drawing problems could ask for a single ’the best’ picture, the social network analysis is a part of data analysis. Its goal is to get insight into the structure and characteristics of a given network, but also how it influences related social processes. What is the goal: exploration of network data (layout editor), presentation of the obtained results (layout viewer), . . . ? What is a GD result: picture or layout ? What is the medium of the result: static picture on ’paper’, interactive layout, . . . ? What kind of user will use the result: simple, advanced, . . . ?
Most methods are ’paper’ oriented. In larger/denser networks there is often too much information to be presented at once. A possible answer are interactive layouts where the user controls what (s)he wants to see. & % Methodenforum der Fakult¨ t f¨ r Sozialwissenschaften, Universit¨ t Wien, 21-22. 6. 2007 a u a S G S
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James Moody: Display of properties – school
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Lothar Krempel
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FAS: The scientific field of Austria
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Katy B¨ rner: Text analysis o
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Jeffrey Johnson: St Marks food web
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Big picture, V. Batagelj, AE’04
subnetwork (n = 5952, m = 18008) of the symmetrized Edinburgh Associative Thesaurus
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Big picture
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several washington
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lung cancer images
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bag
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of fact
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petrol
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canvas
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cookery
about
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novels
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text
author
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mass production yard letter-box seatbelt brakes oil wheel barrow
car
jeep
cubs
land
brake bikes
clothe worst
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pink
dairy
asked
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idiots
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custard
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chick
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galore
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snail remaining
pack
jury rapid
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farm
van
doctor various
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measure
hoist
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frying
searching
refused
natural
contemplating scheme
loneliness motion
demonstrate busby murder
homicide
recorder forget
recollect
obligatory would
glee
long
found
hoar
frost
fornication mating
dumb
infant
brat
fortunate
ten
archaeology
chap
elizabeth
majesty
de gaulle
peg
chrysalis
reagents
honours
celebrated
expensive
colder
hotter
lather
detergent
grand
bachelor
climate
purchase buy
soiled garbage
bin vast rubbish dump
outer
toilet
anonymous tim
infamous rocky
tax
spree
the
b
unpleasantness indefinite article
looking
unnatural
pity
thinking
thoughts
incomplete movement
variety
matt
societies
forgive
remember
doesn’t womb
could
maximum
credible
bruises
maze
discovered
lost
unhappiness bastard
forces
increase
reproduce maniac everlasting
hare deaf spaghetti
laces borne bolognese shoes
playground
offspring
cherubs epoch
seven
18
children
ears
wear
crown coronet
world chart
clothes-line
corridorsdoer
pregnant
props
exceptionally weeks excellent
blues
lotus
degree wash
pakistani
warmer
jailor
spinster
clink drug
featuring cabbage-white
butterfly
flutterby
duckling
prosperity sell blows
perhaps
squash stations possibly trains
steady
ready
y
sits
essence vanilla
a
sequence
pieces bits
series
shame meditation
unfinished
perpetual
sufficient enough suffice
suggestion memory
romantic so
quo
gladness
month year torn
strange
even odd
dejected
labyrinth chemical mend
paradise
illegitimate
fray
annihilate
hearing-aid ager
drag
glare
angels
eight
19 20
kids
twice
once
atlas
geometry
goes
polar bear
greet
greens shape
madam
devil
trinity park
coral
corral
reef sentence
quinine
starring
ugly
elusive
he
foreigners
lavatory
probably sister
maybe
unknown
known
ridiculousstands vigour
income senseless
contrabandsmuggle spending highest
shrews
sues
disgrace
oral
aah
stanza
adequate ooh
hence
therefore
recall positive
status
longing
delight
forgotten homosexual
neuter
illegitimacy
stoat satisfy
weasel
healer
eternal
fierce
socks
boots
forlorn
lids admit
envy
ninety
nine
charles charming
panorama view
upon a time
clothes
fashion
map
group
comes
banner
fashioned helper sir storage mainland pores
apex contours
satan
particle
tit
nice
pauper
phrase beautiful
grotesque gravity
she
elopement recital
nonsense affluence
brother
wogs flatten
railways coons trivial
per cent
futile
useless immaterial
lowest
harold
wilson
sadism
masochism
contraceptive
verse
poem poetry
poet elect
thus
wordsworth plenty
negative
couldn’t
release let go
1968
peculiar
queer unbound
repair
grieving
kinky
jonathan
goliath
march
teen
ferocious
sneak
vivid
heels
confess greyhounds
avarice
greed grow
prince
panoramic
garments
garment hinder
broke
geography monarchy
washing wed
jutland
shade
succour
sun-tan
prison
methedrine
pleasant lovely
metric
hiss
solar pretty
pilgrim
gosh golly
stuff
ben bonny faeces clyde
unimportant
incognito
strength fortitude
irrelevant grandpa consist
taming
rules
statesman politician confirm
finite
prose transmit
ah
opinion
bountiful
declare
nursery
phallus
than regret
saliva
running
1969
gratify
liberal thanks
expect
erotica
faire
david
protest optimism iniquity
hope
honest district
gloom cities
calcium
modern
caress
germanium poisonous
breathtaking
togs quarantine
royal
pegs lessons
penniless
emblem
flag
history
intentions jewels
junction graft
form skin
jazz
elan
captive
tyrant
cell
jail
larger inmate
unmarried
newton system
progressluck
misuse export boo chance treble
abuse
agitate
trip
excursion
sanction
weakness
unready unprepared totally
grandma
regulations
consideration infinite
aft
fore
vote unemployed
poll
rhyme
rhymes oppose
symbol
weird
spit opposite
loth
walking
unusual please goodness
tacks
lust received
savoir
carried
brows
faith
towns
maturity
mid-way pretender
venetian
nursery school merry
snake
holy ghost twenty
vista
garb
flagpole tempo months
union jack mode
laundry molehill
dermis
lecture
lotion
prisoner warder gaol
bigger
ski
improbably
articles
things objects
insignificant
muscle
horrible
stir
twin
outing
rhodesia
shirts
ties
dogfish
catfish
comprise
hoop assume presume la
nosy parker
ablative
dative
beggars
positively
customs
temper
ripped
apology
sorry
yearn
visible
nisi suburb
thank you merci
thankful grateful
metres
pessimism
heeled quaint
sin
urban
wild
schoolroom rattle
enquiries
reptile individual
princess coon
locus
wardrobe
regal
tricolour
hexagonal
answers
questions
laundrette epidermis hemp
terribly pensions
give out convict
tip sputnik
violins stocking neath valley
unlikely
size vale
attractive
leaps
possibility
opportunity
knocks
wholesaledeltoid
terrible
awful
thoroughly
completely
follower disciple
choosers
dole
negativelyexcise
surmise
a daisy
oops
aches
woo
forgot
remembered
gracious
turkish
anticipate
masturbate
yards
signpost
wish
glittering title
gleam
reason
invisible
stroke
frank
family
tickle
folks
shooters
patriotism octopus
narcissus
method mountaineering
marry
chest
drawer
power
horrific sari
quadruped harpsichord penthouse seventh fiasco suite
prisoners
hostage
saves
glen
giant
import
minority
majority
pains
rather
contrary corrosion rust
personalstore
private satiate warehouse
carbonate
unforgivable
motive sparkle
tame
planning
shimmer
thrill
long time no see
lizard
polite well mannered
person
wog
prayer
welcome virgin
reports
mushroomssnooker gleams
shines
fiendish
sense
sixth
satellite whip imprisonment lash
department
excitement
foreigner stranger
toadstools
Pajek
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Temporal networks / September 11
Steve Corman with collaborators from Arizona State University transformed, using his Centering Resonance Analysis (CRA), daily Reuters news (66 days) about September 11th into a temporal network of words coappearance. This network was a challange network for Viszards 2002 session.
Pictures in SVG: 66 days. (SVG viwer)
Every year (from 2002) we have at the Sunbelt conference a Viszards session presenting solutions of Viszards group to a visualizations of selected network or type of networks. &
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Example: Snyder and Kick World Trade
The data are available as a Pajek’s project file SaKtrade.paj The network consists of trade relations (118 vertices, 515 arcs, 2116 edges). The source of the data is the paper: Snyder, David and Edward Kick (1979). The World System and World Trade: An Empirical Exploration of Conceptual Conflicts, Sociological Quaterly, 20,1, 23-36. The project file contains also the (sub)continents partition: 1 - Europe, 2 - North America, 3 - Latin America, 4 - South America, 5 Asia, 6 - Africa, 7 - Oceania.
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Draw / Partition
kod yem alb par lib gua syr saf uru vnr tai arg ins mex gre isr pol usr bul liy jor tha nze tur chd maa els con uga tog nir kod cam gha afg sen ecu ivo jam nir bur vnd car nep rwa rwa nau dah gab gui hai sri rum gab mli upv yem chd car mat nic bol zai tri nau hai hon kmr sie upv mon som mli afg ice syr tun cub cyp liy rum leb bul tur sud lib kuw uru usr cze ege hun sie sri saupol gre pak irq per egy yug chi ghamor alg irn ire braarg cha coltri por fin aus ind ivo can nig bol den mex swe aut swi mla spanor brm ita saf nze uga gua ecu fra net wge isr ven kmr ken uki usa lux bel pan jap tai sen dom phi tha mat eth jor maa cam con ins vnr zai kor cos par els jam hon nic alb
lux spa nor usa net fra jap cze eth can ita col bur ege swe gui indpor nep uki vnd bra wge sau pak den chi cha yug bel swi phi dom irn egy cyp aut aus alg ire leb hun irq fin per nig ven kuw ice cub som sud brm mla tun cos ken pan tog dah lao
mor
kor
lao mon
Draw/Draw Partition Layout/Energy/Kamada-Kawai/Free Layout/Energy/Fruchterman Reingold/2D
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Zoom in
Using right button on the mouse select the zoom area. To restore the standard view select Redraw.
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Fruchterman Reingold / factor = 9
nic cos bol mon els hon hai nep lao rwa gui kod dah tog yem nau vnd chd upv car nir sie mat mli maa afg alb lib par ivo eth cha nig kmr sen uga yug cze egy usr leb bul rum syr liy som tun ege con ven aut cam brm lux usa bel jap uki net wge fra ita ind spa irn isr pol can swi den nor swe aus bra pak gre irq kuw tur sri hun alg cyp sud cub mor sau uru ire rwa gui kod dah tog yem nau vnd chd upv car nir sie mat mli maa afg alb lib par ivo eth cha nig kmr sen uga yug cze egy usr leb bul rum syr liy som tun ege con fin arg dom pan vnr zai jamkor gua ecu chi col phi tri per mla bur ins tha ken mex saf gab tai nze por ice els hon hai nep lao ven aut cam brm lux usa bel jap uki net wge fra ita ind spa irn isr pol can swi den nor swe aus bra pak gre irq kuw tur sri hun alg cyp sud cub mor sau uru ire fin arg cos bol mon nic dom pan vnr zai jamkor gua ecu chi col phi tri per mla bur ins tha ken tai mex saf gab nze por ice
gha
jor
gha
jor
Layout/Energy/Fruchterman Reingold/3D
3D picture / King &
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Matrix representation
Pajek - shadow [0.00,1.00]
usa can cub hai dom jam tri mex gua hon els nic cos pan col ven ecu per bra bol par chi arg uru uki ire net bel lux fra swi spa por wge ege pol aus hun cze ita mat alb yug gre cyp bul rum usr fin swe nor den ice mli sen dah nau nir ivo gui upv lib sie gha tog cam nig gab car chd con zai uga ken bur rwa som eth saf maa mor alg tun liy sud irn tur irq egy syr leb jor isr sau yem kuw afg cha mon tai kod kor jap ind pak brm sri nep tha kmr lao vnd vnr mla phi ins aut nze
Partition/Make Hierarchy [Yes][No] Hierarchy/Make Permutation File/Network/Export Matrix to EPS/Using Permutation [SaKmatrix.EPS][No] GsView: Media/User Defined ... [1300 pt][1300 pt]
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Methodenforum der Fakult¨ t f¨ r Sozialwissenschaften, Universit¨ t Wien, 21-22. 6. 2007 a u a
usa can cub hai dom jam tri mex gua hon els nic cos pan col ven ecu per bra bol par chi arg uru uki ire net bel lux fra swi spa por wge ege pol aus hun cze ita mat alb yug gre cyp bul rum usr fin swe nor den ice mli sen dah nau nir ivo gui upv lib sie gha tog cam nig gab car chd con zai uga ken bur rwa som eth saf maa mor alg tun liy sud irn tur irq egy syr leb jor isr sau yem kuw afg cha mon tai kod kor jap ind pak brm sri nep tha kmr lao vnd vnr mla phi ins aut nze
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Clustering
Better network matrix reorderings can be obtained using clustering:
Cluster/Create Complete Cluster [118] Operations/Dissimilarity*/d5 [1][SaKdendro.EPS] Hierarchy/Make Permutation [select network SaK.net] File/Network/Export Matrix to EPS/Using Permutation [SaKmatrix.EPS][No]
or blockmodeling. P. Doreian, V. Batagelj, A. Ferligoj: Generalized Blockmodeling, CUP, 2004. Amazon.
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Clustering
Pajek - Ward [0.00,135.13]
Pajek - shadow [0.00,1.00]
cyp ice tun cub liy mor alg nig uga ken eth brm tha sud sri gha gre tur egy bul rum syr leb irn irq pak ire aut hun isr sau kuw aus fin por bra arg pol cze usr ege yug ind cha uki fra wge jap net ita usa bel lux swe den swi can nor spa car chd nau tog dah nir gab sie con hai gui mat bol par cam maa yem kod lao mon nep bur rwa vnd som afg mli upv alb kmr jor kor vnr phi nze tai mla ins saf mex col uru per chi ven ecu lib dom zai jam tri pan sen ivo els cos gua hon nic
cyp ice tun cub liy mor alg nig uga ken eth brm tha sud sri gha gre tur egy bul rum syr leb irn irq pak ire aut hun isr sau kuw aus fin por bra arg pol cze usr ege yug ind cha uki fra wge jap net ita usa bel lux swe den swi can nor spa car chd nau tog dah nir gab sie con hai gui mat bol par cam maa yem kod lao mon nep bur rwa vnd som afg mli upv alb kmr jor kor vnr phi nze tai mla ins saf mex col uru per chi ven ecu lib dom zai jam tri pan sen ivo els cos gua hon nic
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Methodenforum der Fakult¨ t f¨ r Sozialwissenschaften, Universit¨ t Wien, 21-22. 6. 2007 a u a
cyp ice tun cub liy mor alg nig uga ken eth brm tha sud sri gha gre tur egy bul rum syr leb irn irq pak ire aut hun isr sau kuw aus fin por bra arg pol cze usr ege yug ind cha uki fra wge jap net ita usa bel lux swe den swi can nor spa car chd nau tog dah nir gab sie con hai gui mat bol par cam maa yem kod lao mon nep bur rwa vnd som afg mli upv alb kmr jor kor vnr phi nze tai mla ins saf mex col uru per chi ven ecu lib dom zai jam tri pan sen ivo els cos gua hon nic
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Reordering clustering
The order of clusters in a hierarchy is not fixed and can be changed.
Pajek - shadow [0.00,1.00]
uki fra wge jap net ita usa bel lux swe den swi can nor spa irn irq pak ire aut hun isr sau kuw aus fin por bra arg pol cze usr ege yug ind cha gre tur egy bul rum syr leb cyp ice tun cub liy mor alg nig uga ken eth brm tha sud sri gha kor vnr phi nze tai mla ins saf mex col uru per chi ven ecu lib dom zai jam tri pan sen ivo els cos gua hon nic car chd nau tog dah nir gab sie con hai gui mat bol par cam maa yem kod lao mon nep bur rwa vnd som afg mli upv alb kmr jor
We see the typical center–periphery structure. &
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uki fra wge jap net ita usa bel lux swe den swi can nor spa irn irq pak ire aut hun isr sau kuw aus fin por bra arg pol cze usr ege yug ind cha gre tur egy bul rum syr leb cyp ice tun cub liy mor alg nig uga ken eth brm tha sud sri gha kor vnr phi nze tai mla ins saf mex col uru per chi ven ecu lib dom zai jam tri pan sen ivo els cos gua hon nic car chd nau tog dah nir gab sie con hai gui mat bol par cam maa yem kod lao mon nep bur rwa vnd som afg mli upv alb kmr jor
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Contraction of cluster
Contraction of cluster C is called a graph G/C, in which all vertices of the cluster C are replaced by a single vertex, say c. More precisely: G/C = (V , L ), where V = (V \ C) ∪ {c} and L consists of lines from L that have both end-vertices in V \ C. Beside these it contains also a ’star’ with the center c and: arc (v, c), if ∃p ∈ L, u ∈ C : p(v, u); or arc (c, v), if ∃p ∈ L, u ∈ C : p(u, v). There is a loop (c, c) in c if ∃p ∈ L, u, v ∈ C : p(u, v). In a network over graph G we have also to specify how are determined the values/weights in the shrunk part of the network. Usually as the sum or maksimum/minimum of the original values.
Operations/Shrink Network/Partition
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Pajek - shadow [0.00,1.00]
usa can cub hai dom jam tri mex gua hon els nic cos pan col ven ecu per bra bol par chi arg uru uki ire net bel lux fra swi spa por wge ege pol aus hun cze ita mat alb yug gre cyp bul rum usr fin swe nor den ice mli sen dah nau nir ivo gui upv lib sie gha tog cam nig gab car chd con zai uga ken bur rwa som eth saf maa mor alg tun liy sud egy irn tur irq syr leb jor isr sau yem kuw afg cha mon tai kod kor jap ind pak brm sri nep tha kmr lao vnd vnr mla phi ins aut nze
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Contracted clusters – international trade
S. America
Europe
Africa
Asia N. America
Australia
usa can cub hai dom jam tri mex gua hon els nic cos pan col ven ecu per bra bol par chi arg uru uki ire net bel lux fra swi spa por wge ege pol aus hun cze ita mat alb yug gre cyp bul rum usr fin swe nor den ice mli sen dah nau nir ivo gui upv lib sie gha tog cam nig gab car chd con zai uga ken bur rwa som eth saf maa mor alg tun liy sud egy irn tur irq syr leb jor isr sau yem kuw afg cha mon tai kod kor jap ind pak brm sri nep tha kmr lao vnd vnr mla phi ins aut nze
L. America
Snyder and Kick’s international trade. Matrix display of dense networks. w(Ci , Cj ) = n(Ci , Cj ) n(Ci ) · n(Cj )
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Computing the weights
File / Pajek Project File / Read [SaKtrade.paj] Net / Transform / Remove / Loops [No] Net / Transform / Edges -> Arcs [No] Operations / Shrink Network / Partition [1][0] 1 2 3 4 5 6 7 --------------------------------------------#usa 1. 2 30 13 56 42 45 4 #cub 2. 30 74 25 196 20 37 12 #per 3. 12 28 33 124 16 36 5 #uki 4. 55 217 130 695 427 483 41 #mli 5. 42 8 14 406 122 117 11 #irn 6. 43 37 43 444 142 307 30 #aut 7. 4 4 5 39 9 30 2 Partition / Make Permutation [select partition (Sub)continents] Operations / Functional Composition / Partition*Permutation Partition / Count Partition / Make Vector Operations / Vector / Put loops
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. . . Computing the weights
1 2 3 4 5 6 7 -------------------------------------------#usa 1. 4 30 13 56 42 45 4 #cub 2. 30 89 25 196 20 37 12 #per 3. 12 28 40 124 16 36 5 #uki 4. 55 217 130 723 427 483 41 #mli 5. 42 8 14 406 155 117 11 #irn 6. 43 37 43 444 142 337 30 #aut 7. 4 4 5 39 9 30 4 count 2 15 7 29 33 30 2 Vector / Create Identity Vector [7] [select as second vector From partition ...] Vectors / Divide First by Second Operations / Vector / Vector # Network / input Operations / Vector / Vector # Network / output [edit partition - rename vertices] 1 2 3 4 5 6 7 --------------------------------------------N.Am 1. 1.00 1.00 0.93 0.97 0.64 0.75 1.00 L.Am 2. 1.00 0.40 0.24 0.45 0.04 0.08 0.40 S.Am 3. 0.86 0.27 0.82 0.61 0.07 0.17 0.36 Euro 4. 0.95 0.50 0.64 0.86 0.45 0.56 0.71 Afri 5. 0.64 0.02 0.06 0.42 0.14 0.12 0.17 Asia 6. 0.72 0.08 0.20 0.51 0.14 0.37 0.50 Ocea 7. 1.00 0.13 0.36 0.67 0.14 0.50 1.00
In Pajek sequences of commands can be combined into a macro command using
Macro / Record
and
Macro / Recording...
The macro can be activated by
Macro / Play
The sequence for computing the weights w(Ci , Cj ) is saved in the macro weights.
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Subgraph
A subgraph H = (V , L ) of a given graph G = (V, L) is a graph which set of lines is a subset of set of lines of G, L ⊆ L, its vertex set is a subset of set of vertices of G, V ⊆ V, and it contains all end-vertices of L . A subgraph can be induced by a given subset of vertices or lines. It is a spanning subgraph iff V = V.
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Cut-out – induced subgraph: Snyder and Kick – Africa
sie nir gab chd ivo upv gui mor mli sen nig gha lib saf liy egy tog alg zai sud car dah uga som eth ken nau tun con cam maa
rwa bur
Operations/Extract from Network/Partition [6]
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Cut-out: Snyder and Kick Latin America : South America
cos ecu gua els jam hon hai nic col mex pan ven dom tri cub uru bra chi arg par bol per
Operations/Extract from Network/Partition [3,4] Pajek Operations/Transform/Remove lines/Inside clusters [3,4]
The vertices can be manually put on a rectangular grid produced by
[Draw] Move/Grid
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Important vertices in network
It seems that the most important distinction between different vertex indices is based on the view/decision whether the network is considered directed or undirected. This gives us two main types of indices: • directed case: measures of importance; with two subgroups: measures of influence, based on out-going arcs; and measures of support, based on incoming arcs; • undirected case: measures of centrality, based on all lines. For undirected networks all three types of measures coincide. If we change the direction of all arcs (replace the relation with its inverse relation) the measure of influence becomes a measure of support, and vice versa. &
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. . . Important vertices in network
The real meaning of measure of importance depends on the relation described by a network. For example the most ’important’ person for the relation ’ doesn’t like to work with ’ is in fact the least popular person. Removal of an important vertex from a network produces a substantial change in structure/functioning of the network.
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Closeness
Most indices are based on the distance d(u, v) between vertices in a network N = (V, L). Two such indices are radius r(v) = maxu∈V d(v, u) S(v) =
u∈V
total closeness
d(v, u)
These two indices are measures of influence – to get measures of support we have to replace in definitions d(u, v) with d(v, u). If the network is not strongly connected rmax and Smax equal ∞. Sabidussi (1966) introduced a related measure 1/S(v); or in its normalized form closeness cl(v) = n−1 u∈V d(v, u)
D = maxu,v∈V d(v, u) is called the diameter of network. &
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Betweeness
Important are also the vertices that can control the information flow in the network. If we assume that this flow uses only the shortest paths (geodesics) we get a measure of betweeness (Anthonisse 1971, Freeman 1977) b(v) = 1 (n − 1)(n − 2) gu,t (v) gu,t
u,t∈V:gu,t >0 u=v,t=v,u=t
where gu,t is the number of geodesics from u to t; and gu,t (v) is the number of those among them that pass through vertex v. If we know matrices [du,v ] and [gu,v ] we can determine also gu,v (t) by: g ·g du,t + dt,v = du,v u,t t,v gu,v (t) = 0 otherwise For computation of geodesic matrix see Brandes. &
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Hubs and authorities
To each vertex v of a network N = (V, L) we assign two values: quality of its content (authority) xv and quality of its references (hub) yv . A good authority is selected by good hubs; and good hub points to good authorities (see Klienberg). xv =
u:(u,v)∈L
yu
and
yv =
u:(v,u)∈L
xu
Let W be a matrix of network N and x and y authority and hub vectors. Then we can write these two relations as x = WT y and y = Wx. We start with y = [1, 1, . . . , 1] and then compute new vectors x and y. After each step we normalize both vectors. We repeat this until they stabilize.
We can show that this procedure converges. The limit vector x∗ is the principal eigen vector of matrix WT W; and y∗ of matrix WWT . & % Methodenforum der Fakult¨ t f¨ r Sozialwissenschaften, Universit¨ t Wien, 21-22. 6. 2007 a u a S G S
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. . . Hubs and authorities
Similar procedures are used in search engines on the web to evaluate the importance of web pages. PageRank, PageRank / Google, HITS / AltaVista, SALSA, teorija. Examples: Krebs, Krempl.
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Clustering coefficient
Let G = (V, E) be simple undirected graph. Clustering in vertex v is usually measured as a quotient between the number of lines in subgraph G 1 (v) = G(N 1 (v)) induced by the neighbors of vertex v and the number of lines in the complete graph on these vertices: 2|L(G 1 (v))| deg(v) > 1 deg(v)(deg(v) − 1) C(v) = 0 otherwise We can consider also the size of vertex neighborhood by the following correction deg(v) C1 (v) = C(v) ∆
where ∆ is the maximum degree in graph G. This measure attains its largest value in vertices that belong to an isolated clique of size ∆. & % Methodenforum der Fakult¨ t f¨ r Sozialwissenschaften, Universit¨ t Wien, 21-22. 6. 2007 a u a S G S
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Dense groups
Several notions were proposed in attempts to formally describe dense groups in graphs. Clique of order k is a maximal complete subgraph (isomorphic to Kk ), k ≥ 3. s-plexes, LS sets, lambda sets, cores, . . . For all of them, except for cores, it turned out that they are difficult to detemine.
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Cores and generalized cores
The notion of core was introduced by Seidman in 1983. Let G = (V, E) be a graph. A subgraph H = (W, E|W ) induced by the set W is a k-core or a core of order k iff ∀v ∈ W : degH (v) ≥ k, and H is a maximal subgraph with this property. The core of maximum order is also called the main core. The core number of vertex v is the highest order of a core that contains this vertex. The degree deg(v) can be: in-degree, out-degree, in-degree + out-degree, etc., determining different types of cores. &
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Properties of cores
From the figure, representing 0, 1, 2 and 3 core, we can see the following properties of cores: • The cores are nested: i < j =⇒ Hj ⊆ Hi • Cores are not necessarily connected subgraphs. An efficient algorithm for determining the cores hierarchy is based on the following property: If from a given graph G = (V, E) we recursively delete all vertices, and edges incident with them, of degree less than k, the remaining graph is the k-core. For details see the paper. &
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6-core of Krebs Internet industries
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Generalized cores
The notion of core can be generalized to networks. Let N = (V, E, w) be a network, where G = (V, E) is a graph and w : E → R is a function assigning values to edges. A vertex property function on N, or a pfunction for short, is a function p(v, U ), v ∈ V, U ⊆ V with real values. Let NU (v) = N (v) ∩ U . Besides degrees, here are some examples of p-functions: pS (v, U ) =
u∈NU (v)
w(v, u), where w : E → R+ 0 max w(v, u), where w : E → R
pM (v, U ) = pk (v, U ) =
u∈NU (v)
number of cycles of length k through vertex v in (U, E|U )
The subgraph H = (C, E|C) induced by the set C ⊆ V is a p-core at level t ∈ R iff ∀v ∈ C : t ≤ p(v, C) and C is a maximal such set. &
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Generalized cores algorithm
The function p is monotone iff it has the property C1 ⊂ C2 ⇒ ∀v ∈ V : (p(v, C1 ) ≤ p(v, C2 )) The degrees and the functions pS , pM and pk are monotone. For a monotone function the p-core at level t can be determined, as in the ordinary case, by successively deleting vertices with value of p lower than t; and the cores on different levels are nested t1 < t2 ⇒ Ht2 ⊆ Ht1 The p-function is local iff p(v, U ) = p(v, NU (v)) . The degrees, pS and pM are local; but pk is not local for k ≥ 4. For a local p-function an O(m max(∆, log n)) algorithm for determining the p-core levels exists, assuming that p(v, NC (v)) can be computed in O(degC (v)). For details see the paper. &
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pS -core at level 46 of Geombib network
E.Arkin J.Mitchell M.Bern D.Eppstein M.Goodrich I.Tollis A.Garg L.Vismara
G.diBattista R.Tamassia G.Liotta
D.Dobkin
S.Suri
J.O’Rourke J.Vitter
J.Hershberger
B.Chazelle R.Seidel B.Aronov L.Guibas H.Edelsbrunner M.Sharir R.Pollack J.Pach E.Welzl J.Matousek C.Yap
F.Preparata J.Snoeyink P.Agarwal D.Halperin M.Smid M.Overmars M.vanKreveld M.deBerg O.Schwarzkopf G.Toussaint M.Teillaud P.Bose J.Boissonnat O.Devillers M.Yvinec J.Schwerdt P.Gupta R.Janardan J.Majhi
J.Urrutia
J.Czyzowicz
C.Icking R.Klein
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Cores and generalized cores / Pajek commands
File/Network/Read [Geom.net] Net/Partitions/Core/All Info/Partition Operations/Extract from Network/Partition [13-*] Draw/Draw-Partition Layout/Energy/Kamada-Kawai Options/Values of lines/Similarities Layout/Energy/Kamada-Kawai Operations/Extract from Network/Partition [21] Draw Layout/Energy/Kamada-Kawai Options/Values of lines/Forget Layout/Energy/Kamada-Kawai [select Geom.net] Net/Vector/PCore/Sum/All Info/Vector Vector/Make Partition/by Intervals/Selected Thresholds [45] Info/Partition Operations/Extract from Network/Partition [2] Draw Options/Values of lines/Similarities Layout/Energy/Fruchterman-Reingold
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Connectivity
Vertex u is reachable from vertex v iff there exists a walk with initial vertex v and terminal vertex u. Vertex v is weakly connected with vertex u iff there exists a semiwalk with v and u as its end-vertices. Vertex v is strongly connected with vertex u iff they are mutually reachable. Weak and strong connectivity are equivalence relations. Equivalence classes induce weak/strong components. &
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Weak components
Reordering the vertices of network such that the vertices from the same class of weak partition are put together we get a matrix representation consisting of diagonal blocks – weak components. Most problems can be solved separately on each component and afterward these solutions combined into final solution.
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Reduction (condensation)
If we shrink every strong component of a given graph into a vertex, delete all loops and identify parallel arcs the obtained reduced graph is acyclic. For every acyclic graph an ordering / level function i : V → N exists s.t. (u, v) ∈ A ⇒ i(u) < i(v). &
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Reduction – Example
Net / Components / Strong [1] Operations / Shrink Network / Partition [1][0] Net / Transform / Remove / Loops [yes] Net / Partitions / Depth / Acyclic Partition / Make Permutation Permutation / Inverse select partition [Strong Components] Operations / Functional Composition / Partition*Permutation Partition / Make Permutation select [original network] File / Network / Export Matrix to EPS / Using Permutation
Pajek - shadow [0.00,1.00]
b
k
i e h
c a d e g
j
f g
l a b c d g f
#a
f h
l
i k j
k e h a b d g c k
#e
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Cuts
The standard approach to find interesting groups inside a network was based on properties/weights – they can be measured or computed from network structure (for example Kleinberg’s hubs and authorities). The vertex-cut of a network N = (V, L, p), p : V → R, at selected level t is a subnetwork N(t) = (V , L(V ), p), determined by the set V = {v ∈ V : p(v) ≥ t} and L(V ) is the set of lines from L that have both endpoints in V . The line-cut of a network N = (V, L, w), w : V → R, at selected level t is a subnetwork N(t) = (V(L ), L , w), determined by the set L = {e ∈ L : w(e) ≥ t} and V(L ) is the set of all endpoints of the lines from L . &
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Vertex-cut: Krebs Internet Industries, core=6
Lucent CommerceOne
UPS
SAP Motorola Yahoo! EMC FoundryNetworks HP Sun Oracle EDS CIBER Ariba Peoplesoft divine WebMethods
IBM Siebel
E.piphany BMCSoftware
AT&T
Cisco Novell Intel ExodusComm KPNQwest CacheFlow Loudcloud Compaq RealNetworks Dell RedHat
Vignette
Akamai Inktomi AOL
MSFT
Pajek
Each vertex represents a company that competes in the Internet industry, 1998 do 2001. n = 219, m = 631. red – content, blue – infrastructure, green – commerce. Two companies are linked with an edge if they have announced a joint venture, strategic alliance or other partnership.
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Line-cut: Krebs Internet Industries, w3 ≥ 5
Motorola Oracle IBM HP Cisco AT&T KPNQwest Intel E.piphany CIBER Ariba FoundryNetworks Sun
KPMG Compaq RealNetworks Dell
Akamai Inktomi AOL
MSFT
TerraLycos
FoundryNetworks
RealNetworks
KPMG TerraLycos AOL AT&T Inktomi
Pajek
Compaq
Sun
HP
MSFT
Dell
Akamai IBM Intel Cisco KPNQwest Motorola
Oracle CIBER
Ariba
E.piphany
Pajek
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Cuts / Pajek commands
Vertex-cut:
File/Pajek Project File/Read [Krebs.paj] Net/Partitions/Core/All Partition/Make Vector Draw/Draw-Partition-Vector Layout/Energy/Kamada-Kawai Operations/Extract from Network/Partition [6] [select Types ... as First partition] [select All core ... as Second partition] Partitions/Extract Second from First [6] Draw/Draw-Partition Layout/Energy/Kamada-Kawai
Line-cut:
[select Krebs ... network] Net/Count/3-Rings/Undirected Info/Network/Line Values Net/Transform/Remove/Lines with Values/lower than [5] Net/Partitions/Degree/All Partition/Make Vector Operations/Extract from Network/Partition [1-*] [select Types ... as First partition] [select All Degree ... as Second partition] Partitions/Extract Second from First [1-*] Draw/Draw-Partition Layout/Energy/Kamada-Kawai
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Simple analysis using cuts
We look at the components of N(t). Their number and sizes depend on t. Usually there are many small components. Often we consider only components of size at least k and not exceeding K. The components of size smaller than k are discarded as ’noninteresting’; and the components of size larger than K are cut again at some higher level. The values of thresholds t, k and K are determined by inspecting the distribution of vertex/arc-values and the distribution of component sizes and considering additional knowledge on the nature of network or goals of analysis. We developed some new and efficiently computable properties/weights. &
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Citation weights
POGGIO-T-1975-V19-P201 KOHONEN-T-1976-V21-P85 KOHONEN-T-1976-V22-P159 ANDERSON-JA-1977-V84-P413 KOHONEN-T-1977-V2-P1065 PFAFFELHUBER-E-1975-V18-P217 AMARI-SI-1977-V26-P175
COOPER-LN-1979-V33-P9 WOOD-CC-1978-V85-P582
PALM-G-1980-V36-P19
BIENENSTOCK-EL-1982-V2-P32 SUTTON-RS-1981-V88-P135 HOPFIELD-JJ-1982-V79-P2554 AMARI-S-1980-V42-P339
ANDERSON-JA-1983-V13-P799
KOHONEN-T-1982-V43-P59
KNAPP-AG-1984-V10-P616
MCCLELLAND-JL-1985-V114-P159
CARPENTER-GA-1987-V37-P54
GROSSBERG-S-1987-V11-P23
HECHTNIELSEN-R-1987-V26-P1892
GROSSBERG-S-1988-V1-P17
HECHTNIELSEN-R-1987-V26-P4979
SEJNOWSKI-TJ-1988-V241-P1299
HECHTNIELSEN-R-1988-V1-P131
BROWN-TH-1988-V242-P724
KOHONEN-T-1990-V78-P1464
BROWN-TH-1990-V13-P475
TREVES-A-1991-V2-P371 HASSELMO-ME-1994-V7-P13 HASSELMO-ME-1993-V16-P218
HASSELMO-ME-1994-V14-P3898
BARKAI-E-1994-V72-P659
HASSELMO-ME-1995-V67-P1
HASSELMO-ME-1995-V15-P5249
The citation network analysis started in 1964 with the paper of Garfield et al. In 1989 Hummon and Doreian proposed three indices – weights of arcs that are proportional to the number of different source-sink paths passing through the arc. We developed algorithms to efficiently compute these indices. Main subnetwork (arc cut at level 0.007) of the SOM (selforganizing maps) citation network (4470 vertices, 12731 arcs). See paper.
Pajek
GLUCK-MA-1997-V48-P481
ASHBY-FG-1999-V6-P363
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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 a complete graph Kr , r ≥ k ≥ 3 we have wk (Kr ) = (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. For example: all r-cliques of a network belong to r−2edge cut for the weight w3 . We can assign to a given graph a triangular network in which every line of the original graph gets as its weight the number of triangles that contain it. The triangular weights provide us, combined with islands, with a very efficient way to identify dense parts of a graph. & % Methodenforum der Fakult¨ t f¨ r Sozialwissenschaften, Universit¨ t Wien, 21-22. 6. 2007 a u a S G S
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Triangular connectivity
Related to triangular network is the notion of triangular connectivity
that can be used to operationalize the notion of strong ties. These notions can be generalized to short cycle connectivity (see paper).
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Edge-cut at level 16 of triangular network of Erd˝ s o collaboration graph
WORMALD, NICHOLAS C. LASKAR, RENU C. SHELAH, SAHARON
MCKAY, BRENDAN D.
HEDETNIEMI, STEPHEN T.
MAGIDOR, MENACHEM
KLEITMAN, DANIEL J. CHUNG, FAN RONG K. SAKS, MICHAEL E. LINIAL, NATHAN FRANKL, PETER GRAHAM, RONALD L.
FUREDI, ZOLTAN
HENNING, MICHAEL A. SPENCER, JOEL H. ALON, NOGA OELLERMANN, ORTRUD R. LOVASZ, LASZLO KOMLOS, JANOS GODDARD, WAYNE D. ALAVI, YOUSEF CHARTRAND, GARY HARARY, FRANK KUBICKI, GRZEGORZ SCHWENK, ALLEN JOHN ROSA, ALEXANDER
ARONOV, BORIS PACH, JANOS POLLACK, RICHARD M.
TUZA, ZSOLT BABAI, LASZLO SZEMEREDI, ENDRE BOLLOBAS, BELA AJTAI, MIKLOS RODL, VOJTECH NESETRIL, JAROSLAV GYARFAS, ANDRAS SCHELP, RICHARD H. LEHEL, JENO CHEN, GUANTAO FAUDREE, RALPH J.
without Erd˝ s, o n = 6926, m = 11343
STINSON, DOUGLAS ROBERT MULLIN, RONALD C. COLBOURN, CHARLES J.
JACOBSON, MICHAEL S.
PHELPS, KEVIN T.
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Directed 3-rings
In directed networks there are two types of 3-rings:
cyclic
transitive
The 3-rings weights were implemented in Pajek in May 2002. &
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Edge-cut at level 11 of transitive network of ODLIS dictionary graph
serial American Library Directory transaction log suggestion box review charge series call number fixed location blanket order vendor entry library issue Library Literature journal publishing American Library Association /ALA/ Books in Print /BIP/ homepage round table librarian catalog bibliographic record condition Oak Knoll dust jacket half-title library binding table of contents /TOC/ invoice new book book size publisher binding cover text parts of a book front matter endpaper collation folio page book published price dummy plate imprint work bibliography editor title index copyright edition fiction abstract title page International Standard Book Number /ISBN/ printing colophon layout frequency publication periodical
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Islands
If we represent a given or computed value of vertices / lines as a height of vertices / lines and we immerse the network into a water up to selected level we get islands. Varying the level we get different islands. Islands are very general and efficient approach to determine the ’important’ subnetworks in a given network.
We developed very efficient algorithms to determine the islands hierarchy and to list all the islands of selected sizes. See details. &
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Islands - Reuters terror news
airline united_airlines american_airlines nuclear flight arabic-language train pilot power organization saudi chemical uniform scare inhale firefighter police anthrax skin phone arab cell help space plea edmund air cheyenne louisiana south base air_force pfc barksdale nebraska team exchange stock effort edward fire smoke plaugher worker large postal chief state emergency district death rescue aid financial thousand market center buildng toll offutt wyoming florida city mayor mayor_giuliani landmark conference american debris miss dept people late new_york specialist 110-story pentagon north tower attack world_trade_ctr the_worst act terrorist apparent morning anti-american terror group responsibility military trace jonn mighty twin case officer airliner jet fbi afghanistan man knife-wielding world commercial hijacker hijack agent official support country plane deadly week strike east thursday pakistan special war united_states terrorism force airport saudi-born bin_laden leader boston congressional pakistani herald plant attendant
weapon contain car manual passenger service member
newspaper
rental
suspect call
dissident
taliban
necessary
embassy business wednesday
headquarters action africa
washington
tuesday news
Using CRA S. Corman and K. Dooley produced the Reuters terror news network that is based on all stories released during 66 consecutive days by the news agency Reuters concerning the September 11 attack on the US. The vertices of a network are words (terms); there is an edge between two words iff they appear in the same text unit. The weight of an edge is its frequency. It has n = 13332 vertices and m = 243447 edges.
Pajek
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Islands – US patents
As an example, let us look at Nber network of US Patents. It has 3774768 vertices and 16522438 arcs (1 loop). We computed SPC weights in it and determined all (2,90)-islands. The reduced network has 470137 vertices, 307472 arcs and for different k: C2 =187610, C5 =8859,C30 =101, C50 =30 islands. Rolex
[1] 0 139793 [11] 190 125 [21] 17 16 [31] 12 3 [41] 1 3 [51] 2 3 [61] 0 0 [71] 0 0 [81] 2 0 29670 9288 3966 1827 997 578 362 250 104 71 47 37 36 33 21 23 8 7 13 10 10 5 5 5 7 3 3 3 2 6 6 2 4 1 5 2 1 1 1 1 3 2 0 0 0 0 0 1 0 0 1 0 0 2 0 0 1 1 0 0 0 1 0 0 0 0 0 1 2 0 0 7
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Island Theme size distribution size distribution
q
q
10000
q q q q
freq
q q q q
100
q
q q q q qq qq qq
q q
q q q
q q q q q q q q q q q qq
qq q
q qq q
qq
q qq
q q qq q q
qq q q qq
1 2
5
10 size
20
50
100
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Main path and main island of Patents
5855814 5683624 5543077
5124824 5171469 5283677 4456712 5555116
5374374
5016988 5122295 5016989
5308538
4957349 4877547 4820839 4832462
5171469 5122295 4877547 4797228
4630896 4659502 4621901 4770503 4795579 4797228 4752414 4721367
4709030
4719032
4695131
4710315
4713197
4704227
4657695
4710315
4583826 4510069 4558151 4550981
4659502
4526704 4502974 4514044
4550981 4526704 4472293 4422951
4415470 4400293 4386007 4460770 4480117 4472293 4472592 4455443
4419263
4422951
4386007
4349452 4368135 4340498
4340498
4387039 4387038 4293434 4290905 4361494 4302352 4330426 4357078
4302352 4229315 4149413 4082428
4154697 4113647 4130502 4032470 4083797 4082428 4118335 4195916 4229315 4261652
4202791
4149413
4198130
4011173
4013582 4029595 4077260 4017416 3975286 4011173 4000084
3960752
3954653 3947375 3960752 3876286
3872140 2544659 3675987
3881806 3731986 3795436
3891307 3697150 3636168 3767289 3666948 3322485 3773747 3796479 3795436 2682562 3691755
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Liquid crystal display
Table 1: Patents on the liquid-crystal display patent 2544659 2682562 3322485 3636168 3666948 3675987 3691755 3697150 3731986 3767289 3773747 3795436 3796479 3872140 3876286 3881806 3891307 3947375 3954653 3960752 3975286 4000084 4011173 4013582 4017416 4029595 4032470 4077260 4082428 date author(s) and title Mar 13, 1951 Dreyer. Dichroic light-polarizing sheet and the like and the formation and use thereof Jun 29, 1954 Wender, et al. Reduction of aromatic carbinols May 30, 1967 Williams. Electro-optical elements utilazing an organic nematic compound Jan 18, 1972 Josephson. Preparation of polynuclear aromatic compounds May 30, 1972 Mechlowitz, et al. Liquid crystal termal imaging system having an undisturbed image on a disturbed background Jul 11, 1972 Rafuse. Liquid crystal compositions and devices Sep 19, 1972 Girard. Clock with digital display Oct 10, 1972 Wysochi. Electro-optic systems in which an electrophoreticlike or dipolar material is dispersed throughout a liquid crystal to reduce the turn-off time May 8, 1973 Fergason. Display devices utilizing liquid crystal light modulation Oct 23, 1973 Aviram, et al. Class of stable trans-stilbene compounds, some displaying nematic mesophases at or near room temperature and others in a range up to 100◦ C Nov 20, 1973 Steinstrasser. Substituted azoxy benzene compounds Mar 5, 1974 Boller, et al. Nematogenic material which exhibit the Kerr effect at isotropic temperatures Mar 12, 1974 Helfrich, et al. Electro-optical light-modulation cell utilizing a nematogenic material which exhibits the Kerr effect at isotropic temperatures Mar 18, 1975 Klanderman, et al. Liquid crystalline compositions and method Apr 8, 1975 Deutscher, et al. Use of nematic liquid crystalline substances May 6, 1975 Suzuki. Electro-optical display device Jun 24, 1975 Tsukamoto, et al. Phase control of the voltages applied to opposite electrodes for a cholesteric to nematic phase transition display Mar 30, 1976 Gray, et al. Liquid crystal materials and devices May 4, 1976 Yamazaki. Liquid crystal composition having high dielectric anisotropy and display device incorporating same Jun 1, 1976 Klanderman, et al. Liquid crystal compositions Aug 17, 1976 Oh. Low voltage actuated field effect liquid crystals compositions and method of synthesis Dec 28, 1976 Hsieh, et al. Liquid crystal mixtures for electro-optical display devices Mar 8, 1977 Steinstrasser. Modified nematic mixtures with positive dielectric anisotropy Mar 22, 1977 Gavrilovic. Liquid crystal compounds and electro-optic devices incorporating them Apr 12, 1977 Inukai, et al. P-cyanophenyl 4-alkyl-4’-biphenylcarboxylate, method for preparing same and liquid crystal compositions using same Jun 14, 1977 Ross, et al. Novel liquid crystal compounds and electro-optic devices incorporating them Jun 28, 1977 Bloom, et al. Electro-optic device Mar 7, 1978 Gray, et al. Optically active cyano-biphenyl compounds and liquid crystal materials containing them Apr 4, 1978 Hsu. Liquid crystal composition and method patent 4083797 4113647 4118335 4130502 4149413 Table 2: Patents on the liquid-crystal display date 1978 1978 1978 1978 1979 author(s) and title Oh. Nematic liquid crystal compositions Coates, et al. Liquid crystalline materials Krause, et al. Liquid crystalline materials of reduced viscosity Eidenschink, et al. Liquid crystalline cyclohexane derivatives Gray, et al. Optically active liquid crystal mixtures and liquid crystal devices containing them Eidenschink, et al. Liquid crystalline hexahydroterphenyl derivatives Coates, et al. Liquid crystal compounds Boller, et al. Liquid crystal mixtures Sato, et al. Nematic liquid crystalline materials Krause, et al. Liquid crystalline cyclohexane derivatives Gray, et al. Liquid crystal compounds and materials and devices containing them Kanbe. Ester compound Deutscher, et al. Liquid crystal compounds Eidenschink, et al. Fluorophenylcyclohexanes, the preparation thereof and their use as components of liquid crystal dielectrics Eidenschink, et al. Cyclohexylbiphenyls, their preparation and use in dielectrics and electrooptical display elements Sugimori. Halogenated ester derivatives Osman, et al. Cyclohexylcyclohexanoates Carr, et al. Liquid crystal compounds containing an alicyclic ring and exhibiting a low dielectric anisotropy and liquid crystal materials and devices incorporating such compounds Osman, et al. Anisotropic cyclohexyl cyclohexylmethyl ethers Osman. Anisotropic compounds with negative or positive DC-anisotropy and low optical anisotropy Krause, et al. Liquid crystalline naphthalene derivatives Fukui, et al. 4-(Trans-4’-alkylcyclohexyl) benzoic acid 4’”-cyano-4”-biphenylyl esters Sugimori, et al. Trans-4-(trans-4’-alkylcyclohexyl)-cyclohexane carboxylic acid 4’”-cyanobiphenyl ester Romer, et al. Liquid crystalline cyclohexylphenyl derivatives Eidenschink, et al. Liquid crystalline fluorine-containing cyclohexylbiphenyls and dielectrics and electro-optical display elements based thereon Praefcke, et al. Liquid crystalline cyclohexylcarbonitrile derivatives Sugimori, et al. Liquid crystal benzene derivatives Takatsu, et al. Nematic halogen Compound Christie, et al. Bismaleimide triazine composition Petrzilka, et al. Liquid crystal mixture Sugimori, et al. High temperature liquid crystal substances of four rings and liquid crystal compositions containing the same Takatsu, et al. Nematic liquid crystalline compounds Takatsu, et al. Nematic liquid crystalline compounds Sugimori, et al. High temperature liquid-crystalline ester compounds Eidenschink, et al. Cyclohexane derivatives patent 4514044 4526704 4550981 4558151 4583826 4621901 4630896 4657695 4659502 4695131 4704227 4709030 4710315 4713197 4719032 4721367 4752414 4770503 4795579 4797228 4820839 4832462 4877547 4957349 5016988 5016989 5122295 5124824 5171469 5283677 5308538 5374374 5543077 5555116 5683624 5855814 Table 3: Patents on the liquid-crystal display date author(s) and title Apr 30, 1985 Gunjima, et al. 1-(Trans-4-alkylcyclohexyl)-2-(trans-4’-(p-sub stituted phenyl) cyclohexyl)ethane and liquid crystal mixture Jul 2, 1985 Petrzilka, et al. Multiring liquid crystal esters Nov 5, 1985 Petrzilka, et al. Liquid crystalline esters and mixtures Dec 10, 1985 Takatsu, et al. Nematic liquid crystalline compounds Apr 22, 1986 Petrzilka, et al. Phenylethanes Nov 11, 1986 Petrzilka, et al. Novel liquid crystal mixtures Dec 23, 1986 Petrzilka, et al. Benzonitriles Apr 14, 1987 Saito, et al. Substituted pyridazines Apr 21, 1987 Fearon, et al. Ethane derivatives Sep 22, 1987 Balkwill, et al. Disubstituted ethanes and their use in liquid crystal materials and devices Nov 3, 1987 Krause, et al. Liquid crystal compounds Nov 24, 1987 Petrzilka, et al. Novel liquid crystal mixtures Dec 1, 1987 Schad, et al. Anisotropic compounds and liquid crystal mixtures therewith Dec 15, 1987 Eidenschink, et al. Nitrogen-containing heterocyclic compounds Jan 12, 1988 Wachtler, et al. Cyclohexane derivatives Jan 26, 1988 Yoshinaga, et al. Liquid crystal device Jun 21, 1988 Eidenschink, et al. Nitrogen-containing heterocyclic compounds Sep 13, 1988 Buchecker, et al. Liquid crystalline compounds Jan 3, 1989 Vauchier, et al. 2,2’-difluoro-4-alkoxy-4’-hydroxydiphenyls and their derivatives, their production process and their use in liquid crystal display devices Jan 10, 1989 Goto, et al. Cyclohexane derivative and liquid crystal composition containing same Apr 11, 1989 Krause, et al. Nitrogen-containing heterocyclic esters May 23, 1989 Clark, et al. Liquid crystal devices Oct 31, 1989 Weber, et al. Liquid crystal display element Sep 18, 1990 Clerc, et al. Active matrix screen for the color display of television pictures, control system and process for producing said screen May 21, 1991 Iimura. Liquid crystal display device with a birefringent compensator May 21, 1991 Okada. Liquid crystal element with improved contrast and brightness Jun 16, 1992 Weber, et al. Matrix liquid crystal display Jun 23, 1992 Kozaki, et al. Liquid crystal display device comprising a retardation compensation layer having a maximum principal refractive index in the thickness direction Dec 15, 1992 Hittich, et al. Liquid-crystal matrix display Feb 1, 1994 Sagawa, et al. Liquid crystal display with ground regions between terminal groups May 3, 1994 Weber, et al. Supertwist liquid-crystal display Dec 20, 1994 Weber, et al. Supertwist liquid-crystal display Aug 6, 1996 Rieger, et al. Nematic liquid-crystal composition Sep 10, 1996 Ishikawa, et al. Liquid crystal display having adjacent electrode terminals set equal in length Nov 4, 1997 Sekiguchi, et al. Liquid crystal composition Jan 5, 1999 Matsui, et al. Liquid crystal compositions and liquid crystal display elements Apr 11, Sep 12, Oct 3, Dec 19, Apr 17,
4154697 May 15, 1979 4195916 Apr 1, 1980 4198130 Apr 15, 1980 4202791 May 13, 1980 4229315 Oct 21, 1980 4261652 Apr 14, 1981 4290905 4293434 4302352 Sep 22, 1981 Oct 6, 1981 Nov 24, 1981
4330426 May 18, 1982 4340498 4349452 4357078 4361494 4368135 Jul 20, 1982 Sep 14, 1982 Nov 2, 1982 Nov 30, 1982 Jan 11, 1983
4386007 May 31, 1983 4387038 Jun 7, 1983 4387039 4400293 4415470 4419263 4422951 4455443 4456712 4460770 4472293 4472592 4480117 4502974 4510069 Jun 7, 1983 Aug 23, 1983 Nov 15, 1983 Dec 6, 1983 Dec Jun Jun Jul Sep 27, 19, 26, 17, 18, 1983 1984 1984 1984 1984
Sep 18, 1984 Oct 30, 1984 Mar 5, 1985 Apr 9, 1985
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Islands – The Edinburgh Associative Thesaurus
n = 23219, m = 325624, transitivity weight
HAPPEN JUST YET PAYMENT PAID PAY COINS HAPPENED NEVERTHELESS NOW INCREASE MONEY UNPAID THRIFTY DEALER LOOT DEFICIT THRIFT OFTEN BUT WHY AS SOON THEREFORE ANYWAY PROBABLY NOT NO PLEASE MORE MONEY OFFER PROVIDE REPAY PROPERTY RECEIPT STOLE
NOTWITHSTANDING
MONIES AGAIN ALREADY
BELIEVE
REFUSE
MEANWHILE
SOMETIME
COULD
POWERFUL INHUMAN LECTURER STUDYING RESPONSIBLE TEACHER ENGINEERING TEACHING SCHOOL EDUCATION HOMEWORK WORK TRAINING LECTURES SCIENCE STUDY RESEARCH ACTIVITY MATHS KINDNESS ADORABLE LESSONS LOVE GIRL ATTRACTIVE DELIGHTFUL FLIRT BEAUTIFUL LOVELY MYSTERIOUS BELOVED CHARM NICE PROFESSION MAN ELEGANCE HAIRY BOSS CHAIRMAN
LEARNING
SCIENTIFIC
SHAPELY
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Islands / Pajek commands
File/Network/Read [eatRS.net] Net/Partitions/Islands/Generate Network with Islands [On] Net/Partitions/Islands/Line Weights Simple [2 50] Partition/Canonical Partition - Decreasing Frequencies Info/Partition Operations/Extract from Network/Partition [1-38] Draw/Draw-Partition-Vector Layout/Energy/Kamada-Kawai/Free [manually distribute components over the available space] Options/Transform/Fit area
The procedure for ’triangular islands’ is similar
File/Network/Read [eatRS.net] Net/Count/3-Rings/Directed/Transitive Net/Partitions/Islands/Generate Network with Islands [On] Net/Partitions/Islands/Line Weights Simple [2 50] ...
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Internet Movie Database http://www.imdb.com/
12th Annual Graph Drawing Contest, 2005. The IMDB network is bipartite (2-mode) and has
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1324748 = 428440 + 896308 vertices and 3792390 arcs.
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Bipartite cores
The subset of vertices C ⊆ V is a (p, q)-core in a bipartite (2-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 bipartite 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 bipartite 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 • big jumps from C(p − 1, q) and C(p, q − 1) to C(p, q). &
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Table (p, q : n1 , n2 ) for Internet Movie Database
1 1590: 1590 1 | 22 24: 1854 1153 | 43 14: 29 83 2 516: 788 3 | 23 23: 47 56 | 44 14: 29 83 3 212: 1705 18 | 24 23: 34 39 | 45 13: 30 95 4 151: 4330 154 | 25 22: 42 53 | 46 13: 29 94 5 131: 4282 209 | 26 22: 31 38 | 47 12: 29 101 6 115: 3635 223 | 27 22: 31 38 | 48 12: 28 100 7 101: 3224 244 | 28 20: 36 53 | 49 12: 26 95 8 88: 2860 263 | 29 20: 35 52 | 50 11: 27 111 9 77: 3467 393 | 30 19: 35 59 | 51 11: 26 110 10 69: 3150 428 | 31 19: 35 59 | 52 11: 16 79 11 63: 2442 382 | 32 19: 34 57 | 53 10: 35 162 12 56: 2479 454 | 33 18: 34 62 | 54 10: 35 162 13 50: 3330 716 | 34 18: 34 62 | 55 10: 34 162 14 46: 2460 596 | 35 18: 33 61 | 56 10: 34 162 15 42: 2663 739 | 36 17: 33 65 | 57 9: 35 187 16 39: 2173 678 | 37 16: 33 75 | 58 9: 33 180 17 35: 2791 995 | 38 16: 30 73 | 59 9: 33 180 18 32: 2684 1080 | 39 16: 29 70 | 60 9: 32 178 19 30: 2395 1063 | 40 15: 29 77 | 61 9: 31 177 20 28: 2216 1087 | 41 15: 28 76 | 62 9: 31 177 21 26: 1988 1087 | 42 15: 28 76 | 63 8: 31 202
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(247,2)-core and (27,22)-core
Zhukov, Boris (I) Wright, Charles (II) Wilson, Al (III) Wight, Paul Wickens, Brian White, Leon Warrior Warrington, Chaz Ware, David (II) Waltman, Sean Walker, P.J. von Erich, Kerry Vaziri, Kazrow Van Dam, Rob Valentine, Greg Vailahi, Sione Tunney, Jack Traylor, Raymond Tenta, John Taylor, Terry (IV) Taylor, Scott (IX) Tanaka, Pat Tajiri, Yoshihiro Szopinski, Terry Storm, Lance Steiner, Scott Steiner, Rick (I) Solis, Mercid Snow, Al Smith, Davey Boy Slaughter, Sgt. Simmons, Ron (I) Shinzaki, Kensuke Shamrock, Ken Senerca, Pete Scaggs, Charles Savage, Randy Saturn, Perry Sags, Jerry Ruth, Glen Runnels, Dustin Rude, Rick Rougeau, Raymond Rougeau Jr., Jacques Rotunda, Mike Ross, Jim (III) Rock, The Roberts, Jake (II) Rivera, Juan (II) Rhodes, Dusty (I) Reso, Jason Reiher, Jim Reed, Bruce (II) Race, Harley Prichard, Tom Powers, Jim (IV) Poffo, Lanny Plotcheck, Michael Piper, Roddy Pfohl, Lawrence Pettengill, Todd Peruzovic, Josip Palumbo, Chuck (I) Page, Dallas Ottman, Fred Orton, Randy Okerlund, Gene Nowinski, Chris Norris, Tony (I) Nord, John Neidhart, Jim Nash, Kevin (I) Muraco, Don Morris, Jim (VII) Morley, Sean Morgan, Matt (III) Mooney, Sean (I) Moody, William (I) Miller, Butch Mero, Marc McMahon, Vince McMahon, Shane Matthews, Darren (II) Martin, Andrew (II) Martel, Rick Marella, Robert Marella, Joseph A. Manna, Michael Lothario, Jose Long, Teddy LoMonaco, Mark Lockwood, Michael Levy, Scott (III) Levesque, Paul Michael Lesnar, Brock Leslie, Ed Leinhardt, Rodney Layfield, John Lawler, Jerry Lawler, Brian (II) Laurinaitis, Joe Laughlin, Tom (IV) Lauer, David (II) Knobs, Brian Knight, Dennis (II) Killings, Ron Kelly, Kevin (VIII) Keirn, Steve Jones, Michael (XVI) Johnson, Ken (X) Jericho, Chris Jarrett, Jeff (I) Jannetty, Marty James, Brian (II) Jacobs, Glen Jackson, Tiger Hyson, Matt Hughes, Devon Huffman, Booker Howard, Robert William Howard, Jamie Houston, Sam Horowitz, Barry Horn, Bobby Hollie, Dan Hogan, Hulk Hickenbottom, Michael Heyman, Paul Hernandez, Ray Henry, Mark (I) Hennig, Curt Helms, Shane Hegstrand, Michael Heenan, Bobby Hebner, Earl Hebner, Dave Heath, David (I) Hayes, Lord Alfred Hart, Stu Hart, Owen Hart, Jimmy (I) Hart, Bret Harris, Ron (IV) Harris, Don (VII) Harris, Brian (IX) Hardy, Matt Hardy, Jeff (I) Hall, Scott (I) Guttierrez, Oscar Gunn, Billy (II) Guerrero, Eddie Guerrero Jr., Chavo Gray, George (VI) Goldberg, Bill (I) Gill, Duane Gasparino, Peter Garea, Tony Funaki, Sho Fujiwara, Harry Frazier Jr., Nelson Foley, Mick Flair, Ric Finkel, Howard Fifita, Uliuli Fatu, Eddie Farris, Roy Eudy, Sid Enos, Mike (I) Eaton, Mark (II) Eadie, Bill Duggan, Jim (II) Douglas, Shane DiBiase, Ted DeMott, William Davis, Danny (III) Darsow, Barry Cornette, James E. Copeland, Adam (I) Constantino, Rico Connor, A.C. Cole, Michael (V) Coage, Allen Coachman, Jonathan Clemont, Pierre Clarke, Bryan Chavis, Chris Centopani, Paul Cena, John (I) Canterbury, Mark Candido, Chris Calaway, Mark Bundy, King Kong Buchanan, Barry (II) Brunzell, Jim Brisco, Gerald Bresciano, Adolph Bloom, Wayne Bloom, Matt (I) Blood, Richard Blanchard, Tully Blair, Brian (I) Blackman, Steve (I) Bischoff, Eric Bigelow, Scott ’Bam Bam’ Benoit, Chris (I) Batista, Dave Bass, Ron (II) Barnes, Roger (II) Backlund, Bob Austin, Steve (IV) Apollo, Phil Anoai, Solofatu Anoai, Sam Anoai, Rodney Anoai, Matt Anoai, Arthur Angle, Kurt AndrØ the Giant Anderson, Arn Albano, Lou Al-Kassi, Adnan Ahrndt, Jason Adams, Brian (VI) Young, Mae (I) Wright, Juanita Wilson, Torrie Vachon, Angelle Stratus, Trish Runnels, Terri Robin, Rockin’ Psaltis, Dawn Marie Moretti, Lisa Moore, Jacqueline (VI) Moore, Carlene (II) Mero, Rena McMichael, Debra McMahon, Stephanie Martin, Judy (II) Martel, Sherri Laurer, Joanie Keibler, Stacy Kai, Leilani Hulette, Elizabeth Guenard, Nidia Garca, LiliÆn Ellison, Lillian Dumas, Amy
’WWF Smackdown!’ ’WWE Velocity’ ’Sunday Night Heat’ ’Raw Is War’ WWF Vengeance WWF Unforgiven WWF Rebellion WWF No Way Out WWF No Mercy WWF Judgment Day WWF Insurrextion WWF Backlash WWE Wrestlemania XX WWE Wrestlemania X-8 WWE Vengeance WWE Unforgiven WWE SmackDown! Vs. Raw WWE No Way Out WWE No Mercy WWE Judgment Day WWE Armageddon Wrestlemania X-Seven Wrestlemania X-8 Wrestlemania 2000 Survivor Series Summerslam Royal Rumble No Way Out King of the Ring Invasion Fully Loaded
Survivor Series
Royal Rumble
Taylor, Scott (IX) Van Dam, Rob Matthews, Darren (II) LoMonaco, Mark Hughes, Devon Huffman, Booker Heyman, Paul Hebner, Earl McMahon, Stephanie Keibler, Stacy Wight, Paul Simmons, Ron (I) Senerca, Pete Ross, Jim (III) Rock, The Reso, Jason McMahon, Vince McMahon, Shane Martin, Andrew (II) Levesque, Paul Michael Layfield, John Lawler, Jerry Jericho, Chris Jacobs, Glen Hardy, Matt Hardy, Jeff (I) Gunn, Billy (II) Guerrero, Eddie Copeland, Adam (I) Cole, Michael (V) Calaway, Mark Bloom, Matt (I) Benoit, Chris (I) Austin, Steve (IV) Anoai, Solofatu Angle, Kurt Stratus, Trish Dumas, Amy
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(2,516)-Hard core
Hot Tight Asses 6 Hot Tight Asses 20 Hot Sweet Honey Hot Shots Hot Nights at the Blue Note Cafe Hot In the City Hospitality Sweet Hooter Heaven Honey Buns Hollywood Starlets Hollywood Exposed 2 Holly Does Hollywood Holiday for Angels Hindlick Maneuver, The Hienie’s Heroes Hidden Obsessions Hershe Highway 4 Herman’s Bed Heads and Tails Head Games Head Clinic Hawaii Vice Part III: Beyond the Badge Hawaii Vice 6 Hawaii Vice 5 Hawaii Vice 2 Harlequin Affair Hard to Handle Hard Rider Hard as a Rock Happy Endings Guilty by Seduction Greatest American Blonde Grand Opening Graduation from F.U. Gorgeous Good Girls Do Going Pro Goin’ Down Slow Goddess of Love Gluteus to the Maximus Glen or Glenda? Glamour Girl 5 Give Me Your Soul... Girls’ Club, The Girls Who Love to Suck Girls of the Double D 9 Girls of the Double D 13 Girls of the Double D 11 Girls of the A Team, The Girls of Paradise 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 Ginger Snacks Ginger On the Rocks Ginger Lynn: The Movie Ginger Lynn Non-Stop Ginger Lynn and Co. Ginger in Ecstasy Ginger Effect, The Ghost Town Gettin’ Ready Gang Bangs II Gang Bangs Gang Bang Wild Style 2 Gang Bang Nymphette Gang Bang Jizz Queens Gang Bang Jizz Jammers Gang Bang Girl 22 Gang Bang Girl 20 Gang Bang Girl 17 Gang Bang Girl 13 Gang Bang Girl 12 Gang Bang Face Bath 4 Gang Bang Face Bath 3 Gang Bang Face Bath 2 Gang Bang Face Bath Gang Bang Cummers Games Couples Play Future Voyeur Full Throttle Girls 1: Boredom Pulled the Trigger Full Nest Full Moon Fever Friends and Lovers: The Sequel Fresh Meat French Doll Forbidden Cravings Forbidden Bodies For Your Thighs Only For the Money 1 Fluffer, The Flesh Shopping Network Flashback First Annual XRCO Adult Film Awards Firm Offer Firefoxes Fire in the Hole Fine Art of Cunnilingus, The Film Buff Filet-o-Breast Femmes Ørotiques, Les Femme Vanessa, La Fast Girls Farmer’s Daughter 2 Fantasy Inc. Fame Is a Whore On Butt Row Eyewitness Nudes Extreme Sex 4: The Experiment Extreme Sex 2: The Dungeon Extreme Sex 1: The Club Exhibitionist, The Executive Suites Every Woman Has a Fantasy 3 Eternity Eternal Lust 2 Escort to Ecstasy Erotic Starlets 12: Crystal Lee Erotic Newcummers 4 Erotic Explosions 2 Entertainment L.A. Style Encore Enchantress Edge of Heat 2 Edge of Heat Ecstasy Easy Way, The Earth Girls Are Sleazy E. TV DØj Vu Dreams of Candace Hart Dreams in the Forbidden Zone Dream Machine, The Dream Lust Dragon Lady 4: Tales from the Bed 3, The Double Penetrations 7 Double Penetrations 6 Double Penetrations 2 Double Penetration 5 Double Penetration 4 Double Penetration 2 Double Penetration Dirty Prancing Dirty Pictures Dirty Movies Dirty Looks Dirty Dreams Dickman and Throbbin Diamond in the Rough Diamond Collection Double X 74 Diamond Collection Double X 10 Diamond Collection 79 Diamond Collection 67 Diamond Collection 64 Diamond Collection 61 Dial F for Fantasy Dial a Nurse Dial A for Anal Devil Made Her Do It, The Devil in Miss Jones 5: The Inferno, The Devil in Miss Jones 4: The Final Outrage Devil in Miss Jones 3: A New Beginning, The Desktop Dolls DeRenzy Tapes Delinquents On Butt Row Deeper, Harder, Faster Deep Throat Girls Deep Obsession Deep Inside Victoria Paris Deep Inside Vanessa del Rio Deep Inside Traci Deep Inside Shanna McCullough Deep Inside Samantha Strong Deep Inside Racquel Darrian Deep Inside P.J. Sparxx Deep Inside Nina Hartley Deep Inside Missy Deep Inside Kelly O’Dell Deep Inside Ginger Lynn Deep Inside Brittany O’Connell Deep Inside Barbii Deep Inside Ariana Deep In Angel’s Ass Deep Cover Deep Cheeks 3 Decadence Debutante, The Dear Bridgette Darker Side of Shayla, The Darker Side of Shayla 2, The Curse of the Catwoman Cumshot Revue 5 Cumshot Revue 3 Cumshot Revue 2 Cumming of Ass Cumback Pussy 9: Your Ass Is Mine! Cumback Pussy 8 Cumback Pussy 6 Cumback Pussy 4 Get Some Cum for Me Carol Cream of Cumback Pussy Crazed Covergirl Country Girl Coming on Strong Coming of Age Come Hither, Cum Ginger Club Head Club Ginger Club DV8 2 Cherry Cheeks Cheerleader Academy Cheeks 2: The Bitter End Cheek to Cheek 2: The Newest Cheeks on the Block Checkmate Charmed and Dangerous Chameleon, The Certifiably Anal Centerfolds Caught In the Middle Caught in the Act Caught From Behind 9 Caught From Behind 6 Caught From Behind 4 Caught From Behind 2: The Sequel Caught From Behind 10 Cat Alley Car Wash Angels Captain Butt’s Beach Canned Heat California Native Cajun Heat Butts of Steel Butties Butt Sisters, The Busted Burning Desire Burgundy Blues Bun for the Money Bun Busters British Are Coming, The Bringing Up the Rear Bride, The Breaking It Brazilian Connection, The Brat Force Bottoms Up! Series 8 Bottom Line, The Bottom Dweller Part Deux, The Bottom Dweller 5: In Search of... Booty Mistress Booty Ho 3 Boobs Butts and Bloopers 1 Boiling Point Boiling Desires Body of Innocence Body Music Bod Squad, The Blowin’ the Whistle Bloopers 2 Bloopers Blondes Who Blow Blondage Blame It On Ginger Black Valley Girls 2 Black Valley Girls Black Throat Black Stockings Billionaire Girls Club Bigger They Come Big Pink, The Big Melons 4 Big Melons 31 Big Melons 26 Big Mellons 25 Beyond Thunderbone Beyond Reality: Mischief in the Making Between the Cheeks 3 Between the Cheeks 2 Between the Cheeks Best Rears of Our Lives, The Best of the Vivid Girls #30 Best of the Dark Bros., Vol. 2, The Best of the Dark Bros., The Best of Talk Dirty, Vol. 1, The Best of Shane 1 Best of Loose Ends Best of Double Penetration Best of Diamond Collection 11 Best of Christy Canyon Best of Caught from Behind 2 Best of Caught from Behind Best of Amber Lynn Beefeaters Beaverly Hills Cop Beaver and Buttcheeks Beat Goes On, The Bazooka County 3 Battle of the Titans Battle of the Superstars Bare Market Bare Elegance Barbii Unleashed Barbara Dare’s Bad Ball Street Badgirls 2: Strip Search Bad Backroad to Paradise Backing in 3 Backfield in Motion Backdoor to Hollywood 11 Backdoor Summer II Backdoor Brides Part 2 Backdoor Brides Backdoor Bonanza 9 Backdoor Bonanza 13 Back to Nature Back Doors, The Bachelor Party Baby Face 2 Babe Watch Aussie Exchange Girls Asspiring Actresses Ass Openers 8 Ass Openers 6 Ass Openers 4 Ass Openers 3 Ass Openers 10 Ass Openers 1 Ass Lovers Special Ass Busters Inc. Ass Backwards Art of Desire Aroused Army Brat 2 Arizona Gold Anything Goes Another Rear View Animal in Me Angels of Mercy Angelica Angel Rising Angel Puss Analizer, The Anal Trashy Ass Anal Thunder 2 Anal Taboo Anal Sweetheart Anal Sluts and Sweethearts Anal Secrets Anal Riders 106 Anal Queen Anal Princess Anal Mystique Anal Lover Anal Justice Anal Innocence 2 Anal Inferno Anal Hounds and Bitches Anal Hellraiser 2 Anal Encounters 2 Anal Encounters 1 Anal Ecstasy Anal Delights 3 Anal Deep Rider Anal Crack Master Anal Climax 3 Anal Attitude Anal Anarchy Anal Adventures of Suzy Super Slut American Pie American Beauty Amber Lynn: The Totally Awesome Amazing Tails 5 Amazing Tails 4 Amazing Tails 3 Amazing Tails 2 Amazing Tails 1 All the Best, Barbara Alice in Hollyweird Al Terego’s Double Anal Alternatives Aja After Midnight Afro Erotica 15 Adventures of Tracy Dick: The Case of the Missing Stiff Adventures of Billy Blues, The Adult 45 Above the Knee A-List, The A Is for Asia 4F Dating Service 40 Something 1-800-934-Boob
North, Peter (I)
Wallice, Marc
Byron, Tom
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
Young Nurses In Lust Young Girls in Tight Jeans Young and Naughty Year of the Sex Dragon, The XTV 2 XTV 1 X-rated Bloopers and Outtakes X-rated Blondes X Dreams Wrapped Up WPINK-TV 3 Woman in Pink, The Within and Without You With Love from Susan With Love from Ginger Witching Hour, The Wire Desire Wings of Passion Willing Women Wild Women 61: Rachel Ryan Wild Women 32: Summer Rose Wild Women 11: Tanya Foxx Wild Weekend Wild in the Wilderness Wild Buck Wild Bananas On Butt Row Wild and Wicked 3 Wicked Whispers Wicked Ways #2 Wicked One Wicked As She Seems Whore, The Whore of the Worlds Whore House Who Killed Holly Hollywood? White Bunbusters Whispered Lies Where the Sun Never Shines What Gets Me Hot! Wet Dreams Reel Fantasies West Coast Girls We Love to Tease Way They Were, The Wacky World of X-Rated Bloopers Voyeur, The Voyeur’s Favorite Blowjobs and Anals 8, The Voodoo Lust: The Possession Visions of Desire Virgin Dreams Video Tramp Victoria and Company Vegas: Snake Eyes Vagina Town Up Your Ass 5 Up ’n Coming Unforgivable Unchain My Heart Unbelievable Orgies Turnabout True Legends of Adult Cinema: The Modern Video Era True Legends of Adult Cinema: The Erotic 80’s Trashy Lady Tracie Lords Toys 4 Us 2 Touch of Mischief Touch Me Torrid Without a Cause Top It Off Top 25 Adult Stars of All Time, The Too Good to Be True Tomboy To the Rear Tits Ahoy Tip of the Tongue Tight Squeeze Tight Ends in Motion Thrill Street Blues Three-way Lust Three by Three Those Lynn Girls This Is Your Sex Life Texas Crude Terms of Endowment Terminal Case of Love Temptation Eyes Teasers Tease, The Tawnee Be Good Taste of Victoria Paris Taste of Tawnee, A Taste of Ariel Tarnished Knight Talk Dirty to Me, Part III Talk Dirty to Me 9 Takin’ It to the Limit Take My Wife, Please! Take Me Tailspin 1 Tails of Perversity 3 Tails of Perversity 2 Tails of Perversity Tailiens 2 Tailiens Tailhouse Rock Tailgunners Swedish Erotica 74 Swedish Erotica 56 Swedish Erotica 54 Surfside Sex Superstars of Sex: Racquel Darrian Super Tramp 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
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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]
Different result (because of multiple lines)
Net/Components/Weak [2] Draw/Draw-Partition Net/Transform/Remove/Multiple lines/Single line Net/Partitions/Core/2-Mode [27 22] Operations/Extract from Network/Partition [Yes 1] Draw/Draw-Partition
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4-rings and analysis of 2-mode networks
In bipartite (2-mode) network there are no 3-rings. The densest substructures are complete bipartite subgraphs Kp,q . They contain many 4-rings.
w4 (Kp,q ) =
p 2
q 2
The 4-rings weights were implemented in Pajek only recently, in August 2005.
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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 Schoenberg, Jeremy Hauer, Brent Charlie Brown and Snoopy Show
Voyeur, Vince Boy, T.T. Davis, Mark (V)
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 Race for Your Life, Charlie Brown Be My Valentine, Charlie Brown It’s Magic, Charlie Brown Melendez, Bill You’re a Good Sport, Charlie Brown It’s a Mystery, Charlie Brown It’s an Adventure, Charlie Brown Momberger, Hilary It’s Flashbeagle, Charlie Brown Play It Again, Charlie Brown Is This Goodbye, Charlie Brown? Charlie Brown Thanksgiving There’s No Time for Love, Charlie Brown You’re Not Elected, Charlie Brown Snoopy Come Home It’s the Easter Beagle, Charlie Brown
Robbins, Peter (I) Shea, Christopher (I) Altieri, Ann
Dough, Jon
Sanders, Alex (I) North, Peter (I) Michaels, Sean
Mendelson, Karen Stratford, Tracy Dryer, Sally
Horner, Mike
Drake, Steve (I)
Boy Named Charlie Brown
Byron, Tom
Silvera, Joey
West, Randy (I)
Jeremy, Ron
Wallice, Marc
Savage, Herschel
Shea, Stephen
Thomas, Paul (I)
Pajek
Pajek
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Example: Island for w4 / Polizeiruf 110 and Starkes Team
Maranow, Maja Starkes Team, Ein ’Affre Semmeling, Die’ Starkes Team - Eins zu Eins, Ein Starkes Team - Kollege Mrder, Ein Starkes Team - Sicherheitsstufe 1, Ein Martens, Florian Starkes Team - Erbarmungslos, Ein Starkes Team - Das Bombenspiel, Ein Starkes Team - Blutsbande, Ein Starkes Team - Tdliche Rache, Ein 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 Starkes Team - Kleine Fische, groe Fische, Ein Starkes Team - Roter Schnee, Ein Starkes Team - Der Todfeind, Ein Starkes Team - Mordlust, Ein Winkler, Wolfgang Starkes Team - Das groe Schweigen, Ein Starkes Team - Der schne Tod, Ein Bademsoy, Tayfun Starkes Team - Der Verdacht, Ein Starkes Team - Trume und Lgen, Ein 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 Polizeiruf 110 - Kurschatten Polizeiruf 110 - Tote erben nicht Polizeiruf 110 - Der Pferdemrder Polizeiruf 110 - Henkersmahlzeit Polizeiruf 110 - Todsicher Polizeiruf 110 - Der Spieler Polizeiruf 110 - Mordsfreunde 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 Polizeiruf 110 - Doktorspiele Polizeiruf 110 - Jugendwahn
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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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Pattern searching
If a selected pattern determined by a given graph does not occur frequently in a sparse network the straightforward backtracking algorithm applied for pattern searching finds all appearences of the pattern very fast even in the case of very large networks. Pattern searching was successfully applied to searching for patterns of atoms in molecula (carbon rings) and searching for relinking marriages in genealogies.
Michael/Zrieva/ Francischa/Georgio/ Junius/Georgio/ Anucla/Zrieva/ Nicola/Ragnina/ Nicoleta/Zrieva/ Marinus/Zrieva/ Maria/Ragnina/
Damianus/Georgio/ Legnussa/Babalio/ Nicolinus/Gondola/ Franussa/Bona/
Junius/Zrieva/ Margarita/Bona/
Lorenzo/Ragnina/ Slavussa/Mence/
Sarachin/Bona/ Nicoletta/Gondola/
Marin/Gondola/ Magdalena/Grede/
Marinus/Bona/ Phylippa/Mence/
Three connected relinking marriages in the genealogy (represented as a p-graph) of ragusan noble families. A solid arc indicates the is a son of relation, and a dotted arc indicates the is a daughter of relation. In all three patterns a brother and a sister from one family found their partners in the same other family.
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Ore-graph
m-grandmother m-grandfather f-grandfather f-grandmother
mother
father
stepmother
sister-in-law
brother
I
wife
sister
daughter-in-law
son
son-in-law
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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p-graph
grandson
son & daughter-in-law
son-in-law & daughter
sister
I & wife
brother & sister-in-law
father & stepmother
father & mother
f-grandfather & f-grandmother
m-grandfather & m-grandmother
In p-graph vertices represent individuals or couples. In the case that a person is not married yet (s)he is represented by a vertex, otherwise person is represented with the partner in a common vertex. There are only arcs in p-graphs – they point from children to their parents, representing the relations FiC is a daughter of and MiC is a son of ; where FiC ≡ female in the couple; and MiC ≡ male in the couple.
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Relinking patterns in p-graphs
A2 A3
All possible relinking marriages in pgraphs with 2 to 6 vertices. Patterns are labeled as follows:
A4.2
A4.1
B4
• first character – number of first vertices: A – single, B – two, C – three. • second character: number of vertices in pattern (2, 3, 4, 5, or 6).
C6
A5.1
A5.2 B5
A6.3 A6.2 A6.1
• last character: identifier (if the two first characters are identical). Patterns denoted by A are exactly the blood marriages. In every pattern the number of first vertices equals to the number of last vertices.
B6.4 B6.1 B6.2 B6.3
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Frequencies normalized with number of couples in p-graph × 1000.
pattern A2 A3 A4.1 B4 A4.2 A5.1 A5.2 B5 A6.1 A6.2 A6.3 C6 B6.1 B6.2 B6.3 B6.4 Sum Loka 0.07 0.07 0.85 3.82 0.00 0.64 0.00 1.34 1.98 0.00 0.00 0.71 0.00 1.91 3.32 0.00 14.70 Silba 0.00 0.00 2.26 11.28 0.00 3.16 0.00 4.96 12.63 0.90 0.00 5.41 0.45 17.59 13.53 0.00 72.17 Ragusa 0.00 0.00 1.50 10.49 0.00 2.00 0.00 23.48 1.00 0.00 0.00 9.49 1.00 31.47 40.96 2.50 123.88 Turcs 0.00 0.00 159.71 98.28 0.00 36.86 0.00 46.68 169.53 0.00 0.00 36.86 0.00 130.22 113.02 7.37 798.53 Royal 0.00 2.64 18.45 6.15 0.00 11.42 0.00 7.03 11.42 0.88 0.00 4.39 0.00 10.54 11.42 0.00 84.36
Most of the relinking marriages happened in the genealogy of Turkish nomads; the second is Ragusa while in other genealogies they are much less frequent.
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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 = 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 ci,j =
k∈K
ai,k · bk,j
In the case when I = K = J we are dealing with ordinary one-mode networks (with square matrices). & % Methodenforum der Fakult¨ t f¨ r Sozialwissenschaften, Universit¨ t Wien, 21-22. 6. 2007 a u a S G S
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Fast sparse matrix multiplication
The standard matrix multiplication has the complexity O(|I| · |K| · |J|) – it is (usually) 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 A NB (k): neighbors of vertex k in network B
In general the multiplication of large sparse networks is a ’dangerous’ operation since the result can ’explode’ – it is not sparse. & % Methodenforum der Fakult¨ t f¨ r Sozialwissenschaften, Universit¨ t Wien, 21-22. 6. 2007 a u a S G S
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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 we can prove: 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. And after 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∗ ). 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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2-mode network analysis by conversion to 1-mode network
Often we transform a 2-mode network into an ordinary (1-mode) network N1 = (U, E1 , w1 ) or/and N2 = (V, E2 , w2 ), where E1 and w1 are (1) determined by the matrix A(1) = AAT , auv = z∈V auz · aT . Evidently zv auv = avu . There is an edge {u, v} ∈ E1 in N1 iff N (u) ∩ N (v) = ∅. Its (1) weight is w1 (u, v) = auv . The network N2 is determined in a similar way by the matrix A(2) = AT A. The networks N1 and N2 are analyzed using standard methods.
(1) (1)
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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. &
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. . . 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)}). We define [qi × qj ]T = qj × qi . It holds 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 ]. &
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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 (RuthDELmain.csv). The rows are the projects participants (idents) and colomns correspond to different their properties. We produced from this table three two-mode networks using J¨ rgen Pfeffer’s Text2Pajek program: u • project.net – idents × projects = P • country.net – idents × countries = C • institution.net – idents × institutions = U |idents| = 8869, |projects| = 933, |countries| = 60. &
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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 – projects × institutions W = PT • Countries.net – countries × countries S = CT U C W
• Institutions.net – institutions × institutions Q = WT • ...
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EU projects – deleted projects
Some projects (27) from original data set have to be deleted – in the final data set RuthDELmain 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; not implemented yet in Pajek). 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.
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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.
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PSI FUR PRODUKTE UND SYS.E DER INFORMATIONSTECH. BICC GENERAL CABLE COLOPLAST A/S 25525 FRAUENHOFER INST. FUER PRODUKTIONSTECH. UND AUTOMATISIERUNG LMS UMWELTSYS.E, DIPL. ING. DR. HERBERT BACK IST-2000-29207 MTU AERO ENGINES BUURSKOV DE ZENTRUM FUER LUFT UND RAUMFAHRT E.V. EUROCOPTER S. DASSAULT AVIATION SNECMA MOTEURS SA VOLKSWAGEN AG EA TECH. LTD 506503 MECALOG SARL 506257 INST. NAT. DE RECHERCHE SUR LES TRANSPORTS ET LEUR SCURIT ARMINES TQT SRL ESI SOFTWARE SA CHALMERS TEKNISKA HOEGSKOLA DAIMLER CHRYSLER AG 28283
EADS DE
BAE SYSTEMS AIRBUS UK LIMITED AIRBUS DEUTSCHLAND 502917 502909 NAT. TEC. UNIV. OF ATHENS
TESSITURA LUIGI SANTI SPA
INST. FUER TEXTIL UND VERFAHRENSTECH. DENKENDORF KBC MANUFAKTUR, KOECHLIN, BAUMGARTNER UND CIE. AG 29817 TRUMPF-BLUSEN-KLEIDER WALTER GIRNER UND CO. KG
INST. SUPERIOR TECNICO
C. R. FIAT S.C.P.A. NL ORG. FOR APPLIED SCIENTIFIC RESEARCH - TNO
502842 AIRBUS FRANCE SAS ALENIA AERONAUTICA SPA G4RD-CT-2002-00836 G4MA-CT-2002-00022 STICHTING NATIONAAL LUCHT OFFICE NAT. DETUDES ET DE REC. AEROSPATIALES BRPR987001 G4RD-CT-2000-00178 502896 G4RD-CT-2001-00403 G4RD-CT-2002-00795 502889 G4RD-CT-2000-00395
BARTENBACH 501084 POLYMAGE SARL
BARCO NV MSO CONCEPT INNOVATION + SOFTWARE
ROSENHEIMER GLASTECH. ENK6-CT-2002-30023 RUDOLF BRAUNS AND CO. KG SHERPA ENGINEERING SARL CATALYSE SARL EVG3-CT-2002-80012
7210-PR/163 CENTRE DE RECH. METALLURG. 7210-PR/233 INST. DE RECHERCHES DE LA SIDERURGIE FR T3.2/99 VOEST-ALPINE STAHL
7215-PP/031
THYSSENKRUPP STAHL A.G. DISENO DE SISTEMAS EN SILICIO
CENTRE FOR EUROP. ECONOMIC SMT4982223 7210-PR/095 CSTB UNIV. DER BUNDESWEHR MUENCHEN BUILDING RESEARCH LANDIS & GYR - EUROPE AG OESTERREICHISCHER BERGRETTUNGSDIENST IFEN GES. FUER SATELLITENNAVIGATION JOE3980089 WYKES ENGINEERING COMPANY IST-2000-30158 CINAR LTD. ENERGY RESEARCH CENTRE NL ENEL.IT SSAB TUNNPL¯T 7215-PP/034 LH AGRO EAST S.R.O. TECHNOFARMING S.R.L. INST. CARTOGRAFIC DE CATALUNYA BAYER. ROTES KREUZ HELP SERVICE REMOTE SENSING LESPROJEKT SLUZBY S.R.O. IST-2000-28177 T3.5/99 RESEARCH INST. OF THE FINNISH ECONOMY QLK6-CT-2002-02292 THE AARHUS SCHOOL OF BUSINESS MEFOS, FOUNDATION FOR METALLURGICAL RESEARCH HPSE-CT-2002-00143 ILEVO AB IST-2001-35358 JERNKONTORET FONDAZIONE ENI - ENRICO MATTEI HPSE-CT-2002-00108
CHIPIDEA - MICROELECTRONICA, S.A. UNIV. PANTHEON-ASSAS - PARIS II
BRITISH STEEL UNIV. OF MACEDONIA 7210-PR/142 JOR3980200 FRAUENHOFER INST. FUER AGRO-SAT CONSULTING MATERIALFLUSS UND LOGISTIK DATASYS S.R.O. UNIV. OF ABERDEEN ENK5-CT-2000-00335 ORAD HI TEC SYS. POLAND CENTRE DE ROBOTIQUE FRIMEKO INT. AB CRE GROUP LTD. MJM GROUP, A.S. INOX PNEUMATIC AS BBL DFA DE FERNSEHNACHRICHTEN AGENTUR TPS TERMISKA PROCESSER AB PROLEXIA KOMMANDITGES. HAMBURG 1 A.S.M. S.A. ZAMISEL D.O.O IST-1999-56418 FERNSEHEN BETEILIGUNGS & CO INGENIORHOJSKOLEN HELSINGOR TEKNIKUM DPME ROBOTICS AB GATE5 AG 511758 LKSOFTWARE UAB LKSOFT BALTIC BRST985352 ALBERTSEN & HOLM AS SVETS & TILLBEHOR AB SUPERELECTRIC DI CARLO PAGLIALUNGA & C. SAS IST-1999-57451 OK GAMES DI ALESSANDRO CARTA UNIV. DE ZARAGOZA INDUSTRIAS ROYO
IST-2000-30082 WISDOM TELE VISION SPORTART
ASM - DIMATEC INGENIERIA FFT ESPANA TECH. DE AUTOMOCION, EDAG ENGINEERING + DESIGN
ENERGITEKNIK HEATEX AB BROD THOMASSON GUNNESTORPS SMIDE & MEKANISKA AB
YAHOO! DEOSAUHING EETRIUKSUS
Pajek
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Analysis of Countries.net
Kazakhstan Japan Georgia India Uzbekistan Belarus Iceland Tunisia Jordan Lebanon Azerbaijan Liechtenstein Morocco Ecuador Afghanistan
To obtain picture in which the stronger lines cover weaker lines we have to sort them
Net/Transform/Sort lines/Line values/Ascending
Algeria
Armenia
Canada
France United Kingdom Italia The Netherlands Turkey Greece Germany Portugal Spain Switzerland Malta
Russian F. Finland Moldavia China
Thailand
Israel
Turkmenistan Estonia
Cyprus
Sweden Slovakia
Denmark
Austria Ukraine Serbia-Montenegro Macedonia Hungary Romania Albania
USA Poland Croatia Luxembourg Latvia
Belgium
Norway
Slovenia Ireland
Bulgaria Czech R. Lithuania
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 – for the New Year 2006 Andrej added this option in Pajek. We get a typical center-periphery structure. &
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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 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 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 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
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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. COVENTRY U. FACULTY OF ELECTRICAL ENG. FED.UNITARY ENTERPRISE FL-SOFT V/JENS DAEDALUS INFORMATICS LTD ALL-RUSSIAN SCIENTIFIC CENTER JOERGEN OESTERGAARD U.PARIS VI PIERRE ET MARIE CURIE BELIMO AUTOMATIONU.LA LAGUNA SOFIISKI U.SVETI KLIMENT OHRIDSKI THE U.COURT OF THE U.OF ABERDEEN SIEMENS BUILDING TECH.S AG SENTIENT MACHINE RESEARCH B.V. AS.CONOCIMIENTO U.OULU I.OF INFORMATION TECH.S STICHTING NEURALE NETWERKEN DATAMED HEALTHCARE INF.SYS. NOTTINGHAM TRENT U. U.E DE COIMBRA T.U.CLAUSTHAL MOMATEC U.CRETE CITY U.LONDON U.ULSTER GOETHE U.FRANKFURT AM MAIN U.WIEN ALLOGG AB RAUTARUUKKI OY SOFTECO SISMAT U.WALES, ABERYSTWYTH DE MONTFORT U. I.DALLE MOLLE DI STUDI SULLIA U.JYVASKYLA BULGARIAN ACAD. U.ZAGREB U.CYPRUS OF SCIENCES U.OF CHEMICAL TECH. BOURNEMOUTH U. ASS.RECH.SCIENTIFIQUE AND METALLURGY ENTE PER LE KINGS COLLEGE LONDON ELITE EUROP.LAB NUOVE TECNOLOGIE STICHTING U.NYENRODE FOR INTELLIGENT TECH. I.NAT.POLITEC.DE TOULOUSE I.FUER NATURSTOFF-FORSCHUNG E.V. U.AMSTERDAM U.GIRONA U.GENT AUSTRIAN I.AI START ENGINEERING JSCO T.U.DELFT I.NAT.DE RECHERCHE SUR ENERGY RESEARCH C.NL LES TRANSPORTS ET LEUR SECURITE U.GRANADA HELSINKI T.U. TEKNILLINEN KORKEAKOULU NAT.U.IRELAND,MAYNOOTH U.TWENTE TSS-TRANSPORT SIMULATION SYS.S.L. U.KLINIKUM AACHEN KATHOLIEKE U.LEUVEN U.BRISTOL FRIEDRICH-SCHILLER-U.JENA CONSEJO SUP.DE ERASMUS U.ROTTERDAM DEP.OF ENVIRONMENT, QINETIQ INVEST.CIENTIFICAS AABO AKADEMI U GKSS TRANSPORT AND THE REGIONS - FORSCHUNGSZENTRUM GEESTHACHT JOZEF STEFAN I. U.P.MADRID MANNESMANN VDO AG U.MARIBOR EUROP.SPACE AGENCY U.PAUL SABATIER DE TOULOUSE III U.PAISLEY TECHSOFT ENGINEERING S.R.O. TECNOLOGIAS CAE AVANZADAS S.L. U.STRATHCLYDE U.VALLADOLID U.P.CATALUNYA U.LEEDS U.S.GENOVA U.NOTTINGHAM C.SVILUPPO MATERIALI HERMSDORFER I.FUER TECH. PT.TORINO PT.BARI U.MANCHESTER OXFORD BROOKES U. DAIMLER CHRYSLER AG U.DORTMUND U.S.PADOVA SAFE TECH. BAE SYSTEMS LOUGHBOROUGH T.U. NOKIA MOBILE PHONES LTD CZECH T.U.PRAGUE AVIO S.P.A. FOKKER SPACE BV BRITISH TELEC. NAT.T.U.ATHENS T.U.V KOSICIACH ANAKON I.SUPERIOR TECNICO NCODE INT. U.SHEFFIELD FUNDACION LABEIN POLISH ACAD.OF SCIENCES FINITE ELEMENT ANALYSIS LTD. RISOE NAT.LAB TUN ABDUL RAZAK RESEARCH C.LTD. U.GREENWICH DANMARKS T.U. PRINCIPIA INGENIEROS CONSULTORES U.C.LOUVAIN FUNDACION INASMET STAVANGER U.COLLEGE CRANFIELD U. SULZER MARKETS AND TECH.AG, IFP SICOMP AB SULZER INNOTEC CHALMERS TEKNISKA HOEGSKOLA DAMT LTD SKF R&D COMPANY B.V. CAESAR SYSTEMS LTD U.S.NAPOLI A.U.THESSALONIKI C.INT.LENGINYERIA NL ORG.FOR APPLIED SCIENTIFIC RESEARCH-TNO C.R.FIAT S.C.P.A. CAD - FEM INTES - INGENIEURGES. FUER TECH.SOFTWARE U.DURHAM U.S.TRIESTE NAFEMS LTD. ENGIN SOFT TRADING SRL
MARITIME HYDRAULICS AS INBIS TECH.LTD FEMSYS LTD SOFISTK AG
C.NAT.DE LA RECHERCHE SCIENTIFIQUE
ABS CONSULTING ALTAIR ENGINEERING NLSE VERENIGDE SCHEEPSBOUW BUREAUS B.V. MERITOR HEAVY VEHICLE BRAKING SYS.- UK LTD CREA CONSULTANTS LTD TRL PD&E AUTOMOTIVE B.V. ACCESS E.V. NEW TECH.ENGINEERING LTD ROCKFIELD SOFTWARE LTD. U.NEWCASTLE UPON TYNE BEHR &CO. AIRBUS FRANCE SAS NAT.NUCLEAR CORP.LTD. U.GLASGOW FEMCOS INGENIEURBUERO MBH ST MECANICA APLICADA S.L. GERMAN AEROSPACE CENTRE FRAUENHOFER I.FUER INGENIEURBUERO FUER BIOMEDICAL ENGINEERING TRAGWERKSPLANUNG KATHOLIEKE HOGESCHOOL SINT-LIEVEN ROYAL I.OF TECH. DUNLOP STANDARD AEROSPACE GROUP U.HANNOVER GIFFORD AND PARTNERS LTD. MECAS S.R.O. VOLVO AERO CORP.AB ATOS ORIGIN ENG. LULEAA T.U. STRUCTURAL INTEGRITY CORK I.OF TECHNOLOGY ASSESSMENTS LTD HAHN-SCHICKARD-GES. NORUT TEKNOLOGI A.S. WS ATKINS CONSULTANTS LTD. EASI ENGINEERING U.E DO MINHO NUMERICAL ANALYSIS U.SPLIT AND DESIGN&CO KG WILDE AND TECHNISCHLTD PARTNERS ADVIESBUREAU N.V. INTEGRATED DESIGN & RANDOM LOADINGANALYSIS CONSULTANTS LTD DESIGN FEGS AWE PLC DC MERKLE UND PARTNER WHITE&PARTNERS LTD DR THELLEN EATEC LTD SAMTECH SA LEUVEN MEASUREMENTS AND SYS.INT.NV
MSC SOFTWARE QUEENS U.BELFAST
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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Pajek - Ward [0.00,1376.93]
ALLOGG AB AS. CONOCI ASS. RECH. AUSTRIAN I BELIMO AUT COVENTRY U DAEDALUS I DATAMED H FACULTY OF FED. UNITA FL-SOFT V/ INST. OF I GOETHE UNI MIT-MANAGE MOMATEC NAT. UNIV. NOTTINGHAM POLITECNIC SENTIENT M SIEMENS BU SOFIISKI U START ENGI STICHTING STICHTING THE UNIV. UNIV. DE L UNIV.SKLIN UNIV. DE G UNIV. PARI UNIV. OF A UNIV. OF C UNIV. OF C UNIV. OF W UNIV. OF Z INST. DALL RAUTARUUKK UNIV.E DE UNIV. OF U SOFTECO SI UNIV. GENT TSS - TRAN UNIV. DE G UNIV. DE V UNIVERZA V UNIV. PAUL TECH. UNIV UNIV. WIEN UNIV. VAN ENERGY RES TEKNILLINE DE MONTFOR CITY UNIVE UNIV. OF J AABO AKADE RISOE NAT. KINGS COLL UNIV. DORT FRIEDRICHUNIV. OF C INST. NAT. BULGARIAN ENTE PER L LOUGHBOROU HELSINKI U CRANFIELD UNIV. CATH OTTO VON G UNIV. DEGL UNIV. OF O UNIV. OF T ELITE EURO ERASMUS UN INST. FUER UNIV. PARI BRITISH TE BOURNEMOUT UNIV. OF B UNIV. OF S DANMARKS T DAIMLER CH INST. NAT. INST. SUPE POLITECNIC POLISH ACA UNIV. POLI NAT. TEC. CONSEJO SU KATHOLIEKE UNIV. TWEN TECH. UNIV UNIV. POLI CENTRE NAT TEC. UNIV. UNIV. OF P FUNDACION JOZEF STEF CZECH TECH UNIV. OF N UNIV. OF S BAE SYSTEM UNIV. OF M ABS CONSUL ALTAIR ENG ANAKON ATOS ORIGI AWE PLC BEHR & CO CAESAR SYS CORK INST. CREA CONSU D C WHITE DAMT LTD DEP. OF E DR THELLEN DUNLOP STA EASI ENGIN EATEC LTD FEMSYS LTD FOKKER SPA FORSCHUNGS GIFFORD AN HAHN-SCHIC HERMSDORFE INBIS TECH INGENIEURB INTEGRATED KATHOLIEKE MANNESMANN MARITIME H MECAS S.R. MERITOR HE MERKLE UND NAFEMS LTD NCODE INT. NLSE VEREN NEW TECH. NOKIA MOBI NORUT TEKN NUMERICAL OXFORD BRO PD & E AUT RANDOM LOA SAFE TECH. SKF R & D SOFISTK AG ST MECANIC STAVANGER STRUCTURAL SULZER MAR TECHNISCH TECHSOFT E TECNOLOGIA TUN ABDUL UNIV.E DO UNIV. OF S WILDE AND FRAUENHOFE GKSS - FOR NAT. NUCLE UNIV. OF D ENGIN SOFT FEGS IFP SICOMP INTES - IN UNIV. DEGL PRINCIPIA FINITE ELE ROCKFIELD WS ATKINS ROYAL INST UNIV. HANN UNIV. DEGL UNIV. OF G UNIV. OF N CAD - FEM C. SVILUPP TRL LEUVEN MEA A. UNIV. T GERMAN AER CENTRE INT UNIV. DEGL QINETIQ UNIV. OF L FUNDACION ACCESS E.V EUROP. SPA UNIV. OF G SAMTECH SA VOLVO AERO AVIO S.P.A MSC SOFTWA LULEAA UNI QUEENS UNI AIRBUS FRA C. R. FIAT NL ORG. FO CHALMERS T
Pajek - shadow [0.00,6.00]
ALLOGG AB AS. CONOCI ASS. RECH. AUSTRIAN I BELIMO AUT COVENTRY U DAEDALUS I DATAMED H FACULTY OF FED. UNITA FL-SOFT V/ INST. OF I GOETHE UNI MIT-MANAGE MOMATEC NAT. UNIV. NOTTINGHAM POLITECNIC SENTIENT M SIEMENS BU SOFIISKI U START ENGI STICHTING STICHTING THE UNIV. UNIV. DE L UNIV.SKLIN UNIV. DE G UNIV. PARI UNIV. OF A UNIV. OF C UNIV. OF C UNIV. OF W UNIV. OF Z INST. DALL RAUTARUUKK UNIV.E DE UNIV. OF U SOFTECO SI UNIV. GENT TSS - TRAN UNIV. DE G UNIV. DE V UNIVERZA V UNIV. PAUL TECH. UNIV UNIV. WIEN UNIV. VAN ENERGY RES TEKNILLINE DE MONTFOR CITY UNIVE UNIV. OF J AABO AKADE RISOE NAT. KINGS COLL UNIV. DORT FRIEDRICHUNIV. OF C INST. NAT. BULGARIAN ENTE PER L LOUGHBOROU HELSINKI U CRANFIELD UNIV. CATH OTTO VON G UNIV. DEGL UNIV. OF O UNIV. OF T ELITE EURO ERASMUS UN INST. FUER UNIV. PARI BRITISH TE BOURNEMOUT UNIV. OF B UNIV. OF S DANMARKS T DAIMLER CH INST. NAT. INST. SUPE POLITECNIC POLISH ACA UNIV. POLI NAT. TEC. CONSEJO SU KATHOLIEKE UNIV. TWEN TECH. UNIV UNIV. POLI CENTRE NAT TEC. UNIV. UNIV. OF P FUNDACION JOZEF STEF CZECH TECH UNIV. OF N UNIV. OF S BAE SYSTEM UNIV. OF M C. R. FIAT NL ORG. FO CHALMERS T SAMTECH SA VOLVO AERO AVIO S.P.A MSC SOFTWA LULEAA UNI QUEENS UNI AIRBUS FRA CAD - FEM C. SVILUPP TRL LEUVEN MEA A. UNIV. T GERMAN AER CENTRE INT UNIV. DEGL QINETIQ UNIV. OF L FUNDACION ACCESS E.V EUROP. SPA UNIV. OF G UNIV. DEGL UNIV. OF G UNIV. OF N FINITE ELE ROCKFIELD WS ATKINS ROYAL INST UNIV. HANN FRAUENHOFE GKSS - FOR NAT. NUCLE UNIV. OF D ENGIN SOFT FEGS IFP SICOMP INTES - IN UNIV. DEGL PRINCIPIA ABS CONSUL ALTAIR ENG ANAKON ATOS ORIGI AWE PLC BEHR & CO CAESAR SYS CORK INST. CREA CONSU D C WHITE DAMT LTD DEP. OF E DR THELLEN DUNLOP STA EASI ENGIN EATEC LTD FEMSYS LTD FOKKER SPA FORSCHUNGS GIFFORD AN HAHN-SCHIC HERMSDORFE INBIS TECH INGENIEURB INTEGRATED KATHOLIEKE MANNESMANN MARITIME H MECAS S.R. MERITOR HE MERKLE UND NAFEMS LTD NCODE INT. NLSE VEREN NEW TECH. NOKIA MOBI NORUT TEKN NUMERICAL OXFORD BRO PD & E AUT RANDOM LOA SAFE TECH. SKF R & D SOFISTK AG ST MECANIC STAVANGER STRUCTURAL SULZER MAR TECHNISCH TECHSOFT E TECNOLOGIA TUN ABDUL UNIV.E DO UNIV. OF S WILDE AND ALLOGG AB AS. CONOCI ASS. RECH. AUSTRIAN I BELIMO AUT COVENTRY U DAEDALUS I DATAMED H FACULTY OF FED. UNITA FL-SOFT V/ INST. OF I GOETHE UNI MIT-MANAGE MOMATEC NAT. UNIV. NOTTINGHAM POLITECNIC SENTIENT M SIEMENS BU SOFIISKI U START ENGI STICHTING STICHTING THE UNIV. UNIV. DE L UNIV.SKLIN UNIV. DE G UNIV. PARI UNIV. OF A UNIV. OF C UNIV. OF C UNIV. OF W UNIV. OF Z INST. DALL RAUTARUUKK UNIV.E DE UNIV. OF U SOFTECO SI UNIV. GENT TSS - TRAN UNIV. DE G UNIV. DE V UNIVERZA V UNIV. PAUL TECH. UNIV UNIV. WIEN UNIV. VAN ENERGY RES TEKNILLINE DE MONTFOR CITY UNIVE UNIV. OF J AABO AKADE RISOE NAT. KINGS COLL UNIV. DORT FRIEDRICHUNIV. OF C INST. NAT. BULGARIAN ENTE PER L LOUGHBOROU HELSINKI U CRANFIELD UNIV. CATH OTTO VON G UNIV. DEGL UNIV. OF O UNIV. OF T ELITE EURO ERASMUS UN INST. FUER UNIV. PARI BRITISH TE BOURNEMOUT UNIV. OF B UNIV. OF S DANMARKS T DAIMLER CH INST. NAT. INST. SUPE POLITECNIC POLISH ACA UNIV. POLI NAT. TEC. CONSEJO SU KATHOLIEKE UNIV. TWEN TECH. UNIV UNIV. POLI CENTRE NAT TEC. UNIV. UNIV. OF P FUNDACION JOZEF STEF CZECH TECH UNIV. OF N UNIV. OF S BAE SYSTEM UNIV. OF M C. R. FIAT NL ORG. FO CHALMERS T SAMTECH SA VOLVO AERO AVIO S.P.A MSC SOFTWA LULEAA UNI QUEENS UNI AIRBUS FRA CAD - FEM C. SVILUPP TRL LEUVEN MEA A. UNIV. T GERMAN AER CENTRE INT UNIV. DEGL QINETIQ UNIV. OF L FUNDACION ACCESS E.V EUROP. SPA UNIV. OF G UNIV. DEGL UNIV. OF G UNIV. OF N FINITE ELE ROCKFIELD WS ATKINS ROYAL INST UNIV. HANN FRAUENHOFE GKSS - FOR NAT. NUCLE UNIV. OF D ENGIN SOFT FEGS IFP SICOMP INTES - IN UNIV. DEGL PRINCIPIA ABS CONSUL ALTAIR ENG ANAKON ATOS ORIGI AWE PLC BEHR & CO CAESAR SYS CORK INST. CREA CONSU D C WHITE DAMT LTD DEP. OF E DR THELLEN DUNLOP STA EASI ENGIN EATEC LTD FEMSYS LTD FOKKER SPA FORSCHUNGS GIFFORD AN HAHN-SCHIC HERMSDORFE INBIS TECH INGENIEURB INTEGRATED KATHOLIEKE MANNESMANN MARITIME H MECAS S.R. MERITOR HE MERKLE UND NAFEMS LTD NCODE INT. NLSE VEREN NEW TECH. NOKIA MOBI NORUT TEKN NUMERICAL OXFORD BRO PD & E AUT RANDOM LOA SAFE TECH. SKF R & D SOFISTK AG ST MECANIC STAVANGER STRUCTURAL SULZER MAR TECHNISCH TECHSOFT E TECNOLOGIA TUN ABDUL UNIV.E DO UNIV. OF S WILDE AND
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What else?
In 2005 we introduced in Pajek also support for multi-relational networks that combined with temporal networks enable analysis of new kinds of networks – such as KEDS networks (Kansas Event Data System or Tabari). You can use URLs in description of vertices (Nov 2005). Still in the implementation phase is the hierarchical clustering in large networks based on Ferligoj and Batagelj (1982 and 1983).
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Selected Books on SNA
• J. P Scott: Social Network Analysis: A Handbook. SAGE Publications, 2000. Amazon. • A. Degenne, M. Fors´ : Introducing Social Networks. SAGE Publications, 1999. e Amazon. • S. Wasserman, K. Faust: Social Network Analysis: Methods and Applications. CUP, 1994. Amazon. • W. de Nooy, A. Mrvar, V. Batagelj: Exploratory Social Network Analysis with Pajek, CUP, 2005. Amazon. ESNA page. • P. Doreian, V. Batagelj, A. Ferligoj: Generalized Blockmodeling, CUP, 2004. Amazon. • E. Lazega: The Collegial Phenomenon: The Social Mechanisms of Cooperation among Peers in a Corporate Law Partnership. OUP, 2001. Amazon. • P.J. Carrington, J. Scott, S. Wasserman (Eds.): Models and Methods in Social Network Analysis. CUP, 2005. Amazon. • U. Brandes, T. Erlebach (Eds.): Network Analysis: Methodological Foundations. LNCS, Springer, Berlin 2005. Amazon.
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Courses on NA
• James Moody, The Ohio State University • Steve Borgatti, UCINET • Barry Wellman, University of Toronto • Douglas White, University of California Irvine • Lada Adamic, University of Michigan • Mark Newman, University of Michigan • Jon Kleinberg, Cornell University • Robert A. Hanneman, University of California, Riverside; workshop • Noah Friedkin, University of California, Santa Barbara • John Levi Martin, University of Wisconsin, Madison • Vladimir Batagelj, University of Ljubljana • Andrej Mrvar, University of Ljubljana &
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Software for SNA
UCINET, NetDraw Visone Negopy NetworkX BGL/Python See also the INSNA list and recent overview by M. Huisman and M.A.J. van Duijn. Pajek SNA/R InFlow prefuse Netminer StOCNET GUESS JUNG
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