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Mark Lombardi

Introductionto Networks

Anuska FerligojVladimir Batagelj

University of Ljubljana

European Science FoundationQMSS Workshop

Ljubljana, Thursday, June 30, 2005, 14:00–15:30

version: June 29, 2005 / 01 : 43

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Outline1 Some names in the development of SNA . . . . . . . . . . . . . . . . . . . . . . 1

2 Some important events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Selected Books on SNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4 Roman roads (Peutinger) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

5 Moreno: Who shall survive? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

6 Development of DNA (Garfield) . . . . . . . . . . . . . . . . . . . . . . . . . . 6

7 Hijackers (Krebs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

8 Wall Street Follies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

9 They Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

10 Lombardi’s networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

11 Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

13 Pajek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

14 Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

15 Graph / Sets – NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

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16 Graph / Neighbors – NET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

17 Graph / Matrix – MAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

18 Vertex Properties / CLU, VEC, PER . . . . . . . . . . . . . . . . . . . . . . . . . 18

20 Pajek’s Project File / PAJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

21 Special graphs – path, cycle, star, complete . . . . . . . . . . . . . . . . . . . . 21

22 Representations of properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

24 Size of network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

25 How to get a network? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

26 Complete and ego-centered networks . . . . . . . . . . . . . . . . . . . . . . . 26

27 Network measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

28 Multiple networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

30 Two-mode networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

32 Temporal networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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Some names in the development of SNA

Moreno

• Moreno (1934, 1953, 1960) -sociometry

• Lewin (1936)

• Warner and Lunt (1941)

• Heider (1946)

• Bavelas (1948) - centrality

• Homans (1950)

• Cartwright and Harary (1956)

• Nadel (1957) - social structure,social positions, roles

• Mitchell (1969)

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Some important events

• International Association ofSocial Network Analysis - INSNA, 1978

• Journal: Social Networks, 1978

• Newsletter: Connections, 1978

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Selected Books on SNA• J. P Scott: Social Network Analysis: A Handbook. SAGE Publications, 2000. Amazon.

• A. Degenne, M. Forse: Introducing Social Networks. SAGE Publications, 1999.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 amongPeers in a Corporate Law Partnership. OUP, 2001. Amazon.

• P.J. Carrington, J. Scott, S. Wasserman (Eds.): Models and Methods in Social NetworkAnalysis. CUP, 2005. Amazon.

• V. Batagelj, A. Mrvar: Pajek – Analysis and Visualization of Large Networks. in Junger,M., Mutzel, P., (Eds.) Graph Drawing Software. Springer, Berlin 2003, p. 77-103.Amazon.

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Roman roads (Peutinger)

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Moreno: Who shall survive?

K: 1: 2:

3: 4: 5:

6: 7: 8:

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Wall Street Follies

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They Rule

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Lombardi’s networks

Mark Lombardi(1951-2000)transformed businessrelations into art.

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Networks

Alexandra Schuler/ Marion Laging-Glaser:Analyse von Snoopy Comics

A network is based on two sets – setof vertices (nodes), that representthe selected units, and set of lines(links), that represent ties betweenunits. They determine a graph. Aline can be directed – an arc, orundirected – an edge.Additional data about vertices orlines can be known – their prop-erties (attributes). For example:name/label, type, value, . . .

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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, A is the set of arcs,E is the set of edges, and L = E ∪ A is the set of lines.n = |V|, m = |L|

• P vertex value functions / properties: p : V → A

• W line value functions / weights: w : L → B

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Pajek

Pajek is a program, for Windows, for anal-ysis and visualization of large networks hav-ing some ten or houndred of thousands ofvertices.In Slovenian language pajek means spider.

The latest version of Pajek is freely available, for noncommercial use, atits home page:

http://vlado.fmf.uni-lj.si/pub/networks/pajek/

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Graph

unit, actor – vertex, nodetie, link – line, edge, arcarc = directed line, (a, d)a is the initial vertex,d is the terminal vertex.edge = undirected line, (c: d)c and d are end vertices.

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Graph / Sets – NET

V = a, b, c, d, e, f, g, h, i, j, k, l

A = (a, b), (a, d), (a, f), (b, a),

(b, f), (c, b), (c, c), (c, g),

(c, g), (e, c), (e, f), (e, h),

(f, k), (h, d), (h, l), (j, h),

(l, e), (l, g), (l, h)

E = (b: e), (c: d), (e: g), (f : h)

G = (V,A, E)

L = A ∪ E

A = ∅ – undirected graph; E = ∅ – directed graph.

Pajek: local: GraphSet; TinaSet;WWW: GraphSet / net; TinaSet / net, picture picture.

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Graph / Neighbors – NET

NA(a) = b, d, fNA(b) = a, fNA(c) = b, c, g, gNA(e) = c, f, hNA(f) = kNA(h) = d, lNA(j) = hNA(l) = e, g, h

NE(e) = b, gNE(c) = dNE(f) = h

Pajek: local: GraphList; TinaList;WWW: GraphList / net; TinaList / net.

N(v) = NA(v) ∪NE(v), also Nout(v), Nin(v)

Star in v, S(v) is the set of all lines with v as their initial vertex.

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Graph / Matrix – MATa b c d e f g h i j k l

a 0 1 0 1 0 1 0 0 0 0 0 0

b 1 0 0 0 1 1 0 0 0 0 0 0

c 0 1 1 1 0 0 2 0 0 0 0 0

d 0 0 1 0 0 0 0 0 0 0 0 0

e 0 1 1 0 0 1 1 1 0 0 0 0

f 0 0 0 0 0 0 0 1 0 0 1 0

g 0 0 0 0 1 0 0 0 0 0 0 0

h 0 0 0 1 0 1 0 0 0 0 0 1

i 0 0 0 0 0 0 0 0 0 0 0 0

j 0 0 0 0 0 0 0 1 0 0 0 0

k 0 0 0 0 0 0 0 0 0 0 0 0

l 0 0 0 0 1 0 1 1 0 0 0 0

Pajek: local: GraphMat; TinaMat, picture picture;WWW: GraphMat / net; TinaMat / net, paj.

Graph G is simple if in the corresponding matrix all entries are 0 or 1.

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Vertex Properties / CLU, VEC, PERAll three types of files have the same structure:

*vertices n n is the number of verticesv1 vertex 1 has value v1

. . .vn

CLUstering – partition of vertices – nominal or ordinal data about verticesvi ∈ IN : vertex i belongs to the cluster vi;

VECtor – numeric data about verticesvi ∈ IR : the property has value vi on vertex i;

PERmutation – ordering of verticesvi ∈ IN : vertex i is at the vi-th position.

When collecting the network data consider to provide as much propertiesas possible.

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Example: Wolfe Monkey Data

inter.net inter.net sex.clu age.vec rank.per*Vertices 20 1 "m01" 2 "m02" 3 "m03" 4 "m04" 5 "m05" 6 "f06" 7 "f07" 8 "f08" 9 "f09" 10 "f10" 11 "f11" 12 "f12" 13 "f13" 14 "f14" 15 "f15" 16 "f16" 17 "f17" 18 "f18" 19 "f19" 20 "f20" *Edges 1 2 2 1 3 10 1 4 4 1 5 5 1 6 5 1 7 9 1 8 7 1 9 4 1 10 3 1 11 3 1 12 7 1 13 3 1 14 2 1 15 5 1 16 1 1 17 4 1 18 1 2 3 5 2 4 1 2 5 3 2 6 1 2 7 4 2 8 2 2 9 6 2 10 2 2 11 5 2 12 4 2 13 3 2 14 2 2 15 2 2 16 6 2 17 3 2 18 1 2 19 1 3 4 8 3 5 9 3 6 5 3 7 11 3 8 7 3 9 8 3 10 8 3 11 14 3 12 17 3 13 9 3 14 11 3 15 11 3 16 5 3 17 9 3 18 4

*Vertices 20 1 "m01" 2 "m02" 3 "m03" 4 "m04" 5 "m05" 6 "f06" 7 "f07" 8 "f08" 9 "f09" 10 "f10" 11 "f11" 12 "f12" 13 "f13" 14 "f14" 15 "f15" 16 "f16" 17 "f17" 18 "f18" 19 "f19" 20 "f20" *Edges 1 2 2 1 3 10 1 4 4 1 5 5 1 6 5 1 7 9 1 8 7 1 9 4 1 10 3 1 11 3 1 12 7 1 13 3 1 14 2 1 15 5 1 16 1 1 17 4 1 18 1 2 3 5 2 4 1 2 5 3 2 6 1 2 7 4 2 8 2 2 9 6 2 10 2 2 11 5 2 12 4 2 13 3 2 14 2 2 15 2 2 16 6 2 17 3 2 18 1 2 19 1 3 4 8 3 5 9 3 6 5 3 7 11 3 8 7 3 9 8 3 10 8 3 11 14 3 12 17 3 13 9 3 14 11 3 15 11 3 16 5 3 17 9 3 18 4

*vertices 20 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

*vertices 20 15 10 10 8 7 15 5 11 8 9 16 10 14 5 7 11 7 5 15 4

*vertices 20 1 2 3 4 5 10 11 6 12 9 7 8 18 19 20 13 14 15 16 17

. . .

Important notes: 0 is not allowed as vertex number. Pajek doesn’t support Unix text files –

lines should be ended with CR LF.

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Pajek’s Project File / PAJAll types of data can be combined into a single file – Pajek’s project filefile.paj.

The easiest way to do this is:

• read all data files in Pajek,

• compute some additional data,

• delete (dispose) some data,

• save all as a project file with File/Project file/Save.

Next time you can restore everything with a singleFile/Project file/Read.

Pajek supports also multiple, two-mode and temporal networks.

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Special graphs – path, cycle, star, complete

Graphs: path P5, cycle C7, star S8 in complete graph K7.

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Representations of propertiesProperties of vertices P and lines W can be measured in different scales:numerical, ordinal and nominal. They can be input as data or computedfrom the network.

In Pajek numerical properties of vertices are represented by vectors,nominal properties by partitions or as labels of vertices. Numericalproperty can be displayed as size of vertex (figure), as its coordinate; and anominal property as color or shape of the figure, or as a vertex label.

We can assign in Pajek numerical values to links. They can be displayedas 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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Display of properties – school (Moody)

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Size of networkThe size of a network/graph is expressed by two numbers: number ofvertices n = |V| and number of lines m = |L|.

In a simple undirected graph (no parallel edges, no loops) m ≤ 12n(n− 1);

and in a simple directed graph (no parallel arcs) m ≤ n2.

The quotient γ = mmmax

is a density of graph.

Small networks (some tens vertices) – can be represented by a picture andanalyzed by many algorithms (UCINET, NetMiner).

Also middle size networks (some hundreds vertices) can still be representedby a picture (!?), but some analytical procedures can’t be used.

Large networks (several thousands vertices) are too big to be displayed;special algorithms are needed for their analysis (Pajek).

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How to get a network?Collecting data about the network N = (V,L,P,W) we have first todecide, what are the units (vertices) – network boundaries, when are twounits related – network completness, and which properties of vertices/lineswe shall consider.

Several networks are already available in computer readable form or can beconstructed from such data.

For large sets of units we often can’t measure the complete network.Therefore we limit the data collection to selected units and their neighbors.We get ego-centered networks.

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Complete and ego-centered networks

Pajek

Egos Alters

COMPLETE NETWORK EGO-CENTEREDNETWORKS

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Network measurementsHow to measure networks (questionaires, interviews, observations, archiverecords, experiments, . . . )?

What is the quality of measured networks (reliability and validity)?

Some answers to these questions will be given by Ferligoj, Kogovsek andHlebec at the seminar next week.

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Multiple networksMultiple or multi-relational networks on the same set of vertices wereimplemented in Pajek only recently (November 2004). Examples ofsuch networks are: Transportation system in a city (stations, lines);WordNet (words, semantic relations: synonymy, antonymy, hyponymy,meronymy,. . . ), KEDS networks (states, relations between states: Visit,Ask information, Warn, Expel person, . . . ), . . .

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. . . Multiple networks

The relation can be assigned to a line as follows:

• add to a keyword for description of lines (*arcs, *edges,

*arcslist, *edgeslist, *matrix) the number of relationfollowed by its name:

*arcslist :3 "sent a letter to"

All lines controlled by this keyword belong to the specified relation.(Sampson, SampsonL)

• Any line controlled by *arcs or *edges can be assigned to selectedrelation by starting its description by the number of this relation.

3: 47 14 5

Line with endpoints 47 and 14 and weight 5 belongs to relation 3.

KEDS / Gulf

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Two-mode networksIn a two-mode network N = (U ,V,L,P,W) the set of vertices consistsof two disjoint sets of vertices U and V , and all the lines from L have oneend-vertex in U and the other in V . Often also a weight w : L → IR ∈ W isgiven; if not, we assume w(u, v) = 1 for all (u, v) ∈ L.

A two-mode network can also be described by a rectangular matrixA = [auv]U×V .

auv =

wuv (u, v) ∈ L0 otherwise

Examples: (persons, societies, years of membership), (buyers/consumers,goods, quantity), (parlamentarians, problems, positive vote), (persons,journals, reading).

A two-mode network is announced by *vertices n nU .

Authors and works.

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Deep South

Classical example of two-mode networkare Southern women (Davis 1941).Davis.paj. Freeman’s overview.

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Temporal networksIn a temporal network the presence/activity of vertex/line can changethrough time. Pajek supports two types of descriptions of temporalnetworks based on presence and on events.

Moody:Drug users in Colorado Springs, 5 years

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Temporal network

Temporal network

NT = (V,L,P,W, T )

is obtained if the time T is attached to an ordinary network. T is a set oftime points t ∈ T .

In temporal network vertices v ∈ V and lines l ∈ L are not necessarilypresent or active in all time points. If a line l(u, v) is active in time point t

then also its endpoints u and v should be active in time t.

We will denote the network consisting of lines and vertices active in timet ∈ T by N (t) and call it the time slice in time point t. To get time slices inPajek useNet / Transform / Generate in time

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Temporal networks – presence

*Vertices 31 "a" [5-10,12-14]2 "b" [1-3,7]3 "e" [4-*]*Edges1 2 1 [7]1 3 1 [6-8]

Vertex a is present in time points 5, 6, 7, 8,9, 10 and 12, 13, 14.Edge (1 : 3) is present in time points 6, 7,8.

* means ’infinity’.A line is present, if both its end-verticesare present.

Time.net.

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Temporal networks / September 11

Steve Corman with collabora-tors from Arizona State Uni-versity transformed, using hisCentering Resonance Analysis(CRA), daily Reuters news (66days) about September 11th intoa temporal network of wordscoappearance.

Pictures in SVG: 66 days.

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