Network Analysis with Pajek

Transcription

1 Network Analysis with Pajek Vladimir Batagelj University of Ljubljana Local Development PhD Program version: September 10, 2009 / 07 : 37

2 V. Batagelj: Network Analysis with Pajek 2 Outline 1 Networks Graph Graph / Sets NET Graph / Neighbors NET Graph / Matrix MAT Vertex Properties / CLU, VEC, PER Pajek s Project File / PAJ Special graphs path, cycle, star, complete Representations of properties Some related operations Types of networks Temporal networks Multiple networks Two-mode networks Approaches to large networks Degrees Example: Snyder and Kick World Trade

3 V. Batagelj: Network Analysis with Pajek 3 32 Clusters, clusterings, partitions, hierarchies Subgraph Walks Equivalence relations and Partitions Connectivity Cuts k-connectivity Triangular and short cycle connectivities Islands Important vertices in network Network centralization measures Citation networks Genealogies Pattern searching Triads Dense groups Cores and generalized cores Analysis of two-mode networks Two-mode cores

4 V. Batagelj: Network Analysis with Pajek Directed 4-rings Multiplication of networks Networks from data tables EU projects on simulation Searching on the Web of Science Last version available at: Based on my article: Batagelj V. (2009) Large-scale Social Network Analysis. in Meyers, R.A. ed. Encyclopedia of Complexity and Systems Science, Springer, Heidelberg.

5 V. Batagelj: Network Analysis with Pajek 1 Alexandra Schuler/ Marion Laging-Glaser: Analyse von Snoopy Comics 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,... Network = Graph + Data The data can be measured or computed.

6 V. Batagelj: Network Analysis with Pajek 2 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

7 V. Batagelj: Network Analysis with Pajek 3 Graph unit, actor vertex, node tie, link line, edge, arc arc = 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.

8 V. Batagelj: Network Analysis with Pajek 4 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.

9 V. Batagelj: Network Analysis with Pajek 5 Graph / Neighbors NET N A (a) = {b, d, f} N A (b) = {a, f} N A (c) = {b, c, g, g} N A (e) = {c, f, h} N A (f) = {k} N A (h) = {d, l} N A (j) = {h} N A (l) = {e, g, h} N E (e) = {b, g} N E (c) = {d} N E (f) = {h} Pajek: local: GraphList; TinaList; WWW: GraphList / net; TinaList / net. N(v) = N A (v) N E (v), also N out (v), N in (v) Star in v, S(v) is the set of all lines with v as their initial vertex.

10 V. Batagelj: Network Analysis with Pajek 6 Graph / Matrix MAT a b c d e f g h i j k l a b c d e f g h i j k l 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.

11 V. Batagelj: Network Analysis with Pajek 7 Vertex Properties / CLU, VEC, PER All three types of files have the same structure: *vertices n n is the number of vertices v 1 vertex 1 has value v 1... v n CLUstering partition of vertices nominal or ordinal data about vertices v i N : vertex i belongs to the cluster v i ; VECtor numeric data about vertices v i R : the property has value v i on vertex i; PERmutation ordering of vertices v i N : vertex i is at the v i -th position. When collecting the network data consider to provide as much properties as possible.

12 12 "f12" 13 "f13" 14 "f14" V. Batagelj: Network Analysis with Pajek 15 "f15" 8 Example: 20 "f20" 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 "f16" 17 "f17" 18 "f18" 19 "f19" *Edges Corvinus University of Budapest, 1 17 September , *vertices *vertices *vertices Important notes: 0 is not allowed as vertex number. Pajek doesn t support Unix text files lines should be ended with CR LF.

13 V. Batagelj: Network Analysis with Pajek 9 Pajek s Project File / PAJ All types of data can be combined into a single file Pajek s project file file.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 single File/Project file/read. Wolfe network as Pajek s project file (PDF/paj).

14 V. Batagelj: Network Analysis with Pajek 10 Special graphs path, cycle, star, complete Graphs: path P 5, cycle C 7, star S 8 in complete graph K 7.

15 V. Batagelj: Network Analysis with Pajek 11 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).

16 V. Batagelj: Network Analysis with Pajek 12 Some related operations Operations/Vector/Put Coordinate Net/Vector/Get Coordinate [Draw] Options [Draw] Layout/Energy/Kamada-Kawai [Draw] Export/2D/EPS-PS

17 V. Batagelj: Network Analysis with Pajek 13 Display of properties school (Moody)

18 V. Batagelj: Network Analysis with Pajek 14 Types of networks Besides ordinary (directed, undirected, mixed) networks some extended types of networks are also used: 2-mode networks, bipartite (valued) graphs networks between two disjoint sets of vertices. multi-relational networks. temporal networks, dynamic graphs networks changing over time. specialized networks: representation of genealogies as p-graphs; Petri s nets,... The network (input) file formats should provide means to express all these types of networks. All interesting data should be recorded (respecting privacy).

19 V. Batagelj: Network Analysis with Pajek 15 ' $ Temporal networks In a temporal network the presence/activity of vertex/line can change through time. Pajek supports two types of descriptions of temporal networks based on presence and on events. Moody: Drug users in Colorado Springs, 5 years s s l y s y s s s & % * 6

20 V. Batagelj: Network Analysis with Pajek 16 Temporal network Temporal network N T = (V, L, P, W, T ) is obtained if the time T is attached to an ordinary network. T is a set of time points t T. In temporal network vertices v V and lines l L are not necessarily present 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 time t T by N (t) and call it the time slice in time point t. To get time slices in Pajek use Net / Transform / Generate in time

21 V. Batagelj: Network Analysis with Pajek 17 Temporal networks presence *Vertices 3 1 "a" [5-10,12-14] 2 "b" [1-3,7] 3 "e" [4-*] *Edges [7] [6-8] Time.net. 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-vertices are present.

22 V. Batagelj: Network Analysis with Pajek 18 Temporal networks events Event TI t TE t AV vns HV v SV v DV v AA uvs HA uv SA uv DA uv AE uvs HE uv SE uv DE uv CV vs CA uvs CE uvs CT uv CD uv PE uvs AP uvs DP uv EP uvs Explanation initial events following events happen when time point t starts end events following events happen when time point t is finished add vertex v with label n and properties s hide vertex v show vertex v delete vertex v add arc (u,v) with properties s hide arc (u,v) show arc (u,v) delete arc (u,v) add edge (u:v) with properties s hide edge (u:v) show edge (u:v) delete edge (u:v) change property of vertex v to s change property of arc (u,v) to s change property of edge (u:v) to s change (un)directedness of line (u,v) change direction of arc (u,v) replace pair of arcs (u,v) and (v,u) by single edge (u:v) with properties s add pair of arcs (u,v) and (v,u) with properties s delete pair of arcs (u,v) and (v,u) replace edge (u:v) by pair of arcs (u,v) and (v,u) with properties s s can be empty. In case of parallel lines :k denotes the k-th line HE: hides the third edge connecting vertices 14 and 37. *Vertices 3 *Events TI 1 AV 2 "b" TE 3 HV 2 TI 4 AV 3 "e" TI 5 AV 1 "a" TI 6 AE TI 7 SV 2 AE TE 7 DE 1 2 DV 2 TE 8 DE 1 3 TE 10 HV 1 TI 12 SV 1 TE 14 DV 1 Time.tim. Friends.tim. File/Time Events Network

23 V. Batagelj: Network Analysis with Pajek 19 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. Pictures in SVG: 66 days.

24 V. Batagelj: Network Analysis with Pajek 20 Multiple networks Multiple or multi-relational networks on the same set of vertices were implemented in Pajek in November Examples of such 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,... ),...

25 V. Batagelj: Network Analysis with Pajek 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 relation followed 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 selected relation by starting its description by the number of this relation. 3: Line with endpoints 47 and 14 and weight 5 belongs to relation 3. KEDS / Gulf

26 V. Batagelj: Network Analysis with Pajek 22 Multi-relational temporal network KEDS/WEIS % Recoded by WEISmonths, Sun Nov 28 21:57: % from keds/data.dir/balk.html *vertices "AFG" [1-*] 2 "AFR" [1-*] 3 "ALB" [1-*] 4 "ALBMED" [1-*] 5 "ALG" [1-*] "YUGGOV" [1-*] 319 "YUGMAC" [1-*] 320 "YUGMED" [1-*] 321 "YUGMTN" [1-*] 322 "YUGSER" [1-*] 323 "ZAI" [1-*] 324 "ZAM" [1-*] 325 "ZIM" [1-*] *arcs :0 "*** ABANDONED" *arcs :10 "YIELD" *arcs :11 "SURRENDER" *arcs :12 "RETREAT"... *arcs :223 "MIL ENGAGEMENT" *arcs :224 "RIOT" *arcs :225 "ASSASSINATE TORTURE" *arcs 224: [4] YUG KSV 224 (RIOT) RIOT-TORN 212: [4] YUG ETHALB 212 (ARREST PERSON) ALB ETHNIC JAILED IN YUG 224: [4] ALB ETHALB 224 (RIOT) RIOTS 123: [4] ETHALB KSV 123 (INVESTIGATE) PROBING... 42: [175] GER CYP 042 (ENDORSE) GAVE SUPPORT 212: [175] UNWCT BOSSER 212 (ARREST PERSON) SENTENCED TO PRISON 43: [175] VAT EUR 043 (RALLY) RALLIED 13: [175] UNWCT BOSSER 013 (RETRACT) CLEARED 121: [175] UNWCT BAL 121 (CRITICIZE) CHARGES 122: [175] SER UNWCT 122 (DENIGRATE) TESTIFIED 121: [175] BOSSER UNWCT 121 (CRITICIZE) ACCUSED Kansas Event Data System KEDS

27 V. Batagelj: Network Analysis with Pajek 23 Two-mode networks In a two-mode network N = (U, V, L, P, W) the set of vertices consists of two disjoint sets of vertices U and V, and all the lines from L have one end-vertex in U and the other in V. Often also a weight w : L R W is given; if not, we assume w(u, v) = 1 for all (u, v) L. A two-mode network can also be described by a rectangular matrix A = [a uv ] U V. { wuv (u, v) L a uv = 0 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 n U. Authors and works.

28 V. Batagelj: Network Analysis with Pajek 24 Deep South Classical example of two-mode network are the Southern women (Davis 1941). Davis.paj. Freeman s overview.

29 V. Batagelj: Network Analysis with Pajek 25 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 identify smaller (sub) networks that can be analyzed further using more sophisticated methods.

30 V. Batagelj: Network Analysis with Pajek 26 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. initial vertex v indeg(v) = 0 terminal vertex v outdeg(v) = 0 n = 12, m = 23, indeg(e) = 3, outdeg(e) = 5, deg(e) = 6 indeg(v) = outdeg(v) = A + 2 E, deg(v) = 2 L E 0 v V v V v V

31 V. Batagelj: Network Analysis with Pajek 27 Degree distribution Random graph degree distribution, n=100000, degav=30 US Patents degree distribution freq freq 1 e+00 1 e+02 1 e+04 1 e deg deg Real-life networks are usually not random in the Erdős/Renyi sense. The analysis of their distributions gave a new view about their structure Watts (Small worlds), Barabási (nd/networks, Linked).

32 V. Batagelj: Network Analysis with Pajek 28 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, 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. For latest data see NBER / Feenstra; and for an analysis see Science / Hidalgo et. al.

33 V. Batagelj: Network Analysis with Pajek 29 Draw / Partition 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 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 Draw/Draw Partition Layout/Energy/Kamada-Kawai/Free Layout/Energy/Fruchterman Reingold/2D

34 V. Batagelj: Network Analysis with Pajek 30 Zoom in Using right button on the mouse select the zoom area. To restore the standard view select Redraw.

35 V. Batagelj: Network Analysis with Pajek 31 Fruchterman Reingold / factor = 9 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 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 Layout/Energy/Fruchterman Reingold/3D 3D picture / King

36 V. Batagelj: Network Analysis with Pajek 32 Clusters, clusterings, partitions, hierarchies A nonempty subset C V is called a cluster (group). A nonempty set of clusters C = {C i } forms a clustering. Clustering C = {C i } is a partition iff C = i C i = V and i j C i C j = Clustering C = {C i } is a hierarchy iff C i C j {, C i, C j } Hierarchy C = {C i } is complete, iff C = V; and is basic if for all v C also {v} C.

37 V. Batagelj: Network Analysis with Pajek 33 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

38 V. Batagelj: Network Analysis with Pajek 34 Pajek - shadow [0.00,1.00] Contracted clusters international trade 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 usa can cub hai dom jam mex gua hon els nic cos pan col ven ecu per bra bol par chi arg uru uki net bel lux ire fra swi spa por wge ege pol aus hun cze mat alb yug gre cyp bul rum usr swe nor den ice mli sen dah nau nir ivo gui upv sie gha tog cam nig gab car chd con zai uga ken bur rwa som eth saf maa mor alg tun sud egy syr leb irn tur irq jor sau yem kuw afg cha mon kod kor jap ind pak brm nep tha kmr lao vnd vnr mla phi ins aut nze tri ita fin lib liy isr tai sri Asia Europe Australia S. America N. America L. America Africa Snyder and Kick s international trade. Matrix display of dense networks. w(c i, C j ) = n(c i, C j ) n(c i ) n(c j )

39 V. Batagelj: Network Analysis with Pajek 35 Clustering Pajek - Ward [0.00,135.13] 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 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

40 V. Batagelj: Network Analysis with Pajek 36 The order of clusters in a hierarchy is not fixed and can be changed. Reordering clustering 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 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 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 dom zai jam 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 liy lib tri We see the typical center periphery structure.

41 V. Batagelj: Network Analysis with Pajek 37 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.

42 V. Batagelj: Network Analysis with Pajek 38 Cut-out induced subgraph: Snyder and Kick Africa sie nir gab ivo maa chd upv gui mli mor cam nig sen lib gha tun con nau saf egy liy tog alg zai car sud dah som uga eth ken rwa bur Operations/Extract from Network/Partition 6

43 V. Batagelj: Network Analysis with Pajek 39 Cut-out: Snyder and Kick Latin America : South America cos gua els ecu col mex per bra jam pan chi par hon hai nic ven dom tri arg uru bol cub Operations/Extract from Network/Partition 3,4 Pajek Operations/Transform/Remove lines/inside clusters 3,4 [Draw] Move/Grid

44 V. Batagelj: Network Analysis with Pajek 40 Walks length s of the walk s is the number of lines it contains. s = (j, h, l, g, e, f, h, l, e, c, b, a) s = 11 A walk is closed iff its initial and terminal vertex coincide. If we don t consider the direction of the lines in the walk we get a semiwalk or chain. trail walk with all lines different path walk with all vertices different cycle closed walk with all internal vertices different A graph is acyclic if it doesn t contain any cycle.

45 V. Batagelj: Network Analysis with Pajek 41 Shortest paths A shortest path from u to v is also called a geodesic from u to v. Its length is denoted by d(u, v). If there is no walk from u to v then d(u, v) =. d(j, a) = (j, h, d, c, b, a) = 5 d(a, j) = ˆd(u, v) = max(d(u, v), d(v, u)) is a distance: ˆd(v, v) = 0, ˆd(u, v) = ˆd(v, u), ˆd(u, v) ˆd(u, t) + ˆd(t, v). The diameter of a graph equals to the distance between the most distant pair of vertices: D = max u,v V d(u, v).

46 V. Batagelj: Network Analysis with Pajek 42 Shortest paths black lack back blank clack rack lace balk wack lick bank lank bask blink clank click rick race late bale walk bilk lice wick bane lane link bast clink chick rice rate hale wale bile wink bine wane line bait wast cline chink chic rite whale wile wine wait chine chit write while whine whit white DICT28.

47 V. Batagelj: Network Analysis with Pajek 43 Equivalence relations and Partitions A relation R on V is an equivalence relation iff it is reflexive v V : vrv, symmetric u, v V : urv vru, and transitive u, v, z V : urz zrv urv. Each equivalence relation determines a partition into equivalence classes [v] = {u : vru}. Each partition C determines an equivalence relation urv C C : u C v C. k-neighbors of v is the set of vertices on distance k from v, N k (v) = {u v : d(v, u) = k}. The set of all k-neighbors, k = 0, 1,... of v is a partition of V. k-neighborhood of v, N (k) (v) = {u v : d(v, u) k}.

48 1 V. Batagelj: Network Analysis with Pajek 44 Motorola s neighborhood MasterCard MasterCard Fujitsu Fujitsu MapInfo MapInfo ComputerAssociates ComputerAssociates Nextel 1 1 Nextel 1 1 iplanet iplanet 1 1 #### Sun Cisco AOL IBM GM Motorola 365 #### Sun 1 Cisco AOL 1 1 IBM 1 GM Motorola 3COM COM 1 2 MSFT 6 MSFT BaltimoreTechnologies 1 1 BaltimoreTechnologies Unisys Unisys AvantGo 1 AvantGo Yahoo! Yahoo! The thickness of edges is a square root of its value.

49 V. Batagelj: Network Analysis with Pajek 45 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.

50 V. Batagelj: Network Analysis with Pajek 46 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.

51 V. Batagelj: Network Analysis with Pajek 47 Special graphs bipartite, tree A graph G = (V, L) is bipartite iff its set of vertices V can be partitioned into two sets V 1 and V 2 such that every line from L has one end-vertex in V 1 and the other in V 2. A weakly connected graph G is a tree iff it doesn t contain loops and semicycles of length at least 3.

52 V. Batagelj: Network Analysis with Pajek 48 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).

53 V. Batagelj: Network Analysis with Pajek 49 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 Pajek Permutation - shadow [0.00,1.00] b k i e c h a d e g f #a g j l a b c f h l d g i f k k j #e i e h a b c d g i j l f k

54 V. Batagelj: Network Analysis with Pajek 50 Bow-tie structure of the Web graph Let S be the largest strong component in network N ; W the weak component containing S; I the set of vertices from which S can be reached; O the set of vertices reachable from S; T (tubes) set of vertices (not in S) on paths from I to O; R = W \(I S O T ) (tendrils); and D = V \ W. The partition {I, S, O, T, R, D} Kumar &: The Web as a graph is called the bow-tie partition of V.

55 V. Batagelj: Network Analysis with Pajek 51 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 : L 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.

56 V. Batagelj: Network Analysis with Pajek 52 Vertex-cut: Krebs Internet Industries, core=6 Lucent CommerceOne UPS Yahoo! AT&T Motorola Ariba EMC Oracle Peoplesoft Sun FoundryNetworks EDS divine WebMethods IBM HP Siebel E.piphany BMCSoftware Cisco Novell Intel ExodusComm KPNQwest CacheFlow CIBER SAP Vignette RealNetworks Loudcloud Compaq RedHat Dell Akamai Inktomi MSFT AOL Pajek Each vertex represents a company that competes in the Internet industry, 1998 do 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. Pajek

57 V. Batagelj: Network Analysis with Pajek 53 Line-cut: Krebs Internet Industries, w 3 5 Motorola CIBER AT&T Ariba Oracle Sun FoundryNetworks IBM HP Cisco Intel KPNQwest E.piphany RealNetworks Compaq Dell KPMG Akamai Inktomi MSFT AOL TerraLycos FoundryNetworks RealNetworks KPMG TerraLycos AOL Inktomi AT&T Pajek Compaq Sun HP MSFT Dell Akamai Intel IBM Cisco KPNQwest Oracle Ariba Motorola CIBER E.piphany Pajek Pajek

58 V. Batagelj: Network Analysis with Pajek 54 Line-cut in EAT File/Network/read eatrs.net Info/Network/Line values... >= 70 Net/Transform/Remove/Lines with Value/lower than 70 Net/Partitions/Degree/All Operations/Extract from Network/Partition 1-* Net/Components/Weak Draw/Draw-Partition ARMY GO GOD RUNGS SUET YOLK QUENCH PACIFIC HERS GOOD HOMEWARD HELP HIM SALVATION GET SET ALMIGHTY LADDER PUDDING EGG THIRST OCEAN HIS BAD BOUND ASSIST HER AQUAINTANCE WEST PUSSY MORSE MONEY ANSWER LOVE WINDSOR EITHER CHOCOLATE RICH BOY FRIEND EAST CAT CODE PURSE QUESTION AFFECTION CASTLE OR CADBURY S POOR GIRL DING MOO EVE TREASURE VEIN CRAVAT TIDY DOGS NOAH TRAFALGAR HE STARK WIFE DONG COW ADAM TROVE JUGULAR TIE NEAT KENNELS ARK SQUARE SHE NAKED HUSBAND PINS DOWN NOSE RESERVOIR SHOVE WRONG HONG CORE MEN HATTER MEEK ICE ADHERE NEEDLES UP NOSTRILS WATER PUSH RIGHT KONG APPLE WOMEN MAD MILD RINK STICK FISH BOYS NECK RAVE PONG NO HOLSTER ANSWERS NOOK EVER READY COMPREHEND UNTRUTH MACKEREL GIRLS NAPE RANT PING YES GUN QUESTIONS CRANNY BATTERY UNDERSTAND LIE FOOD YELLOW LOBE OFTEN ULCER SILL JIGSAW TRIANGLE NOURISHMENT JAUNDICE EAR FREQUENTLY DUODENAL WINDOW PUZZLE ISOSCELES VEHICLE TRANSFUSION FLOCK EDAM SHAGGY THANK WEARY FIR LOAF CRUSTS CAR BLOOD SHEEP CHEESE DOG YOU TIRED TREE BREAD MOTOR DONOR FLOCKS CHEDDAR BARKING ME FATIGUED ELM LOAVES OUTBOARD

59 V. Batagelj: Network Analysis with Pajek 55 We look at the components of N (t). Simple analysis using cuts 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.

60 V. Batagelj: Network Analysis with Pajek 56 Biconnectivity Vertices u and v are biconnected iff they are connected (in both directions) by two independent (no common internal vertex) paths. Biconnectivity determines a partition of the set of lines. A vertex is an articulation vertex iff its deletion increases the number of weak components in a graph. A line is a bridge iff its deletion increases the number of weak components in a graph.

61 V. Batagelj: Network Analysis with Pajek 57 k-connectivity Vertex connectivity κ of graph G is equal to the smallest number of vertices that, if deleted, induce a disconnected graph or the trivial graph K 1. Line connectivity λ of graph G is equal to the smallest number of lines that, if deleted, induce a disconnected graph or the trivial graph K 1. Whitney s inequality: κ(g) λ(g) δ(g). Graph G is (vertex) k connected, if κ(g) k and is line k connected, if λ(g) k. Whitney / Menger theorem: Graph G is vertex/line k connected iff every pair of vertices can be connected with k vertex/line internally disjoint (semi)walks.

62 V. Batagelj: Network Analysis with Pajek 58 Triangular and short cycle connectivities In an undirected graph we call a triangle a subgraph isomorphic to K 3. A sequence (T 1, T 2,..., T s ) of triangles of G (vertex) triangularly connects vertices u, v V iff u T 1 and v T s or u T s and v T 1 and V(T i 1 ) V(T i ), i = 2,... s. It edge triangularly connects vertices u, v V iff a stronger version of the second condition holds E(T i 1 ) E(T i ), i = 2,... s. Vertex triangular connectivity is an equivalence on V; and edge triangular connectivity is an equivalence on E. See the paper.

63 V. Batagelj: Network Analysis with Pajek 59 Triangular network Let G be a simple undirected graph. A triangular network N T (G) = (V, E T, w) determined by G is a subgraph G T = (V, E T ) of G which set of edges E T consists of all triangular edges of E(G). For e E T the weight w(e) equals to the number of different triangles in G to which e belongs. Triangular networks can be used to efficiently identify dense clique-like parts of a graph. If an edge e belongs to a k-clique in G then w(e) k 2.

64 V. Batagelj: Network Analysis with Pajek 60 Edge-cut at level 16 of triangular network of Erdős collaboration graph WORMALD, NICHOLAS C. LASKAR, RENU C. SHELAH, SAHARON MCKAY, BRENDAN D. HEDETNIEMI, STEPHEN T. MAGIDOR, MENACHEM KLEITMAN, DANIEL J. SAKS, MICHAEL E. CHUNG, FAN RONG K. GRAHAM, RONALD L. ARONOV, BORIS LINIAL, NATHAN PACH, JANOS POLLACK, RICHARD M. HENNING, MICHAEL A. FRANKL, PETER SPENCER, JOEL H. ALON, NOGA OELLERMANN, ORTRUD R. LOVASZ, LASZLO KOMLOS, JANOS GODDARD, WAYNE D. FUREDI, ZOLTAN TUZA, ZSOLT ALAVI, YOUSEF BABAI, LASZLO SZEMEREDI, ENDRE CHARTRAND, GARY BOLLOBAS, BELA HARARY, FRANK AJTAI, MIKLOS KUBICKI, GRZEGORZ SCHWENK, ALLEN JOHN without Erdős, n = 6926, m = RODL, VOJTECH NESETRIL, JAROSLAV ROSA, ALEXANDER GYARFAS, ANDRAS SCHELP, RICHARD H. STINSON, DOUGLAS ROBERT LEHEL, JENO CHEN, GUANTAO MULLIN, RONALD C. FAUDREE, RALPH J. COLBOURN, CHARLES J. JACOBSON, MICHAEL S. PHELPS, KEVIN T.

65 V. Batagelj: Network Analysis with Pajek 61 Triangular connectivity in directed graphs If the graph G is mixed we replace edges with pairs of opposite arcs. In the following let G = (V, A) be a simple directed graph without loops. For a selected arc (u, v) A there are only two different types of directed triangles: cyclic and transitive. cyc tra For each type we get the corresponding triangular network N cyc and N tra. The notion of triangular connectivity can be extended to the notion of short (semi) cycle connectivity.

66 V. Batagelj: Network Analysis with Pajek 62 Arc-cut at level 11 of transitive triangular network of ODLIS dictionary American Library Directory transaction log publication periodical serial suggestion box charge library call number review series Library Literature issue frequency colophon journal layout fixed location publishing printing blanket order American Library Association /ALA/ title page Books in Print /BIP/ vendor homepage International Standard Book Number /ISBN/ entry round table published price dummy librarian condition edition catalog plate fiction Oak Knoll bibliographic record imprint abstract dust jacket work book bibliography half-title editor library binding title table of contents /TOC/ index invoice new book text endpaper copyright book size parts of a book front matter collation publisher binding folio cover page Pajek

67 V. Batagelj: Network Analysis with Pajek 63 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. We developed very efficient algorithms to determine the islands hierarchy and to list all the islands of selected sizes. An island is simple iff it has a single peak. See details.

68 V. Batagelj: Network Analysis with Pajek Islands Islands are very general and efficient approach to determine the important subnetworks in a given network. We have to express the goals of our analysis with a related property of the vertices or weight of the lines. Using this property we determine the islands of an appropriate size (in the interval k to K). In large networks we can get many islands which we have to inspect individually and interpret their content. An important property of the islands is that they identify locally important subnetworks on different levels. Therefore they detect also emerging groups.

69 V. Batagelj: Network Analysis with Pajek 65 Islands - Reuters terror news call phone cell help plea louisiana air_force car arab base barksdale smoke man air attendant airline flight arabic-language dept fire rental suspect space contain case offutt nebraska scare plaugher chief edward united_airlines passenger american_airlines aid chemical bin_laden inhale dissident necessary firefighter taliban jet fbi afghanistan airport police anthrax knife-wielding agent world force skin thursday commercial official pakistan officer airliner hijacker special support terrorism country war trace hijack united_states embassy jonn deadly business mighty plane wednesday twin week strike east attack act specialist military 110-story edmund world_trade_ctr headquarters pentagon terrorist the_worst apparent action cheyenne north tower washington morning africa tuesday south anti-american late news wyoming terror florida city new_york group responsibility mayor pfc conference american mayor_giuliani team landmark exchange debris miss people effort train manual rescue postal pilot member worker service uniform power stock emergency state nuclear market plant weapon financial district center saudi saudi-born pakistani organization large newspaper buildng boston herald leader thousand toll death congressional 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 = vertices and m = edges. Pajek

70 V. Batagelj: Network Analysis with Pajek 66 Islands The Edinburgh Associative Thesaurus n = 23219, m = , transitivity weight CHAMPIONSHIP DISTRAUGHT DESPAIR PLAYING TEAM FIVES MISFORTUNE UNHAPPINESS GRIEF DISAPPOINTMENT GOALIE SOCCER PLAYER GAME FOOTBALL SPORT BALL RUGBY TENNIS RUGGER NETBALL GAMES BADMINTON DEPRESSED MOOD MISERY JOY SADNESS SORROW SAD HAPPY JOYFUL HAPPINESS LAUGHTER CONTENTMENT REFEREE BASKETBALL SHORTS HOCKEY GYM BALLS ENJOYMENT LAUGH MERRIMENT ROADS TRUCK LORRIES DRIVERS BIKE BICYCLE MOTOR CYCLE BICYCLES BIKES WHISKY LIQUOR SPIRITS BEER DRUNK KEG BITTER RAILWAYS COACH TRAIN VAN CAR STREET MACHINES LANE SIPPING BRANDY GIN DRINK WHISKEY BEER-MUG RAILWAY MOVING MOBILE STOPPING ROAD SIP DRY MARTINI STOP PATH LEMONADE

71 V. Batagelj: Network Analysis with Pajek Islands The Edinburgh Associative Thesaurus YET NOTWITHSTANDING JUST HAPPEN AGAIN ALREADY MONIES COINS PAYMENT PAID PAY RECEIPT STOLE UNPAID THRIFTY HAPPENED INCREASE MONEY DEALER NEVERTHELESS NOW DEFICIT THRIFT LOOT PROPERTY WHY BUT AS OFTEN BELIEVE NO MORE MONEY OFFER PROVIDE REPAY PLEASE THEREFORE ANYWAY SOON PROBABLY NOT REFUSE MEANWHILE SOMETIME COULD POWERFUL INHUMAN BOSS TEACHING ENGINEERING LECTURER STUDYING TEACHER RESPONSIBLE PROFESSION MAN CHAIRMAN HAIRY ELEGANCE LEARNING SCHOOL EDUCATION MYSTERIOUS BELOVED CHARM NICE HOMEWORK WORK LECTURES TRAINING LOVE GIRL ATTRACTIVE SCIENTIFIC SCIENCE RESEARCH STUDY MATHS LESSONS KINDNESS FLIRT ADORABLE DELIGHTFUL BEAUTIFUL LOVELY ACTIVITY SHAPELY

72 V. Batagelj: Network Analysis with Pajek 68 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.

73 V. Batagelj: Network Analysis with Pajek 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.

74 V. Batagelj: Network Analysis with Pajek 70 Normalization Let p : V R be an index in network N = (V, L, p). If we want to compare indices p over different networks we have to make them comparable. Usually we try to achieve this by normalization of p. Let N N(V), where N(V) is a selected family of networks over the same set of vertices V, p max = max max p N (v) and p min = min min p N (v) N N(V) v V N N(V) v V then we define the normalized index as p (v) = p(v) p min p max p min [0, 1]

75 V. Batagelj: Network Analysis with Pajek 71 Degrees The simplest index are the degrees of vertices. Since for simple networks deg min = 0 and deg max = n 1, the corresponding normalized indices are centrality deg (v) = deg(v) n 1 and similary support indeg (v) = indeg(v) n influence outdeg (v) = outdeg(v) n Instead of degrees in original network we can consider also the degrees with respect to the reachability relation (transitive closure).

76 V. Batagelj: Network Analysis with Pajek 72 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) = max u V d(v, u) total closeness S(v) = u V 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 r max and S max equal. Sabidussi (1966) introduced a related measure 1/S(v); or in its normalized form closeness cl(v) = u V n 1 d(v, u) D = max u,v V d(v, u) is called the diameter of network.

77 V. Batagelj: Network Analysis with Pajek 73 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) u,t V:g u,t >0 u v,t v,u t g u,t (v) g u,t where g u,t is the number of geodesics from u to t; and g u,t (v) is the number of those among them that pass through vertex v. If we know matrices [d u,v ] and [g u,v ] we can determine also g u,v (t) by: g u,t g t,v d u,t + d t,v = d u,v g u,v (t) = 0 otherwise For computation of geodesic matrix see Brandes.

78 V. Batagelj: Network Analysis with Pajek 74 Hubs and authorities To each vertex v of a network N = (V, L) we assign two values: quality of its content (authority) x v and quality of its references (hub) y v. A good authority is selected by good hubs; and good hub points to good authorities (see Kleinberg). x v = y u and y v = u:(u,v) L u:(v,u) L 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 = W T 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 W T W; and y of matrix WW T. x u

79 V. Batagelj: Network Analysis with Pajek 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, theory. Examples: Krebs, Krempl. Net/Paths between 2 vertices/diameter Net/Vector/Centrality/Closeness Net/Vector/Centrality/Betweeness Net/Vector/Important Vertices/1-Mode:Hubs-Authorities Net/Vector/Clustering Coefficients

80 V. Batagelj: Network Analysis with Pajek 76 Clustering 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: C(v) = 2 L(G 1 (v)) deg(v)(deg(v) 1) deg(v) > 1 0 otherwise We can consider also the size of vertex neighborhood by the following correction C 1 (v) = deg(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.

81 V. Batagelj: Network Analysis with Pajek 77 Network centralization measures Network centralization measures measure the extent to which the network supports/is dominated by a single node. They are usually constructed combining the corresponding node values. There are two general approaches. Extremal approach: Let p : V R be an index in network N = (V, L). We introduce the quantities p = max v V p(v) D = v V(p p(v)) D = max{d(n ) : N is a network on V} Then we can define p max (N ) = D(N ) D.

82 V. Batagelj: Network Analysis with Pajek Network centralization measures It can be shown that for simple graphs measure p general symmetric D degree/all deg (v) n 1 n 1 degree/irreflexive deg i(v) (n 1) 2 /n n 2 closeness cl(v) n 1 (n 1)(n 2)/(2n 3) betweenness b(v) n 1 and that among connected graphs in all these cases the star is the maximal graph (p max = 1) and the complete graph is the minimal graph (p max = 0).

83 V. Batagelj: Network Analysis with Pajek Network centralization measures Variational approach The other approach is based on the variance. First we compute the average node centrality p = 1 p(v) n and then define p V (N ) = 1 n v V (p(v) p) 2 v V

84 V. Batagelj: Network Analysis with Pajek 80 Citation networks In a given set of units/vertices U (articles, books, works, etc.) we introduce a citing relation/set of arcs R U U urv v cites u which determines a citation network N = (U, R). A citing relation is usually irreflexive (no loops) and (almost) acyclic. We shall assume that it has these two properties. Since in real-life citation networks the strong components are small (usually 2 or 3 vertices) we can transform such network into an acyclic network by shrinking strong components and deleting loops. Other approaches exist. It is also useful to transform a citation network to its standardized form by adding a common source vertex s / U and a common sink vertex t / U. The source s is linked by an arc to all minimal elements of R; and all maximal elements of R are linked to the sink t. We add also the feedback arc (t, s).

85 V. Batagelj: Network Analysis with Pajek 81 Search path count method The search path count (SPC) method is based on counters n(u, v) that count the number of different paths from s to t through the arc (u, v). To compute n(u, v) we introduce two auxiliary quantities: n (v) counts the number of different paths from s to v, and n + (v) counts the number of different paths from v to t. Then n(u, v) = n (u) n + (v), (u, v) R. After the topological sort of the graph we can compute, using the above relations in topological order, the weights in time of order O(m), m = R. t c (u) = n (u) n + (u)

86 V. Batagelj: Network Analysis with Pajek 82 Properties of SPC weights The values of counters n(u, v) form a flow in the citation network the Kirchoff s vertex law holds: For every vertex u in a standardized citation network incoming flow = outgoing flow: n(v, u) = n(u, v) = n (u) n + (u) v:vru v:urv The weight n(t, s) equals to the total flow through network and provides a natural normalization of weights w(u, v) = n(u, v) n(t, s) 0 w(u, v) 1 and if C is a minimal arc-cut-set (u,v) C w(u, v) = 1. In large networks the values of weights can grow very large. This should be considered in the implementation of the algorithms.

87 V. Batagelj: Network Analysis with Pajek 83 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 COOPER-LN-1979-V33-P9 AMARI-SI-1977-V26-P175 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 GROSSBERG-S-1988-V1-P17 HECHTNIELSEN-R-1987-V26-P1892 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 GLUCK-MA-1997-V48-P481 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, arcs). See paper. ASHBY-FG-1999-V6-P363 Pajek

88 V. Batagelj: Network Analysis with Pajek 84 Islands US patents As an example, let us look at Nber network of US Patents. It has vertices and arcs (1 loop). We computed SPC weights in it and determined all (2,90)-islands. The reduced network has vertices, arcs and for different k: C 2 =187610, C 5 =8859,C 30 =101, C 50 =30 islands. Rolex [1] [11] [21] [31] [41] [51] [61] [71] [81]

89 V. Batagelj: Network Analysis with Pajek 85 IslandTheme size size distribution freq size

90 V. Batagelj: Network Analysis with Pajek 86 Main path and main island of Patents

91 V. Batagelj: Network Analysis with Pajek 87 Liquid crystal display Table 1: Patents on the liquid-crystal display Table 2: Patents on the liquid-crystal display Table 3: Patents on the liquid-crystal display patent 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 date author(s) and title Apr 11, 1978 Oh. Nematic liquid crystal compositions Sep 12, 1978 Coates, et al. Liquid crystalline materials Oct 3, 1978 Krause, et al. Liquid crystalline materials of reduced viscosity Dec 19, 1978 Eidenschink, et al. Liquid crystalline cyclohexane derivatives Apr 17, 1979 Gray, et al. Optically active liquid crystal mixtures and liquid crystal devices containing them May 15, 1979 Eidenschink, et al. Liquid crystalline hexahydroterphenyl derivatives Apr 1, 1980 Coates, et al. Liquid crystal compounds Apr 15, 1980 Boller, et al. Liquid crystal mixtures May 13, 1980 Sato, et al. Nematic liquid crystalline materials Oct 21, 1980 Krause, et al. Liquid crystalline cyclohexane derivatives Apr 14, 1981 Gray, et al. Liquid crystal compounds and materials and devices containing them Sep 22, 1981 Kanbe. Ester compound Oct 6, 1981 Deutscher, et al. Liquid crystal compounds Nov 24, 1981 Eidenschink, et al. Fluorophenylcyclohexanes, the preparation thereof and their use as components of liquid crystal dielectrics May 18, 1982 Eidenschink, et al. Cyclohexylbiphenyls, their preparation and use in dielectrics and electrooptical display elements Jul 20, 1982 Sugimori. Halogenated ester derivatives Sep 14, 1982 Osman, et al. Cyclohexylcyclohexanoates Nov 2, 1982 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 Nov 30, 1982 Osman, et al. Anisotropic cyclohexyl cyclohexylmethyl ethers Jan 11, 1983 Osman. Anisotropic compounds with negative or positive DC-anisotropy and low optical anisotropy May 31, 1983 Krause, et al. Liquid crystalline naphthalene derivatives Jun 7, 1983 Fukui, et al. 4-(Trans-4 -alkylcyclohexyl) benzoic acid 4 -cyano-4 -biphenylyl esters Jun 7, 1983 Sugimori, et al. Trans-4-(trans-4 -alkylcyclohexyl)-cyclohexane carboxylic acid 4 -cyanobiphenyl ester Aug 23, 1983 Romer, et al. Liquid crystalline cyclohexylphenyl derivatives Nov 15, 1983 Eidenschink, et al. Liquid crystalline fluorine-containing cyclohexylbiphenyls and dielectrics and electro-optical display elements based thereon Dec 6, 1983 Praefcke, et al. Liquid crystalline cyclohexylcarbonitrile derivatives Dec 27, 1983 Sugimori, et al. Liquid crystal benzene derivatives Jun 19, 1984 Takatsu, et al. Nematic halogen Compound Jun 26, 1984 Christie, et al. Bismaleimide triazine composition Jul 17, 1984 Petrzilka, et al. Liquid crystal mixture Sep 18, 1984 Sugimori, et al. High temperature liquid crystal substances of four rings and liquid crystal compositions containing the same Sep 18, 1984 Takatsu, et al. Nematic liquid crystalline compounds Oct 30, 1984 Takatsu, et al. Nematic liquid crystalline compounds Mar 5, 1985 Sugimori, et al. High temperature liquid-crystalline ester compounds Apr 9, 1985 Eidenschink, et al. Cyclohexane derivatives patent 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

92 V. Batagelj: Network Analysis with Pajek 88 Genealogies Another example of acyclic networks are genealogies. There are different network representations of genealogies: grandson grandson m-grandmother m-grandfather f-grandfather f-grandmother son & daughter-in-law son-in-law & daughter son & daughter-in-law son-in-law & daughter daughter-in-law son daughter son-in-law mother father stepmother I & wife brother & sister-in-law sister I & wife brother & sister-in-law sister wife I brother sister-in-law sister-in-law brother I wife sister father & stepmother father & mother daughter-in-law son son-in-law daughter father & stepmother father & mother stepmother father mother f-grandfather & f-grandmother m-grandfather & m-grandmother grandson f-grandfather & f-grandmother m-grandfather & m-grandmother f-grandfather f-grandmother m-grandfather m-grandmother Ore graph, p-graph, and bipartite p-graph paper

93 V. Batagelj: Network Analysis with Pajek 89 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/ Damianus/Georgio/ Legnussa/Babalio/ Nicolinus/Gondola/ Franussa/Bona/ Marin/Gondola/ Magdalena/Grede/ Junius/Georgio/ Anucla/Zrieva/ Nicola/Ragnina/ Nicoleta/Zrieva/ Junius/Zrieva/ Margarita/Bona/ Sarachin/Bona/ Nicoletta/Gondola/ Marinus/Bona/ Phylippa/Mence/ Marinus/Zrieva/ Maria/Ragnina/ Lorenzo/Ragnina/ Slavussa/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.

94 V. Batagelj: Network Analysis with Pajek Pattern searching To speed up the search or to consider some additional properties of the pattern, a user can set some additional options: vertices in network should match with vertices in pattern in some nominal, ordinal or numerical property (for example, type of atom in molecula); values of edges must match (for example, edges representing male/female links in the case of p-graphs); the first vertex in the pattern can be selected only from a given subset of vertices in the network.

95 V. Batagelj: Network Analysis with Pajek 91 A2 A4.1 A5.1 A6.1 B6.1 Relinking patterns in p-graphs A3 B4 A4.2 A5.2 B5 A6.3 A6.2 C6 B6.4 B6.2 B6.3 All possible relinking marriages in p- graphs with 2 to 6 vertices. Patterns are labeled as follows: 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). 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.

96 V. Batagelj: Network Analysis with Pajek 92 Frequencies normalized with number of couples in p-graph pattern Loka Silba Ragusa Turcs Royal A A A B A A A B A A A C B B B B Sum 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.

97 V. Batagelj: Network Analysis with Pajek 93 Bipartite p-graphs: Marriage among half-cousins Benjamin Simoniti & Natalija Mavric Benjamin Simoniti Anton Simoniti & Jozefa Mavric Natalija Mavric Alojz Mavric & Angela Zuljan Jozefa Mavric Josip Mavric & Marjuta Zamar Alojz Mavric Josip Mavric & Rezka Zamar Josip Mavric

98 V. Batagelj: Network Analysis with Pajek 94 Triads D 5-021U 6-021C 7-111D 8-111U 9-030T C D Let G = (V, R) be a simple directed graph without loops. A triad is a subgraph induced by a given set of three vertices. There are 16 nonisomorphic (types of) triads. They can be partitioned into three basic types: the null triad 003; dyadic triads 012 and 102; and connected triads: 111D, 201, 210, 300, 021D, 111U, 120D, 021U, 030T, 120U, 021C, 030C and 120C U C

99 V. Batagelj: Network Analysis with Pajek 95 Triadic spectrum Moody Several properties of a graph can be expressed in terms of its triadic spectrum distribution of all its triads. It also provides ingredients for p network models. A direct approach to determine the triadic spectrum is of order O(n 3 ); but in most large graphs it can be determined much faster.

100 V. Batagelj: Network Analysis with Pajek 96 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 K k ), 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.

101 V. Batagelj: Network Analysis with Pajek 97 Cores and generalized cores The notion of core was introduced by Seidman in 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 : deg H (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.

102 V. Batagelj: Network Analysis with Pajek 98 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 = H j H i 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.

103 V. Batagelj: Network Analysis with Pajek Properties of cores The cores, because they can be determined very efficiently, are one among few concepts that provide us with meaningful decompositions of large networks. We expect that different approaches to the analysis of large networks can be built on this basis. For example: we get the following bound on the chromatic number of a given graph G χ(g) 1 + core(g) Cores can also be used to localize the search for interesting subnetworks in large networks since: if it exists, a k-component is contained in a k-core; and a k-clique is contained in a k-core. For details see the paper.

104 V. Batagelj: Network Analysis with Pajek core of Krebs Internet industries

105 V. Batagelj: Network Analysis with Pajek 101 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 p-function for short, is a function p(v, U), v V, U V with real values. Let N U (v) = N(v) U. Besides degrees and (corrected) clustering coefficient, here are some examples of p-functions: p S (v, U) = w(v, u), where w : E R + 0 u N U (v) p M (v, U) = max w(v, u), where w : E R u N U (v) p k (v, U) = 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.

106 V. Batagelj: Network Analysis with Pajek 102 Additional p-functions relative density deg(v, C) p γ (v, C) =, if deg(v) > 0; 0, otherwise max u N(v) deg(u) diversity p δ (v, C) = max deg(u) u N + (v,c) average weight 1 p a (v, C) = N(v, C) u N(v,C) min deg(u) u N + (v,c) w(v, u), if N(v, C) ; 0, otherwise

107 V. Batagelj: Network Analysis with Pajek 103 Generalized cores algorithm The function p is monotone iff it has the property C 1 C 2 v V : (p(v, C 1 ) p(v, C 2 )) The degrees and the functions p S, p M and p k 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 t 1 < t 2 H t2 H t1 The p-function is local iff p(v, U) = p(v, N U (v)). The degrees, p S and p M are local; but p k 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, N C (v)) can be computed in O(deg C (v)). For details see the paper.

108 V. Batagelj: Network Analysis with Pajek 104 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

109 V. Batagelj: Network Analysis with Pajek 105 Cores of orders in Computational Geometry C.Zelle H.A.El-Gindy S.P.Fekete M.E.Houle J.Czyzowicz M.L.Demaine P.Belleville V.Sacristan D.H.Rappaport K.R.Romanik I.Streinu H.Everett F.Hurtado H.Meijer D.Bremner A.Lubiw G.Liotta T.C.Shermer B.Zhu D.M.Avis W.J.Lenhart S.S.Skiena D.M.Mount T.C.Biedl P.K.Bose M.J.vanKreveld E.M.Arkin G.T.Toussaint S.M.Robbins J.Urrutia J.S.B.Mitchell G.T.Wilfong M.Yvinec S.H.Whitesides O.Aichholzer J-M.Robert O.Devillers E.D.Demaine M.T.deBerg Te.Asano S.Lazard N.Katoh D.L.Souvaine I.G.Tollis T.Roos M.Teillaud M.H.Overmars J-R.Sack H.Alt G.Rote H.Imai R.Seidel J.O Rourke M.T.Goodrich D.Halperin J-D.Boissonnat J.Erickson S.Suri J.S.Vitter M.Sharir N.M.Amato K.Mehlhorn R.Pollack B.Chazelle D.Z.Chen S-W.Cheng R.Wenger J.S.Snoeyink O.Schwarzkopf J.E.Hershberger P.K.Agarwal R.Tamassia R.L.S.Drysdale J.Pach F.P.Preparata Q.Huang E.Welzl L.J.Guibas K.Kedem S.J.Fortune J.C.Clements J.Matousek D.Eppstein C-K.Yap D.G.Kirkpatrick H.S.Sawhney S.A.Mitchell B.Aronov J.Ashley D.White E.Trimble H.Edelsbrunner D.P.Dobkin F.Aurenhammer D.T.Lee M.W.Bern D.Steele W.J.Bohnhoff R.R.Lober L.P.Chew A.Aggarwal G.D.Sjaardema S.R.Kosaraju T.K.Dey N.Amenta P.Yanker D.Petkovic G.Lerman T.J.Wilson L.Lopez-Buriek J.R.Hipp P.Plassmann J.Hass E.Sedgwick C.K.Johnson M.Flickner M.Gorkani W.R.Oakes J.Harer D.Letscher C.Grimm S.Parker T.J.Tautges A.Hicks D.Zorin J.Weeks J.Hafner T.L.Edwards W.Niblack S.E.Benzley T.D.Blacker M.Whitely B.Dom

110 V. Batagelj: Network Analysis with Pajek 106 p S -core at level 46 of Geombib network E.Arkin M.Bern D.Eppstein J.Mitchell I.Tollis A.Garg L.Vismara M.Goodrich G.diBattista R.Tamassia D.Dobkin S.Suri J.O Rourke J.Vitter G.Liotta J.Hershberger B.Chazelle R.Seidel B.Aronov L.Guibas J.Snoeyink H.Edelsbrunner M.Sharir P.Agarwal R.Pollack J.Pach D.Halperin F.Preparata P.Gupta E.Welzl J.Matousek C.Yap M.Overmars M.vanKreveld M.deBerg O.Schwarzkopf G.Toussaint P.Bose J.Boissonnat O.Devillers M.Yvinec M.Teillaud M.Smid R.Janardan J.Majhi J.Schwerdt J.Urrutia J.Czyzowicz C.Icking R.Klein

111 V. Batagelj: Network Analysis with Pajek 107 Analysis of two-mode networks A two-mode network or affiliation network is a structure N = (U, V, A, w), where U and V are disjoint sets of vertices, A is the set of arcs with the initial vertex in the set U and the terminal vertex in the set V, and w : A R is a weight. If no weight is defined we can assume a constant weight w(u, v) = 1 for all arcs (u, v) A. The set A can be viewed also as a relation A U V. A two-mode network can be formally represented by rectangular matrix W = [w uv ] U V. w uv = { w(u, v) (u, v) A 0 otherwise Two new direct methods will be presented in this lecture: two-mode cores and 4-rings.

112 V. Batagelj: Network Analysis with Pajek 108 Internet Movie Database 12th Annual Graph Drawing Contest, The IMDB network is two-mode and has = vertices and arcs.

113 V. Batagelj: Network Analysis with Pajek 109 Two-mode cores The subset of vertices C V is a (p, q)-core in a two-mode network N = (V 1, V 2 ; L), V = V 1 V 2 iff a. in the induced subnetwork K = (C 1, C 2 ; L(C)), C 1 = C V 1, C 2 = C V 2 it holds v C 1 : deg K (v) p and v C 2 : deg K (v) q ; b. C is the maximal subset of V satisfying condition a. It holds C(0, 0) = V and (p 1 p 2 ) (q 1 q 2 ) C(p 1, q 1 ) C(p 2, q 2 ) To determine a (p, q)-core the procedure similar to the ordinary core procedure can be used. It can be implemented to run in O(m) time.

114 V. Batagelj: Network Analysis with Pajek 110 Table (p, q : n 1, n 2 ) for Internet Movie Database Interesting (p, q)-cores? Table of cores characteristics n 1 = C 1 (p, q), n 2 = C 2 (p, q) n 1 + n 2 selected threshold or the border line in the (p, q)-table : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 28 76

115 V. Batagelj: Network Analysis with Pajek 111 (247,2)-core and (27,22)-core Survivor Series Royal Rumble 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 Garc a, 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 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

116 V. Batagelj: Network Analysis with Pajek 112 (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... 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117 V. Batagelj: Network Analysis with Pajek rings and analysis of two-mode networks In two-mode network there are no 3-rings. The densest substructures are complete bipartite subgraphs K p,q. They contain many 4-rings. There are ( p 2 )( ) q 2 = 1 p(p 1)q(q 1) 4 4-rings in K p,q ; and each of its edges e has weight w 4 (e) = (p 1)(q 1) The 4-rings weights were implemented in Pajek in August 2005.

118 V. Batagelj: Network Analysis with Pajek 114 There are 4 types of directed 4-rings: 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.

119 V. Batagelj: Network Analysis with Pajek 115 Directed 5-rings In the future we intend to implement in Pajek also weights w 5. Again there are only 4 types of directed 5-rings. cyclic transitive????????

120 V. Batagelj: Network Analysis with Pajek 116 Simple line islands in IMDB for w 4 We obtained simple line islands on vertices. Here is their size distribution. Size Freq Size Freq Size Freq Size Freq

121 V. Batagelj: Network Analysis with Pajek 117 Example: Islands for w 4 / Charlie Brown and Adult Kesten, Brad Brando, Kevin Schoenberg, Jeremy Robbins, Peter (I) Hauer, Brent Shea, Christopher (I) Charlie Brown and Snoopy Show Altieri, Ann Reilly, Earl Rocky Ornstein, Geoffrey Charlie Brown Celebration 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 Mendelson, Karen Be My Valentine, Charlie Brown Stratford, Tracy Dryer, Sally It s Magic, Charlie Brown Melendez, Bill You re a Good Sport, Charlie Brown Boy Named 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? There s No Time for Love, Charlie Brown Charlie Brown Thanksgiving You re Not Elected, Charlie Brown Snoopy Come Home It s the Easter Beagle, Charlie Brown Boy, T.T. North, Peter (I) Byron, Tom Wallice, Marc Morgan, Jonathan (I) Davis, Mark (V) Voyeur, Vince Dough, Jon Sanders, Alex (I) Michaels, Sean Horner, Mike Drake, Steve (I) Silvera, Joey West, Randy (I) Jeremy, Ron Savage, Herschel Shea, Stephen Thomas, Paul (I) Pajek Pajek

122 V. Batagelj: Network Analysis with Pajek 118 Example: Islands for w 4 / Mark Twain and Abid Sergeant Madden Sawak nus el lail Honky Tonk Roaring Twenties, The Hoodlum Saint, The Unconquered Malak el zalem, El Rostom, Hind Soltan, Hoda Fatawa, El Union Pacific Phelps, Lee (I) Big City Flavin, James Saum, Cliff Wells Fargo Star Is Born, A Dunn, Ralph You Can t Take It with You San Quentin Vogan, Emmett Chandler, Eddy Flowers, Bess O Connor, Frank (I) Whole Town s Talking, The Nancy Drew... Reporter Sullivan, Charles (I) Dust Be My Destiny Castle on the Hudson Racket Busters Go Getter, The Women in the Wind Man Who Talked Too Much, The Yankee Doodle Dandy Kid From Kokomo, The They Drive by Night Secret Service of the Air Knockout Meet John Doe Holmes, Stuart Valley of the Giants Kid Galahad They Made Me a Criminal Mower, Jack Naughty But Nice King of the Underworld Bad Men of Missouri Adventures of Mark Twain, The Smashing the Money Ring Hamdi, Imad Pajek Ard el ahlam Sarhan, Shukry Port Said Riad, Hussein Tarik el saada Hub fil zalam Hamama, Faten El Dekn, Tewfik Shawqi, Farid Baad al wedah Massiada, Al Asrar el naas Baba Amin Beyt al Taa Haked, El Osta Hassan, El Ibn al ajar Ana bint min? Murra kulshi, El Mohtal, El Zalamuni el habaieb Ashki limin? Ana zanbi eh? Sittat afarit, al- Fatat el mina Hareb min el ayyam Abu Hadid Elf laila wa laila Souk el selah Nashal, El Maktub alal guebin Fatawat el Husseinia Amir el antikam Abid el gassad Ghaltet ab Abu Dahab Aguazet seif Hamida Batal lil nehaya Namrud, El Ebn el-hetta Nassab, El Zoj el azeb, El Abid el mal Cass el azab Ghazal al-banat Rasif rakam khamsa Laab bil nar, El Iskanderija... lih? Imlak, El Matloub zawja fawran El-Meliguy, Mahmoud Abu Ahmad

123 V. Batagelj: Network Analysis with Pajek 119 Example: Island for w 4 / Polizeiruf 110 and Starkes Team Maranow, Maja Aff re Semmeling, Die Martens, Florian Starkes Team - Erbarmungslos, Ein Starkes Team - M rderisches Wiedersehen, Ein Lansink, Leonard Starkes Team - Roter Schnee, Ein Bademsoy, Tayfun Starkes Team - Der Verdacht, Ein Starkes Team, Ein Starkes Team - Eins zu Eins, Ein Starkes Team - Kollege M rder, Ein Starkes Team - Sicherheitsstufe 1, Ein Starkes Team - Das Bombenspiel, Ein Starkes Team - Blutsbande, Ein Starkes Team - T dliche Rache, Ein Starkes Team - Der Mann, den ich hasse, Ein Starkes Team - Kindertr ume, Ein Starkes Team - Auge um Auge, Ein Schwarz, Jaecki Starkes Team - Lug und Trug, Ein Starkes Team - Der letzte Kampf, Ein Starkes Team - Kleine Fische, gro e Fische, Ein Starkes Team - Der Todfeind, Ein Starkes Team - Mordlust, Ein Starkes Team - Das gro e Schweigen, Winkler, Wolfgang Ein Starkes Team - Der sch ne Tod, Ein Starkes Team - Tr ume und L gen, Ein Starkes Team - Bankraub, Ein Starkes Team - Verraten und verkauft, Ein Starkes Team - Braunauge, Ein Starkes Team - Im Visier des M rders, Ein Starkes Team - Die Natter, Ein Polizeiruf Ein Bild von einem M rder Polizeiruf Kopf in der Schlinge Polizeiruf Zerst rte Tr ume Polizeiruf Angst um Tessa B low Polizeiruf Rosentod Polizeiruf Doktorspiele Polizeiruf Jugendwahn Polizeiruf Hei kalte Liebe Polizeiruf Todsicher Polizeiruf Der Spieler Polizeiruf Mordsfreunde Polizeiruf Kurschatten Polizeiruf Tote erben nicht Polizeiruf Der Pferdem rder Polizeiruf Henkersmahlzeit Lerche, Arnfried Pajek

124 V. Batagelj: Network Analysis with Pajek 120 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 = [w i,j ] with elements: w i,j = w(i, j) for (i, j) E and w i,j = 0 otherwise. Given a pair of compatible networks N A = (I, K, E A, w A ) and N B = (K, J, E B, w B ) with corresponding matrices A I K and B K J we call a product of networks N A and N B a network N C = (I, J, E C, w C ), where E C = {(i, j) : i I, j J, c i,j 0} and w C (i, j) = c i,j for (i, j) E C. The product matrix C = [c i,j ] I J = A B is defined in the standard way c i,j = a i,k b k,j k K In the case when I = K = J we are dealing with ordinary one-mode networks (with square matrices).

125 V. Batagelj: Network Analysis with Pajek 121 Fast sparse matrix multiplication The standard matrix multiplication has the complexity O( I K J ) it is too slow to be used for large networks. For sparse large networks we can multiply faster considering only nonzero elements: for k in K do for i in N A (k) do for j in N B (k) do if c i,j then c i,j := c i,j + a i,k b k,j else new c i,j := a i,k b k,j N A (k): neighbors of vertex k in network N A N B (k): neighbors of vertex k in network N B In general the multiplication of large sparse networks is a dangerous operation since the result can explode it is not sparse.

126 V. Batagelj: Network Analysis with Pajek 122 Complexity of fast sparse matrix multiplication It is easy to see: If at least one of the sparse networks N A and N B has small maximal degree on K then also the resulting product network N C is sparse. A more detailed complexity analysis gives: Let d min (k) = min(deg A (k), deg B (k)), min = max k K d min (k), d max (k) = max(deg A (k), deg B (k)), K(d) = {k K : d max (k) d}, d = argmin d ( K(d) d) and K = K(d ). If for the sparse networks N A and N B the quantities min and d are small then also the resulting product network N C is sparse.

127 V. Batagelj: Network Analysis with Pajek 123 Example: Derived kinship relations From the genealogy we get the relations: F: is a father of, M: is a mother of, E: is a spouse of, and to distinguish between male and female: L: is a male / 1-male, 0-female, and J: is a female / 1-female, 0-male. Other basic relations can be obtained using macros based on identities: is a parent of P = F M is a child of C = P T is a son of S = L C is a daughter of D = J C is a husband of H = L E is a wife of W = J E is a sibling of G = ((F T F ) (M T M)) \ I is a brother of B = L G is a sister of Z = J G is an uncle of U = B P is an aunt of A = Z P is a semi-sibling of G e = (P T P ) \ I and using them other relations can be determined is a grand mother of M 2 = M P is a niece of Ni = D G

128 V. Batagelj: Network Analysis with Pajek 124 Two-mode network analysis by conversion to one-mode network Often we transform a two-mode network N = (U, V, E, w) into an ordinary (one-mode) network N 1 = (U, E 1, w 1 ) or/and N 2 = (V, E 2, w 2 ), where E 1 and w 1 are determined by the matrix W (1) = WW T, w (1) uv = z V w uz w T zv. Evidently w (1) uv = w (1) vu. There is an edge (u : v) E 1 in N 1 iff N(u) N(v). Its weight is w 1 (u, v) = w (1) uv. The network N 2 is determined in a similar way by the matrix W (2) = W T W. The networks N 1 and N 2 are analyzed using standard methods.

129 V. Batagelj: Network Analysis with Pajek 125 Normalizations The normalization approach was developed for quick inspection of (1- mode) networks obtained from two-mode networks a kind of network based data-mining. In networks obtained from large two-mode networks there are often huge differences in weights. Therefore it is not possible to compare the vertices according to the raw data. First we have to normalize the network to make the weights comparable. There exist several ways how to do this. Some of them are presented in the following table. They can be used also on other networks. In the case of networks without loops we define the diagonal weights for undirected networks as the sum of out-diagonal elements in the row (or column) w vv = u w vu and for directed networks as some mean value of the row and column sum, for example w vv = 1 2 ( u w vu + u w uv). Usually we assume that the network does not contain any isolated vertex.

130 V. Batagelj: Network Analysis with Pajek Normalizations Geo uv = w uv wuu w vv GeoDeg uv = w uv degu deg v Input uv = w uv w vv Output uv = w uv w uu Min uv = w uv min(w uu, w vv ) Max uv = w uv max(w uu, w vv ) MinDir uv = { wuv w uu w uu w vv 0 otherwise MaxDir uv = { wuv w vv w uu w vv 0 otherwise After a selected normalization the important parts of network are obtained by line-cuts or islands approaches. Slovenian journals and magazins. Reuters Terror News: GeoDeg, MaxDir, MinDir.

131 V. Batagelj: Network Analysis with Pajek 127 GeoDeg normalization of Reuters terror news network

132 V. Batagelj: Network Analysis with Pajek 128 Networks from data tables A data table T is a set of records T = {T k : k K}, where K is the set of keys. A record has the form T k = (k, q 1 (k), q 2 (k),..., q r (k)) where q i (k) is the value of the property (attribute) q i for the key k.

133 V. Batagelj: Network Analysis with Pajek Networks from data tables Suppose that the property q has the range 2 Q. For example: Authors[WasFau] = { S. Wasserman, K. Faust }, PubYear[WasFau] = { 1994 },... 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 v q(k), and w(k, v) = 1. Also, for properties q i and q j we can define a two-mode network q i q j = (Q i, Q j, E, w) where (u, v) E iff k K : (q i (k) = u q j (k) = v), and w(u, v) = card({k K : (q i (k) = u q j (k) = v)}). It holds [q i q j ] T = q j q i and q i q j = [K q i ] T [K q j ] = [q i K] [K q j ]. We can join a pair of properties q i and q j also with respect to the third property q s : we get a two-mode network [q i q j ]/q s = [q i q s ] [q s q j ].

134 V. Batagelj: Network Analysis with Pajek 130 EU projects on simulation For the meeting The Age of Simulation at Ars Electronica in Linz, January 2006 a dataset of EU projects on simulation was collected by FAS research, Vienna and stored in the form of Excel table (SimPro.csv). The rows are the projects participants (idents) and colomns correspond to different their properties. Three two-mode networks were produced from this table using Jürgen Pfeffer s Text2Pajek program: project.net P = [idents projects] country.net C = [idents countries] institution.net U = [idents institutions] idents = 8869, projects = 933, institutions = 3438, countries = 60.

135 V. Batagelj: Network Analysis with Pajek 131 EU projects network multiplication Since all three networks have the common set (idents) we can derive from them using network multiplication several interesting networks: ProjInst.net W = [projects institutions] = P T U Countries.net S = [countries countries] = C T C Institutions.net Q = [institutions institutions] = W T W... For identifying important parts of ProjInst.net we first computed the 4-rings weights and in the obtained network we determined the line islands. We obtained 101 islands. We extracted 18 islands of the size at least 5. There are two most important islands: aviation companies and car companies.

136 V. Batagelj: Network Analysis with Pajek 132 BICC GENERAL CABLE PSI FUR PRODUKTE UND SYS.E DER INFORMATIONSTECH BARCO NV COLOPLAST A/S FRAUENHOFER INST. FUER PRODUKTIONSTECH. UND AUTOMATISIERUNG LMS UMWELTSYS.E, DIPL. ING. DR. HERBERT BACK TESSITURA LUIGI SANTI SPA INST. FUER TEXTIL UND VERFAHRENSTECH. DENKENDORF KBC MANUFAKTUR, KOECHLIN, BAUMGARTNER UND CIE. AG MSO CONCEPT INNOVATION + SOFTWARE TRUMPF-BLUSEN-KLEIDER WALTER GIRNER UND CO. KG 7210-PR/163 Analysis of ProjInst.net IST MTU AERO ENGINES EADS DE AIRBUS UK LIMITED AIRBUS DEUTSCHLAND NAT. TEC. UNIV OF ATHENS BAE SYSTEMS EUROCOPTER S AIRBUS FRANCE SAS ALENIA AERONAUTICA SPA G4RD-CT G4MA-CT STICHTING NATIONAAL LUCHT OFFICE NAT. DETUDES ET DE REC. AEROSPATIALES BRPR G4RD-CT G4RD-CT G4RD-CT EA TECH. LTD MECALOG SARL INST. NAT. DE RECHERCHE SUR LES TRANSPORTS ET LEUR S CURIT ENK6-CT CENTRE DE RECH. METALLURG PP/ SHERPA ENGINEERING SARL G4RD-CT CATALYSE SARL 7210-PR/233 VOEST-ALPINE STAHL INST. DE RECHERCHES EVG3-CT THYSSENKRUPP STAHL A.G. DE LA SIDERURGIE FR T3.2/99 DISENO DE SISTEMAS EN SILICIO CENTRE FOR EUROP. ECONOMIC SMT ILEVO AB FONDAZIONE ENI - ENRICO MATTEI 7210-PR/095 CSTB IST JERNKONTORET HPSE-CT UNIV. DER BUNDESWEHR MUENCHEN BUILDING RESEARCH CHIPIDEA - MICROELECTRONICA, S.A. LANDIS & GYR - EUROPE AG ENEL.IT UNIV. PANTHEON-ASSAS - PARIS II OESTERREICHISCHER BERGRETTUNGSDIENST SSAB TUNNPL T IFEN GES. FUER SATELLITENNAVIGATION JOE PP/034 RESEARCH INST. OF THE FINNISH ECONOMY IST WYKES ENGINEERING COMPANY LH AGRO EAST S.R.O. QLK6-CT TECHNOFARMING S.R.L. T3.5/99 THE AARHUS SCHOOL OF BUSINESS MEFOS, FOUNDATION FOR INST. CARTOGRAFIC DE CATALUNYA CINAR LTD. HELP SERVICE REMOTE SENSING METALLURGICAL RESEARCH LESPROJEKT SLUZBY S.R.O. HPSE-CT BAYER. ROTES KREUZ ENERGY RESEARCH CENTRE NL IST BRITISH STEEL 7210-PR/142 UNIV. OF MACEDONIA JOR FRAUENHOFER INST. FUER AGRO-SAT CONSULTING MATERIALFLUSS UND LOGISTIK ENK5-CT DATASYS S.R.O. UNIV. OF ABERDEEN ORAD HI TEC SYS. POLAND CRE GROUP LTD. MJM GROUP, A.S. FRIMEKO INT. AB CENTRE DE ROBOTIQUE BBL INOX PNEUMATIC AS DFA DE FERNSEHNACHRICHTEN AGENTUR TPS TERMISKA PROCESSER AB KOMMANDITGES. HAMBURG 1 PROLEXIA FERNSEHEN BETEILIGUNGS & CO A.S.M. S.A. ZAMISEL D.O.O DPME ROBOTICS AB INGENIORHOJSKOLEN HELSINGOR TEKNIKUM IST GATE5 AG INDUSTRIAS ROYO LKSOFTWARE UAB LKSOFT BALTIC BRST SUPERELECTRIC DI WISDOM TELE VISION IST SVETS & TILLBEHOR AB ALBERTSEN & HOLM AS CARLO PAGLIALUNGA & C. SAS SPORTART IST OK GAMES DI ALESSANDRO CARTA ASM - DIMATEC INGENIERIA FFT ESPANA TECH. DE AUTOMOCION, ENERGITEKNIK HEATEX AB YAHOO! DE OSAUHING EETRIUKSUS UNIV. DE ZARAGOZA BROD THOMASSON EDAG ENGINEERING + DESIGN GUNNESTORPS SMIDE & MEKANISKA AB Pajek ARMINES BUURSKOV DE ZENTRUM FUER LUFT UND RAUMFAHRT E.V. DASSAULT AVIATION SNECMA MOTEURS SA TQT SRL INST. SUPERIOR TECNICO C. R. FIAT S.C.P.A. NL ORG. FOR APPLIED SCIENTIFIC RESEARCH - TNO BARTENBACH ROSENHEIMER GLASTECH. ESI SOFTWARE SA CHALMERS TEKNISKA HOEGSKOLA VOLKSWAGEN AG POLYMAGE SARL DAIMLER CHRYSLER AG RUDOLF BRAUNS AND CO. KG

137 V. Batagelj: Network Analysis with Pajek 133 Uzbekistan Armenia India Kazakhstan Georgia Belarus Canada Japan Azerbaijan Analysis of Countries.net Liechtenstein France United Kingdom Russian F. Italia Finland The Netherlands Moldavia China Malta Turkey Germany Greece Thailand Portugal Spain Switzerland Israel Turkmenistan Cyprus Sweden Denmark Austria Estonia USA Slovakia Serbia-Montenegro Ukraine Norway Belgium Macedonia Croatia Poland Slovenia Hungary Luxembourg Ireland Latvia Bulgaria Czech R. Lithuania Afghanistan Iceland Romania Morocco Tunisia Jordan Ecuador Lebanon Algeria Albania Pajek To obtain picture in which the stronger lines cover weaker lines we have to sort them Net/Transform/Sort lines/line values/ascending 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 d 5 (corrected Euclidean distance). The permutation determined by hierarchy can often be improved by changing the positions of clusters. We get a typical center-periphery structure.

138 V. Batagelj: Network Analysis with Pajek 134 Analysis of Countries.net Pajek - shadow [0.00,4.00] Pajek - Ward [0.00, ] Ecuador Thailand Armenia Turkmenist Uzbekistan Moldavia Japan Kazakhstan Azerbaijan India Macedonia Albania Liechtenst Serbia-Mon Iceland Canada Estonia China Belarus Georgia Tunisia Lebanon Jordan Algeria Malta Morocco Afghanista Luxembourg Croatia Latvia Lithuania Cyprus Turkey Bulgaria Ukraine Slovenia Romania Slovakia USA Portugal Denmark Poland Finland Switzerlan Austria Czech R. Ireland Norway Hungary Israel Russian F. Sweden Greece Belgium Spain The Nether France United Kin Germany Italia Ecuador Thailand Armenia Turkmenist Uzbekistan Moldavia Japan Kazakhstan Azerbaijan India Macedonia Albania Liechtenst Serbia-Mon Iceland Canada Estonia China Belarus Georgia Afghanista Morocco Malta Tunisia Lebanon Jordan Algeria Croatia Latvia Lithuania Luxembourg Cyprus Turkey Bulgaria Ukraine Slovenia Romania Slovakia USA Russian F. Israel Hungary Ireland Czech R. Norway Poland Finland Portugal Denmark Switzerlan Austria Sweden Greece Belgium Spain The Nether Italia France United Kin Germany 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

139 V. Batagelj: Network Analysis with Pajek 135 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 DAEDALUS ENTERPRISE INFORMATICS LTD FL-SOFT V/JENS ALL-RUSSIAN SCIENTIFIC U.PARIS CENTER VI PIERRE ET JOERGEN MARIE CURIE OESTERGAARD BELIMO AUTOMATIONU.LA LAGUNA SOFIISKI U.SVETI KLIMENT OHRIDSKI THE U.COURT OF THE U.OF ABERDEEN SENTIENT MACHINE RESEARCH B.V. SIEMENS BUILDING TECH.S AG U.OULU I.OF INFORMATION TECH.S AS.CONOCIMIENTO DATAMED HEALTHCARE INF.SYS. STICHTING NEURALE NETWERKEN NOTTINGHAM TRENT U. U.E DE COIMBRA T.U.CLAUSTHAL MOMATEC U.CRETE U.ULSTER CITY U.LONDON 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. AND METALLURGY ASS.RECH.SCIENTIFIQUE BOURNEMOUTH U. ELITE EUROP.LAB KINGS COLLEGE LONDONENTE PER LE NUOVE TECNOLOGIE FOR INTELLIGENT STICHTING TECH. U.NYENRODE I.FUER NATURSTOFF-FORSCHUNG E.V. I.NAT.POLITEC.DE TOULOUSE U.GIRONA AUSTRIAN I.AI START ENGINEERING JSCO ENERGY RESEARCH C.NL U.GRANADA 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 ERASMUS U.ROTTERDAM CONSEJO SUP.DE AABO AKADEMI U INVEST.CIENTIFICAS U.MARIBOR U.P.MADRID U.PAUL SABATIER DE TOULOUSE III U.VALLADOLID U.P.CATALUNYA U.S.GENOVA PT.TORINO PT.BARI DAIMLER CHRYSLER AG U.DORTMUND LOUGHBOROUGH T.U. BRITISH TELEC. NAT.T.U.ATHENS I.SUPERIOR TECNICO U.SHEFFIELD POLISH ACAD.OF SCIENCES RISOE NAT.LAB DANMARKS T.U. U.C.LOUVAIN U.AMSTERDAM U.GENT CRANFIELD U. C.NAT.DE LA RECHERCHE SCIENTIFIQUE T.U.DELFT I.NAT.DE RECHERCHE SUR LES TRANSPORTS ET LEUR SECURITE HELSINKI T.U. BAE SYSTEMS CZECH T.U.PRAGUE T.U.V KOSICIACH FUNDACION LABEIN C.R.FIAT S.C.P.A. JOZEF STEFAN I. U.PAISLEY U.STRATHCLYDE U.NOTTINGHAM U.MANCHESTER CHALMERS TEKNISKA HOEGSKOLA A.U.THESSALONIKI C.INT.LENGINYERIA NL ORG.FOR APPLIED SCIENTIFIC RESEARCH-TNO CAD - FEM QINETIQ DEP.OF ENVIRONMENT, GKSS TRANSPORT - FORSCHUNGSZENTRUM AND THE REGIONS GEESTHACHT EUROP.SPACE AGENCY MANNESMANN VDO AG TECHSOFT ENGINEERING S.R.O. U.LEEDS TECNOLOGIAS CAE AVANZADAS S.L. C.SVILUPPO MATERIALI HERMSDORFER I.FUER TECH. OXFORD BROOKES U. U.S.PADOVA SAFE TECH. NOKIA MOBILE PHONES LTD AVIO S.P.A. FOKKER SPACE BV ANAKON NCODE INT. FINITE ELEMENT ANALYSIS LTD. TUN ABDUL RAZAK RESEARCH C.LTD. U.GREENWICH PRINCIPIA INGENIEROS CONSULTORES FUNDACION INASMET STAVANGER U.COLLEGE SULZER MARKETS AND TECH.AG, IFP SICOMP AB SULZER INNOTEC U.S.NAPOLI DAMT LTD SKF R&D COMPANY B.V. CAESAR SYSTEMS LTD NAFEMS LTD. INTES - INGENIEURGES. FUER TECH.SOFTWARE ENGIN SOFT TRADING SRL U.DURHAM MARITIME HYDRAULICS AS INBIS TECH.LTD FEMSYS LTD U.S.TRIESTE SOFISTK AG MSC SOFTWARE QUEENS U.BELFAST ABS CONSULTING ALTAIR ENGINEERING SAMTECH SA NLSE VERENIGDE SCHEEPSBOUW BUREAUS B.V. LEUVEN MEASUREMENTS AND SYS.INT.NV MERITOR HEAVY VEHICLE CREA CONSULTANTS BRAKING LTD SYS.- UK LTD TRL ACCESS E.V. PD&E AUTOMOTIVE B.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 ROYAL I.OF TECH. DUNLOP STANDARD KATHOLIEKE HOGESCHOOL SINT-LIEVEN U.HANNOVER AEROSPACE GROUP GIFFORD AND PARTNERS LTD. VOLVO AERO CORP.AB MECAS S.R.O. LULEAA T.U. ATOS ORIGIN ENG. STRUCTURAL INTEGRITY ASSESSMENTS LTD CORK I.OF TECHNOLOGY NORUT TEKNOLOGI HAHN-SCHICKARD-GES. A.S. WS ATKINS CONSULTANTS LTD. EASI ENGINEERING U.E DO MINHO U.SPLIT NUMERICAL ANALYSIS WILDE AND TECHNISCH PARTNERS LTD ADVIESBUREAU N.V. AND DESIGN&CO KG INTEGRATED DESIGN & FEGS RANDOM LOADING AWE ANALYSIS DESIGN PLC CONSULTANTS LTD MERKLE UND PARTNER D C WHITE&PARTNERS LTD DR THELLEN EATEC LTD To identify the most important institutions we first computed p S -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.

140 V. Batagelj: Network Analysis with Pajek 136 AIRBUS FRA Pajek - Ward [0.00, ] C. R. FIAT NL ORG. FO CHALMERS T Analysis of Institutions.net Pajek - shadow [0.00,6.00] ALLOGG AB AS. CONOCI ASS. RECH. AUSTRIAN I BELIMO AUT ALLOGG AB COVENTRY U AS. CONOCI DAEDALUS I ASS. RECH. DATAMED H AUSTRIAN I FACULTY OF BELIMO AUT FED. UNITA COVENTRY U FL-SOFT V/ DAEDALUS I INST. OF I DATAMED H GOETHE UNI FACULTY OF MIT-MANAGE FED. UNITA MOMATEC FL-SOFT V/ NAT. UNIV. INST. OF I NOTTINGHAM GOETHE UNI POLITECNIC MIT-MANAGE SENTIENT M MOMATEC SIEMENS BU NAT. UNIV. SOFIISKI U NOTTINGHAM START ENGI POLITECNIC STICHTING SENTIENT M STICHTING SIEMENS BU THE UNIV. SOFIISKI U UNIV. DE L START ENGI UNIV.SKLIN STICHTING UNIV. DE G STICHTING UNIV. PARI THE UNIV. UNIV. OF A UNIV. DE L UNIV. OF C UNIV.SKLIN UNIV. OF C UNIV. DE G UNIV. OF W UNIV. PARI UNIV. OF Z UNIV. OF A INST. DALL UNIV. OF C RAUTARUUKK UNIV. OF C UNIV.E DE UNIV. OF W UNIV. OF U UNIV. OF Z SOFTECO SI INST. DALL UNIV. GENT RAUTARUUKK TSS - TRAN UNIV.E DE UNIV. DE G UNIV. OF U UNIV. DE V SOFTECO SI UNIVERZA V UNIV. GENT UNIV. PAUL TSS - TRAN TECH. UNIV UNIV. DE G UNIV. WIEN UNIV. DE V UNIV. VAN UNIVERZA V ENERGY RES UNIV. PAUL TEKNILLINE TECH. UNIV DE MONTFOR UNIV. WIEN CITY UNIVE UNIV. VAN UNIV. OF J ENERGY RES AABO AKADE TEKNILLINE RISOE NAT. DE MONTFOR KINGS COLL CITY UNIVE UNIV. DORT UNIV. OF J FRIEDRICH- AABO AKADE UNIV. OF C RISOE NAT. INST. NAT. KINGS COLL BULGARIAN UNIV. DORT ENTE PER L FRIEDRICH- LOUGHBOROU UNIV. OF C HELSINKI U INST. NAT. CRANFIELD BULGARIAN UNIV. CATH ENTE PER L OTTO VON G LOUGHBOROU UNIV. DEGL HELSINKI U UNIV. OF O CRANFIELD UNIV. OF T UNIV. CATH ELITE EURO OTTO VON G ERASMUS UN UNIV. DEGL INST. FUER UNIV. OF O UNIV. PARI UNIV. OF T BRITISH TE ELITE EURO BOURNEMOUT ERASMUS UN UNIV. OF B INST. FUER UNIV. OF S UNIV. PARI DANMARKS T BRITISH TE DAIMLER CH BOURNEMOUT INST. NAT. UNIV. OF B INST. SUPE UNIV. OF S POLITECNIC DANMARKS T POLISH ACA DAIMLER CH UNIV. POLI INST. NAT. NAT. TEC. INST. SUPE CONSEJO SU POLITECNIC KATHOLIEKE POLISH ACA UNIV. TWEN UNIV. POLI TECH. UNIV NAT. TEC. UNIV. POLI CONSEJO SU CENTRE NAT KATHOLIEKE TEC. UNIV. UNIV. TWEN UNIV. OF P TECH. UNIV FUNDACION UNIV. POLI JOZEF STEF CENTRE NAT CZECH TECH TEC. UNIV. UNIV. OF N UNIV. OF P UNIV. OF S FUNDACION BAE SYSTEM JOZEF STEF UNIV. OF M CZECH TECH C. R. FIAT UNIV. OF N NL ORG. FO UNIV. OF S CHALMERS T BAE SYSTEM SAMTECH SA UNIV. OF M VOLVO AERO ABS CONSUL AVIO S.P.A ALTAIR ENG MSC SOFTWA ANAKON LULEAA UNI ATOS ORIGI QUEENS UNI AWE PLC AIRBUS FRA BEHR & CO CAD - FEM CAESAR SYS C. SVILUPP CORK INST. TRL CREA CONSU LEUVEN MEA D C WHITE A. UNIV. T DAMT LTD GERMAN AER DEP. OF E CENTRE INT DR THELLEN UNIV. DEGL DUNLOP STA QINETIQ EASI ENGIN UNIV. OF L EATEC LTD FUNDACION FEMSYS LTD ACCESS E.V FOKKER SPA EUROP. SPA FORSCHUNGS UNIV. OF G GIFFORD AN UNIV. DEGL HAHN-SCHIC UNIV. OF G HERMSDORFE UNIV. OF N INBIS TECH FINITE ELE INGENIEURB ROCKFIELD INTEGRATED WS ATKINS KATHOLIEKE ROYAL INST MANNESMANN UNIV. HANN MARITIME H FRAUENHOFE MECAS S.R. GKSS - FOR MERITOR HE NAT. NUCLE MERKLE UND UNIV. OF D NAFEMS LTD ENGIN SOFT NCODE INT. FEGS NLSE VEREN IFP SICOMP NEW TECH. INTES - IN NOKIA MOBI UNIV. DEGL NORUT TEKN PRINCIPIA NUMERICAL ABS CONSUL OXFORD BRO ALTAIR ENG PD & E AUT ANAKON RANDOM LOA ATOS ORIGI SAFE TECH. AWE PLC SKF R & D BEHR & CO SOFISTK AG CAESAR SYS ST MECANIC CORK INST. STAVANGER CREA CONSU STRUCTURAL D C WHITE SULZER MAR DAMT LTD TECHNISCH DEP. OF E TECHSOFT E DR THELLEN TECNOLOGIA DUNLOP STA TUN ABDUL EASI ENGIN UNIV.E DO EATEC LTD UNIV. OF S FEMSYS LTD WILDE AND FOKKER SPA FRAUENHOFE FORSCHUNGS GKSS - FOR GIFFORD AN NAT. NUCLE HAHN-SCHIC UNIV. OF D HERMSDORFE ENGIN SOFT INBIS TECH FEGS INGENIEURB IFP SICOMP INTEGRATED INTES - IN KATHOLIEKE UNIV. DEGL MANNESMANN PRINCIPIA MARITIME H FINITE ELE ROCKFIELD MECAS S.R. WS ATKINS MERITOR HE ROYAL INST MERKLE UND UNIV. HANN NAFEMS LTD UNIV. DEGL NCODE INT. UNIV. OF G NLSE VEREN UNIV. OF N NEW TECH. CAD - FEM NOKIA MOBI C. SVILUPP NORUT TEKN TRL NUMERICAL LEUVEN MEA OXFORD BRO A. UNIV. T PD & E AUT GERMAN AER RANDOM LOA CENTRE INT SAFE TECH. UNIV. DEGL SKF R & D QINETIQ SOFISTK AG UNIV. OF L ST MECANIC FUNDACION STAVANGER ACCESS E.V STRUCTURAL EUROP. SPA SULZER MAR UNIV. OF G TECHNISCH SAMTECH SA TECHSOFT E VOLVO AERO TECNOLOGIA AVIO S.P.A TUN ABDUL MSC SOFTWA UNIV.E DO LULEAA UNI UNIV. OF S QUEENS UNI 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 FRIEDRICH- UNIV. 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. 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141 V. Batagelj: Network Analysis with Pajek 137 Searching on the Web of Science The Web of Science WoS (ISI/Thomson) allows us to save on a file the records corresponding to our queries. For example, using General search with a query "social network*" we get 6936 hits (27. December 2007). Trying to save them we are informed that we can save at once at most 500 records. We have to save the records by parts on separate files. At the end we concatenate all these files into a single file.