A network N = (V,L,P,W) is determined with four sets:
The set of nodes and the set of links describe its structure and form a graph G = (V,L).
A = ∅ - undirected graph
E = ∅ - directed graph
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)1, (c,g)2, (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
The nodes are indexed starting with 1. In listing of links (arcs or edges) their end-nodes are replaced by their indices.
A line describing a node (vertex) has the following components: index, label, X, Y, Z. If the label contains a space it should be put in quotes “la bel”. Pajek's internal coordinates X, Y, Z are in the range [0,1]. If your coordinates are out of this range (for example geographical longitude and latitude) you can transform them into the Pajek's range in the drawing window using the option
Options/Transform/Fit Area
You can even skip the list of nodes - specifying only the line *vertex
n. In this case the node labels will be created automatically and the coordinates are put on the circle.
A line describing a link has the following components: 1st-node-index, 2nd-node-index, weight. If the weight is omitted the default value 1 is used.
A line starting with % is a comment line.
% Example network % by Vladimir Batagelj, April 2003 *Vertices 12 1 "a" 0.1020 0.3226 2 "b" 0.2860 0.0876 3 "c" 0.5322 0.2304 4 "d" 0.3259 0.3917 5 "e" 0.5543 0.4770 6 "f" 0.1552 0.6406 7 "g" 0.8293 0.3249 8 "h" 0.4479 0.6866 9 "i" 0.8204 0.8203 10 "j" 0.4789 0.9055 11 "k" 0.1175 0.9032 12 "l" 0.7095 0.6475 *Arcs 1 2 2 1 1 4 1 6 2 6 3 2 3 3 3 7 3 7 5 3 5 6 5 8 6 11 8 4 10 8 12 5 12 7 8 12 12 8 *Edges 2 5 3 4 5 7 6 8
GraphSet: PDF / net; TinaSet: PDF / net, PDF picture.
NA(a) = { b, d, f }
NA(b) = { a, f }
NA( c) = { b, c, g, g }
NA(e) = { c, f, h }
NA(f) = { k }
NA(h) = { d, l }
NA(j) = { h }
NA(l) = { e, g, h }
NE(e) = { b, g }
NE( c) = { d }
NE(f) = { h }
N(v) = NA(v) ∪ NE(v)
The line with node's neighbors description has a form: i0, i1, i1, …, ik with a meaning: the node i0 has neighbors i1, i1, …, ik.
*Vertices 12 1 "a" 0.1020 0.3226 2 "b" 0.2860 0.0876 3 "c" 0.5322 0.2304 4 "d" 0.3259 0.3917 5 "e" 0.5543 0.4770 6 "f" 0.1552 0.6406 7 "g" 0.8293 0.3249 8 "h" 0.4479 0.6866 9 "i" 0.8204 0.8203 10 "j" 0.4789 0.9055 11 "k" 0.1175 0.9032 12 "l" 0.7095 0.6475 *Arcslist 1 2 4 6 2 1 6 3 2 3 7 7 5 3 6 8 6 11 8 4 12 10 8 12 5 7 8 *Edgeslist 2 5 3 4 5 7 6 8
Our network can be also described using a matrix.
a b c d e f g h i j k l a 0 1 0 1 0 1 0 0 0 0 0 0 b 1 0 0 0 1 1 0 0 0 0 0 0 c 0 1 1 1 0 0 2 0 0 0 0 0 d 0 0 1 0 0 0 0 0 0 0 0 0 e 0 1 1 0 0 1 1 1 0 0 0 0 f 0 0 0 0 0 0 0 1 0 0 1 0 g 0 0 0 0 1 0 0 0 0 0 0 0 h 0 0 0 1 0 1 0 0 0 0 0 1 i 0 0 0 0 0 0 0 0 0 0 0 0 j 0 0 0 0 0 0 0 1 0 0 0 0 k 0 0 0 0 0 0 0 0 0 0 0 0 l 0 0 0 0 1 0 1 1 0 0 0 0
Graph G is simple if in the corresponding matrix all entries are 0 or 1.
In general, a matrix entry can be any real number. Each matrix line should be written in a single line with at least one space between entries as a separator.
*Vertices 12 1 "a" 0.1020 0.3226 2 "b" 0.2860 0.0876 3 "c" 0.5322 0.2304 4 "d" 0.3259 0.3917 5 "e" 0.5543 0.4770 6 "f" 0.1552 0.6406 7 "g" 0.8293 0.3249 8 "h" 0.4479 0.6866 9 "i" 0.8204 0.8203 10 "j" 0.4789 0.9055 11 "k" 0.1175 0.9032 12 "l" 0.7095 0.6475 *Matrix 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 0 0 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0
In a two-mode network N=((U,V),L,P,W) the set of nodes consists of two disjoint (sub)sets of nodes U and V, and all the links from L have one endnode in U and the other endnode 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 = [auv]U☓V.
Examples:
((persons, societies), years of membership),
((buyers/consumers, goods), quantity),
((parlamentarians, problems), positive vote),
((persons, journals), reading).
A two-mode network is announced by
*vertices
n nU
Authors and works.
The southern women event participation data (Davis)
page.
A multi-relational network is denoted by N =(V, (L1,L2,…,Lk),P,W) and contains different relations Li (sets of links) over the same set of nodes. Also the weights from W are defined on different relations or their union.
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, …), …
The relation can be assigned to a link in two ways:
*arcs
or *edges
can be assigned to selected relation by starting its description by the number of this relation3: 47 14 5
47
and 14
and weight 5
belongs to relation 3
.Multi-relational temporal network – KEDS/WEIS
% Recoded by WEISmonths, Sun Nov 28 21:57:00 2004 % from http://www.ku.edu/~keds/data.dir/balk.html *vertices 325 1 "AFG" [1-*] 2 "AFR" [1-*] 3 "ALB" [1-*] 4 "ALBMED" [1-*] 5 "ALG" [1-*] ... 318 "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: 314 153 1 [4] 890402 YUG KSV 224 (RIOT) RIOT-TORN 212: 314 83 1 [4] 890404 YUG ETHALB 212 (ARREST PERSON) ALB ETHNIC JAILED IN YUG 224: 3 83 1 [4] 890407 ALB ETHALB 224 (RIOT) RIOTS 123: 83 153 1 [4] 890408 ETHALB KSV 123 (INVESTIGATE) PROBING ... 42: 105 63 1 [175] 030731 GER CYP 042 (ENDORSE) GAVE SUPPORT 212: 295 35 1 [175] 030731 UNWCT BOSSER 212 (ARREST PERSON) SENTENCED TO PRISON 43: 306 87 1 [175] 030731 VAT EUR 043 (RALLY) RALLIED 13: 295 35 1 [175] 030731 UNWCT BOSSER 013 (RETRACT) CLEARED 121: 295 22 1 [175] 030731 UNWCT BAL 121 (CRITICIZE) CHARGES 122: 246 295 1 [175] 030731 SER UNWCT 122 (DENIGRATE) TESTIFIED 121: 35 295 1 [175] 030731 BOSSER UNWCT 121 (CRITICIZE) ACCUSED
A temporal network NT =(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 nodes v ∈ V and links l ∈ L are not necessarily present or active in all time points. If a link l(u,v) is active in time point t then also its endnodes u and v should be active in time t.
We will denote the network consisting of links and nodes active in time t ∈ T by N(t) and call it a time slice in time point t. To get time slices in Pajek use
Network/Temporal Network/Generate in time
Temporal networks - presence
*Vertices 3 1 "a" [5-10,12-14] 2 "b" [1-3,7] 3 "e" [4-*] *Edges 1 2 1 [7] 1 3 1 [6-8]
Node 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 link is present, if both its endnodes are present.
Temporal networks - events
Event | Explanation |
---|---|
TI t | initial events – following events happen when |
time point t starts | |
TE t | end events – following events happen when |
time point t is finished | |
AV v n s | add vertex v with label n and properties s |
HV v | hide node v |
SV v | show node v |
DV v | delete node v |
AA u v s | add arc (u,v) with properties s |
HA u v | hide arc (u,v) |
SA u v | show arc (u,v) |
DA u v | delete arc (u,v) |
AE u v s | add edge (u:v) with properties s |
HE u v | hide edge (u:v) |
SE u v | show edge (u:v) |
DE u v | delete edge (u:v) |
CV v s | change property of node v to s |
CA u v s | change property of arc (u,v) to s |
CE u v s | change property of edge (u:v) to s |
CT u v | change (un)directedness of link (u,v) |
CD u v | change direction of arc (u,v) |
PE u v s | replace pair of arcs (u,v) and (v,u) by single edge (u:v) |
with properties s | |
AP u v s | add pair of arcs (u,v) and (v,u) |
with properties s | |
DP u v | delete pair of arcs (u,v) and (v,u) |
EP u v s | replace edge (u:v) by pair of arcs (u,v) and (v,u) |
with properties s |
s can be empty.
In case of parallel links :k denotes the k-th link - HE:3 14 37
hides the third edge linking nodes 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 1 3 1 TI 7 SV 2 AE 1 2 1 TE 7 DE 1 2 DV 2 TE 8 DE 1 3 TE 10 HV 1 TI 12 SV 1 TE 14 DV 1
File/Network/Read Time Events
Pictures in SVG: Terror news - 66 days.
See Pajek manual, page 89-95.