Table of Contents

A *network* **N** = (**V**,**L**,**P**,**W**) is determined with four sets:

- the set of
*nodes***V**, - the set of
*links***L**, - the set of
*node properties***P**, and - the set of
*link properties*(weights)**W**.

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.

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 }

N(v) = N_{A}(v) ∪ N_{E}(v)

The line with node's neighbors description has a form: i_{0}, i_{1}, i_{1}, …, i_{k}
with a meaning: the node i_{0} has neighbors i_{1}, i_{1}, …, i_{k}.

*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** = [a_{uv}]_{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 n_{U}

Authors and works.

The southern women event participation data (Davis)
page.

A **multi-relational network** is denoted by **N** =(**V**, (**L**_{1},**L**_{2},…,**L**_{k}),**P**,**W**)
and contains different relations **L**_{i} (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:

- Any link controlled by
`*arcs`

or`*edges`

can be assigned to selected relation by starting its description by the number of this relation

`3: 47 14 5`

Link with endnodes`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** **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 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.

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