Boudourides et al., Transitions and Trajectories in Temporal Networks with Overlapping Activity Time...

Transitions and Trajectoriesin Temporal Networks with

Overlapping Activity Time Intervals

Extended Abstract

Moses A. Boudourides, Sergios T. Lenis,

Martin Everett & Elisa Bellotti

18 February 2015

Boudourides, Lenis, Everett & Bellotti Transitions and Trajectories in Temporal Networks

Overview

This is an elaboration of Borgatti & Halgin’s analysis oftrajectories in temporal networks in the case that networkactivity times are arranged in overlapping intervals.

To make clear the methodology we are developing, we aregiving here two simplistic examples of two small (artificial)networks.

However, we have written a script in Python, which isimplementing the relevant computations for temporalnetworks of any size.Thus, in the final version of this paper to be presented at theBrighton 2015 Sunbelt Conference, we are going to apply suchan analysis of trajectories for the following two empirical largetemporal networks:

the temporal one–mode network from the data of theCorrelates of War Project (directed by Zeev Maoz) andthe temporal two–mode network of interlocing directorates ofthe SIRF Project (University of Manchester).


Basic Concepts and Notation

By a temporal network we understand an ordinary network,in which edges and vertices are not active or present in alltime, but for certain time points inside a given time period(time interval) T. In general, T ⊂ R+ and time points can beeither isolated points or (sub)intervals (of nonnegative realnumbers).

We denote by V ,E the (finite) sets of vertices and edges,respectively, of a temporal network.

Let (u, v) ∈ E be an arbitrary edge joining two verticesu, v ∈ V . We denote by T(u,v),Tu,Tv ⊂ T the activity timeset of edge (u, v) and vertices u, v , respectively, and weassume the following consistency condition to hold, for alledges (u, v) and vertices u, v :

T(u,v) ⊆ Tu ∩ Tv .


The activity timeline of edge (u, v) is defined as a functionα(u,v) : T(u,v) −→ {0, 1} such that

α(u,v)(t) =

{1, whenever t ∈ T(u,v),

0, whenever t ∈ Tr T(u,v).

Similarly, the activity timeline of vertex u is defined as afunction αu : Tu −→ {0, 1} such that

αu(t) =

{1, whenever t ∈ Tu,0, whenever t ∈ Tr Tu.

Furthermore, given a vertex u and a time point τ ∈ Tu, wewrite uτ in order to denote the (activated) vertex u at time τ .In other words, if τ ∈ T(u,v), i.e., α(u,v)(τ) = 1, thenτ ∈ Tu ∩ Tv , αuτ (τ) = αvτ (τ) = 1, where (uτ , vτ ) is an edgeof the temporal network.


To avoid certain technicalities, let us assume (from now on)that we have a temporal network such that, for any edge orvertex, the activity time set of this edge or vertex is a union ofdisjoint intervals (each one having positive length).

Thus, for any edge e, the activity set Te of e is:

Te =k⋃

n=1

Tn(e).

Above k = k(e) and, for all n = 1, . . . , k , Tn(e) is a closedinterval of the form:

Tn(e) = [tn(e), t ′n(e)],

where tn(e), t ′n(e) ≥ 0, for n = 1, . . . , k, and:

tn(e) < t ′n(e) < tn+1(e).

Tn(e) is called n–th activity subinterval of the activity set Te

and tn(e), t ′n(e) are its left, right end points (respectively).


Note that, in this way, T becomes the convex hull (i.e., theminimum closed interval) containing the union of all activitysubintervals of all edges of the temporal network:

T = conv

( ⋃e∈E

k⋃n=1

Tn(e)

).

Any t ∈ Tr(⋃

e∈E⋃k

n=1 Tn(e)

)is called intermitting

time point.

Furthermore, we denote by Ie the set of all end points of allsubintervals of Te , i.e.,

Ie ={t0(e), t ′0(e), t1(e), t ′1(e), . . . , tk(e), t ′k(e)

}.

Lumping together all sets Ie , for all edges e, one gets thetotal set of all end points of the temporal network

I =⋃e∈EIe .


Definition

Let τ ∈ I. Then τ is called:

intermediate time point when, for every edge e ∈ E , ifτ ∈ Tn(e), for some n, then τ is always an interior point ofTn(e).

co–terminal time point when, for every edge e ∈ E , ifτ ∈ Tn(e), for some n, then τ is always either the left or theright end point of Tn(e) (but always the same for all edges),

t t tτ τ τ

e e e

f f f

intermediate time τ left co–terminal time τ right co–terminal time τ


Definition

Let e, f be two edges and let τ ∈ Te ∪ Tf . Then τ is called:

anti–terminal time point (w.r.t. e, f ) when τ is the right(or left) end point of Tn(e), for some n, and the left (right,resp.) end point of Tm(f ), for some m,

step–like time point (w.r.t. e, f ) when τ is is an interiorpoint of Tn(e), for some n (or an interior point of Tm(f ), forsome m) and an end point of Tm(f ), for some m (or an endpoint of Tn(e), for some n, resp.).

t tτ τ

e e

f f

anti–terminal time τ step–like time τ


Definition of Transitions

Definition

Let u be a vertex of the temporal network and v ,w two neighborsof u. If τ ∈ T(u,v) ∪ T(u,w) is such that τ is either an anti–terminalor a step–like time point (w.r.t. (u, v), (u,w)), then we say that upasses from v to w at time τ through a transition denoted asvτ

uτ−→ wτ .

t tτ τ

Transition vτuτ−→ wτ

(u, v) (u, v)

(u,w) (u,w)


Examples of Transitions

In the following diagramm, there are 12 transitions of uthrough its neighbors v ,w , z :

v2u2−→ w2, v3

u3−→ z3,w3u3−→ z3, v4

u4−→ z4,

w4u4−→ z4, z6

u6−→ v6, z6u6−→ w6, v8

u8−→ w8,

v8u8−→ z8, w10

u10−→ v10,w10u10−→ z10, z10

u10−→ v10.


Definition of Translations

Definition

Let u, v two adjacent vertices in the temporal network. Ifτ, σ ∈ T(u,v), τ < σ, are such that [τ, σ] ⊂ [ti (u, v), t ′i (u, v)], forsome activity subinterval [ti (u, v), t ′i (u, v)], then we say that u

shifts from vτ to vσ through a translation denoted as vτu

99K vσ.

tti τ σ t′i

Translation vτu

99K vσ

(u, v)

Remarks:In a temporal network, neither transitions nor translations make up edges.However, if two vertices are joined by a transition, it is possible (but notnecessary) that these vertices were joined by an edge too.In a temporal network without self–loops, any two vertices joined by atranslation cannot be joined by an edge.


Definition of Trajectories

Definition

In a temporal network, a trajectory of vertex u passing over itsneighbors v ,w , z , . . . is an alternating sequence of vertices,translations and transitions of the form:

[(v0, u, v1), (v1, u1,w1), (w1, u,w2), (w2, u2, z2), . . .]

where v0u

99K v1 is a translation, v1u1−→ w1 is a transition,

w1u

99K w2 is a translation, w2u2−→ z2 is a transition, etc.


Example: Two Trajectories

t0 1 2 3 4

Trajectory [(v0, u, v1), (v1, u1,w1), (w1, u,w2), (w2, u2, v2), (v2, u, v3), (v3, u3,w3), (w3, u,w4)]

(u, v)

(u,w)

t0 1 2 3 4

Trajectory [(v0, u, v3), (v3, u3,w3), (w3, u,w4)]

(u, v)

(u,w)


Example of a Temporal One–Mode Network

u

v

w

z

[0, 4], [6, 8], [10, 12] [2,7],

[10, 12]

[0, 3], [7, 11][3,12]

[2, 4], [6, 10]


Activity Timeline of the One–Mode Network


Transitions of u in the One–Mode Network


Transitions of v in the One–Mode Network


Transitions of w in the One–Mode Network


Transitions of z in the One–Mode Network


The Graph of Transitions of the One–Mode Network

u

v

w

z

w4 ,w

10

w6 ,w

7

u3 ,u4 ,w7 ,w8

w2 ,

w3 ,

u6 ,

u10 ,

w10 ,

w11

v2 , v4 , z7

z3 , v6 , v7 , z11

w 7,w

10

w 2,w

3

u6

u3 , u

4 , u10

u 2, u

8

u 10


Statistics of Trajectories of the One–Mode Network


Example of a Temporal Two–Mode Network

u

w

A

B

C

[0, 4], [6,

8], [10, 12

]

[2, 7],[10,12]

[0, 3], [7,

11]

[3, 12]

[2, 4], [6, 10]


Activity Timeline of the Two–Mode Network


Transitions of u in the Two–Mode Network


Transitions of w in the Two–Mode Network


The Graph of Trajectories of the Two–Mode Network

A B

C

u3, u5,w7, u8

w2,w3, u6, u10,w10,w11

w 6,w

7

w 4,w

10w2 ,w

3

w7 ,w

10


Statistics of Transitions of the Two–Mode Network


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