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

Transitions and Trajectories
in 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 of
trajectories in temporal networks in the case that network
activity times are arranged in overlapping intervals.
To make clear the methodology we are developing, we are
giving here two simplistic examples of two small (artiﬁcial)
networks.
However, we have written a script in Python, which is
implementing the relevant computations for temporal
networks of any size.
Thus, in the ﬁnal version of this paper to be presented at the
Brighton 2015 Sunbelt Conference, we are going to apply such
an analysis of trajectories for the following two empirical large
temporal networks:
the temporal one–mode network from the data of the
Correlates of War Project (directed by Zeev Maoz) and
the temporal two–mode network of interlocing directorates of
the 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 all
time, but for certain time points inside a given time period
(time interval) T. In general, T ⊂ R+ and time points can be
either isolated points or (sub)intervals (of nonnegative real
numbers).
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 vertices
u, v ∈ V . We denote by T(u,v), Tu, Tv ⊂ T the activity time
set of edge (u, v) and vertices u, v, respectively, and we
assume the following consistency condition to hold, for all
edges (u, v) and vertices u, v:
T(u,v) ⊆ Tu ∩ Tv .

The activity timeline of edge (u, v) is deﬁned as a function
α(u,v) : T(u,v) −→ {0, 1} such that
α(u,v)(t) =
1, whenever t ∈ T(u,v),
0, whenever t ∈ T T(u,v).
Similarly, the activity timeline of vertex u is deﬁned as a
function αu : Tu −→ {0, 1} such that
αu(t) =
1, whenever t ∈ Tu,
0, whenever t ∈ T Tu.
Furthermore, given a vertex u and a time point τ ∈ Tu, we
write 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, then
τ ∈ Tu ∩ Tv , αuτ (τ) = αvτ (τ) = 1, where (uτ , vτ ) is an edge
of the temporal network.

To avoid certain technicalities, let us assume (from now on)
that we have a temporal network such that, for any edge or
vertex, the activity time set of this edge or vertex is a union of
disjoint 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 closed
interval of the form:
Tn(e) = [tn(e), tn(e)],
where tn(e), tn(e) ≥ 0, for n = 1, . . . , k, and:
tn(e) < tn(e) < tn+1(e).
Tn(e) is called n–th activity subinterval of the activity set Te
and tn(e), tn(e) are its left, right end points (respectively).

Note that, in this way, T becomes the convex hull (i.e., the
minimum closed interval) containing the union of all activity
subintervals of all edges of the temporal network:
T = conv
e∈E
k
n=1
Tn(e) .
Any t ∈ T e∈E
k
n=1 Tn(e) is called intermitting
time point.
Furthermore, we denote by Ie the set of all end points of all
subintervals of Te, i.e.,
Ie = t0(e), t0(e), t1(e), t1(e), . . . , tk(e), tk(e) .
Lumping together all sets Ie, for all edges e, one gets the
total set of all end points of the temporal network
I =
e∈E
Ie.

Deﬁnition
Let τ ∈ I. Then τ is called:
intermediate time point when, for every edge e ∈ E, if
τ ∈ Tn(e), for some n, then τ is always an interior point of
Tn(e).
co–terminal time point when, for every edge e ∈ E, if
τ ∈ Tn(e), for some n, then τ is always either the left or the
right 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 interior
point of Tn(e), for some n (or an interior point of Tm(f ), for
some m) and an end point of Tm(f ), for some m (or an end
point 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 neighbors
of u. If τ ∈ T(u,v) ∪ T(u,w) is such that τ is either an anti–terminal
or a step–like time point (w.r.t. (u, v), (u, w)), then we say that u
passes from v to w at time τ through a transition denoted as
vτ
uτ
−→ wτ .
t tτ τ
Transition vτ
uτ
−→ wτ
(u, v) (u, v)
(u, w) (u, w)

Examples of Transitions
In the following diagramm, there are 12 transitions of u
through its neighbors v, w, z:
v2
u2
−→ w2, v3
u3
−→ z3, w3
u3
−→ z3, v4
u4
−→ z4,
w4
u4
−→ z4, z6
u6
−→ v6, z6
u6
−→ w6, v8
u8
−→ w8,
v8
u8
−→ z8, w10
u10
−→ v10, w10
u10
−→ z10, z10
u10
−→ v10.

Deﬁnition of Translations
Deﬁnition
Let u, v two adjacent vertices in the temporal network. If
τ, σ ∈ T(u,v), τ < σ, are such that [τ, σ] ⊂ [ti (u, v), ti (u, v)], for
some activity subinterval [ti (u, v), ti (u, v)], then we say that u
shifts from vτ to vσ through a translation denoted as vτ
u
vσ.
tti τ σ ti
Translation vτ
u
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 not
necessary) that these vertices were joined by an edge too.
In a temporal network without self–loops, any two vertices joined by a
translation cannot be joined by an edge.

Deﬁnition of Trajectories
Deﬁnition
In a temporal network, a trajectory of vertex u passing over its
neighbors 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 v0
u
v1 is a translation, v1
u1
−→ w1 is a transition,
w1
u
w2 is a translation, w2
u2
−→ 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
w
4 , w
10
w
6 , w
7
u3,u4,w7,w8
w2,w3,u6,u10,w10,w11
v2 , v4 , z7
z3 , v6 , v7 , z11
w7
, w10
w2
, w3
u6
u3 , u4 , u10
u2
, u8
u10

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
10
w2
,w3
w7
,w10

Statistics of Transitions of the Two–Mode Network

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

Recomendados

Recomendados

Mais conteúdo relacionado

Mais procurados

Mais procurados (18)

Destaque

Destaque (16)

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

Semelhante a Boudourides et al., Transitions and Trajectories in Temporal Networks with Overlapping Activity Time Intervals (20)

Mais de Moses Boudourides

Mais de Moses Boudourides (20)

Último

Último (20)

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