International audienceLa plupart des livres de référence sur les graphes commencent par introduire le même ensemble de concepts élémentaires, puis définissent des concepts plus avancés sur cette base. Nous montrons ici que cette approche peut être généralisée pour traiter non seulement la structure mais aussi la dynamique, de façon unifiée. On obtient un langage pour décrire les interactions temporelles, similaire au langage fourni par la théorie des graphes pour décrire les relations

Latapy, Matthieu

Magnien, Clémence

Viard, Tiphaine

French

Archive Ouverte en Sciences de l'Information et de la Communication

HAL Id: hal-01782899
https://hal.archives-ouvertes.fr/hal-01782899
Submitted on 2 May 2018
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Flots de liens et stream graphs pour la modélisation des
interactions temporelles
Matthieu Latapy, Tiphaine Viard, Clémence Magnien
To cite this version:
Matthieu Latapy, Tiphaine Viard, Clémence Magnien. Flots de liens et stream graphs pour la mod-
élisation des interactions temporelles. ALGOTEL 2018 - 20èmes Rencontres Francophones sur les
Aspects Algorithmiques des Télécommunications, May 2018, Roscoff, France. ￿hal-01782899￿
Flots de liens et stream graphs pour la
modélisation des interactions temporelles †
Matthieu Latapy, Tiphaine Viard et Clémence Magnien
Sorbonne Université, CNRS, Laboratoire d’Informatique de Paris 6, LIP6, F-75005 Paris, France
La plupart des livres de référence sur les graphes commencent par introduire le même ensemble de concepts élémentaires,
puis définissent des concepts plus avancés sur cette base. Nous montrons ici que cette approche peut être généralisée
pour traiter non seulement la structure mais aussi la dynamique, de façon unifiée. On obtient un langage pour décrire
les interactions temporelles, similaire au langage fourni par la théorie des graphes pour décrire les relations.
Mots-clefs : stream graphs, link streams, temporal networks, time-varying graphs, dynamic graphs, dynamic networks,
longitudinal networks, interactions, time, graphs, networks
Basic definitions
Given two sets X and Y , we denote by X×Y the set of ordered pairs (x,y) with x ∈ X and y ∈ Y , and we
denote by X ⊗Y the set of unordered pairs of distinct elements xy = yx with x ∈ X , y ∈ Y , and x 6= y. Then,
a graph G = (V,E) is classicaly defined by a set of nodes V and a set of links E ⊆V ⊗V .
We define a stream graph S = (T,V,W,E) by a finite set of nodes V , a continuous time interval T ⊂R, a
set of temporal nodes W ⊆ T ×V , and a set of links E ⊆ T ×V ⊗V such that (t,uv) ∈ E implies (t,u) ∈W
and (t,v) ∈W .
We denote by Tv the set of time instants at which v is present, by Tuv the set of time instants at which uv is
present, by Vt the set of nodes present at time t, and by Et the set of links present at time t : Tv = {t,(t,v) ∈
W}, Tuv = {t,(t,uv) ∈ E}, Vt = {v,(t,v) ∈W}, and Et = {uv,(t,uv) ∈ E}. Notice that Tuv ⊆ Tu∩Tv.
Stream graphs model interactions between nodes over time, as well as the dynamics of nodes themselves.
For instance, nodes may represent individuals present in a given building and links may represent contacts
between them. Nodes may represent on-line computers and links may represent data exchanges between
them. Such examples are countless.
Induced graph, link stream, and graph-equivalent streams
We define G(S) = ({v,Tv 6= /0},{uv,Tuv 6= /0}) = (
⋃
t∈T Vt ,
⋃
t∈T Et) the graph induced by S. It is the
graph where there is a link between two nodes if they interacted at least once.
If all nodes are present all the time, i.e. Tv = T for all v or, equivalently, Vt = V for all t, then we say
that S is a link stream and we denote it by L = (T,V,E) (with W = T ×V implicitly). Indeed, there is no
dynamics on nodes in this case, and S is fully defined by this triplet.
If in addition for all u and v in V , Tuv ∈ { /0,T}, i.e. all existing links are present all the time, then there is
no significant distinction between S and G(S), and we say that S is a graph-equivalent stream. This gives a
formal ground to the fact that stream graphs generalize graphs : we extend graph concepts to stream graphs
in a way such that, if a stream graph has a given stream graph property and happens to be a graph-equivalent
stream, then this graph has the corresponding graph property.
Example and drawings
We illustrate our definitions with drawings like in Figure 1. We display node names on a vertical axis
on the left, and time on an horizontal axis at the bottom of the drawing. Each node presence times are
†Cet article est un extrait condensé de l’article [3], en cours d’évaluation. Le résumé fourni ici joue aussi le rôle d’introduction.
Matthieu Latapy, Tiphaine Viard et Clémence Magnien
represented by an horizontal dotted line in front of its name, whenever the node is present. Each link
presence times are represented by an horizontal solid line parallel to the two dotted lines of involved nodes,
and a vertical solid line joining these two dotted lines (marked with bullets) when the two nodes start
interacting.
a
b
c
d
0 2 4 6 8 time
a
b
c
d
0 2 4 6 8 time
FIGURE 1: Simple examples of stream graphs and link streams. Left : a stream graph S = (T,V,W,E) with T =
[0,10] ⊆ R, V = {a,b,c,d}, W = [0,10]×{a}∪ ([0,4]∪ [5,10])×{b}∪ [4,9]×{c}∪ [1,3]×{d}, and E = ([1,3]∪
[7,8])×{ab}∪ [4.5,7.5]×{ac}∪ [6,9]×{bc}∪ [2,3]×{bd}. In other words, Ta = [0,10], Tb = [0,4]∪ [5,10], Tc = [4,9],
Td = [1,3], Tab = [1,3]∪ [7,8], Tac = [4.5,7.5], Tbc = [6,9], Tbd = [2,3], and Tad = Tcd = /0. Right : a link stream
L = (T,V,E) with T = [0,10] ⊆ R, V = {a,b,c,d}, and E = ([0,4]∪ [6,9])×{ab} ∪ [2,5]×{ac} ∪ [1,8]×{bc} ∪
[7,10]×{bd}∪ [6,9]×{cd}. In other words, Ta = Tb = Tc = Td = T and Tab = [0,4]∪ [6,9], Tac = [2,5], Tbc = [1,8],
Tbd = [7,10] and Tcd = [6,9].
Size and duration
In order to capture the fact that some nodes may be present for much longer than others, we define the
contribution of node v as nv =
|Tv|
|T | . We define similarly the contribution of a pair of nodes uv as muv =
|Tuv|
|T | .
We then define the number of nodes n and the number of links m in S :
n = ∑
v∈V
nv =
|W |
|T |
and m = ∑
uv∈V⊗V
muv =
|E|
|T |
.
Then, each node contributes to the total number of nodes proportionally to its involvement in S : v in V
accounts for 1 node only if it is present in S all the time. The same holds for links.
In Figure 1, we obtain for S the values n = |Ta|10 +
|Tb|
10 +
|Tc|
10 +
|Td |
10 = 2.6 nodes and m = 1 link. For L, we
obtain n = 4 nodes and m = 2.3 links.
In a link stream L = (T,V,E), nv = 1 for all v and so n = |V |. In a graph-equivalent stream, the number
of nodes and links in the stream are equal to the number of nodes and links in the corresponding graph.
Density
Since the density of graph G = (V,E) is the probability when one takes a random element uv in V ⊗V
that there is a link between u and v in E, we define the density of stream graph S = (T,V,W,E) as the
probability when one takes a random element (t,uv) of T ×V ⊗V such that (t,u) and (t,v) are in W , that
(t,uv) is in E :
δ(S) =
∑
uv∈V⊗V
|Tuv|
∑
uv∈V⊗V
|Tu∩Tv|
=
∫
t∈T
|Et |dt∫
t∈T
|Vt ⊗Vt |dt
The density is the fraction of possible links that do exist, both for graphs and stream graphs.
Notice that ∑uv∈V⊗V |Tuv|=
∫
t∈T |Et |dt = |E|.
For S defined in Figure 1 (left), we obtain δ(S) = 1022 ∼ 0.45, and for L defined in this figure (right),
δ(L) = 2360 ∼ 0.38.
There is in general no relation between the density δ, the number of nodes n and the number of links m
in a stream graph. However, the classical graph relation δ = 2mn(n−1) holds for link streams. In addition, the
density of a graph-equivalent stream is equal to the density of the corresponding graph.
Flots de liens et stream graphs
Substreams and clusters
A graph G′ = (V ′,E ′) is a subgraph of G = (V,E) if V ′ ⊆ V and E ′ ⊆ E. A cluster C of G = (V,E) is a
subset of V . The set of links between nodes in C is E(C) = {uv∈E,u∈C and v∈C}, and G(C) = (C,E(C))
denotes the subgraph of G induced by C.
We say that a stream S′ = (T ′,V ′,W ′,E ′) is a substream of S = (T,V,W,E) if T ′ ⊆ T , V ′ ⊆ V , W ′ ⊆
W , and E ′ ⊆ E. We define a cluster C of S = (T,V,W,E) as a subset of W . We define the set of links
between nodes involved in C as E(C) = {(t,uv) ∈ E,(t,u) ∈ C and (t,v) ∈ C}, and we denote by S(C) =
(T,V,C,E(C)) the substream of S induced by C. See Figure 2.
a
b
c
d
0 2 4 6 8 time
a
b
c
d
0 2 4 6 8 time
FIGURE 2: An example of cluster with its induced substream. Left : the cluster, displayed in blue, is C = ([1,4]∪
[5,8])×{a}∪ [5,9]×{b}∪ [3,8]×{c}. Right : the substream induced by C is S(C) = ([0,10],{a,b,c,d},C,E(C)) with
E(C) = [6,8]×{ab}∪ [3,4]×{ac}∪ [5,8]×{bc}.
Cliques
Just like in graphs, a clique of stream graph S is a cluster C of S of density 1. In other words, all pairs of
nodes involved in C are linked in S whenever both are involved in C. A clique C is maximal if there is no
other clique C′ such that C ⊂C′, and a clique C is compact if there exists [b,e] ⊆ T and X ⊆ V such that
C = [b,e]×X . See Figure 3 for an illustration.
a
b
c
d
0 2 4 6 8 time
FIGURE 3: Examples of maximal compact cliques. We display the two maximal compact cliques involving three
nodes of the link stream L of Figure 1 (right) : [2,4]×{a,b,c} and [7,8]×{b,c,d}. Its other maximal compact cliques
are [0,4]×{a,b}, [6,9]×{a,b}, [2,5]×{a,c}, [1,8]×{b,c}, [7,10]×{b,d}, [6,9]×{c,d} (involving two nodes each).
If one considers a graph-equivalent stream, then its maximal cliques are necessarily compact, and they
correspond exactly to the maximal cliques of its induced graph.
Neighborhood and degree
In the graph G = (V,E), the neighborhood N(v) of v∈V is the cluster N(v) = {u,uv∈ E}, and the degree
d(v) of v is the number of nodes in this cluster, which is equal to the number of links involving v. We then
have ∑v∈V d(v) = 2 ·m.
In a stream graph S = (T,V,W,E), we define the neighborhood of a node v as the cluster N(v) =
{(t,u),(t,uv) ∈ E} and the degree d(v) of v as the number of nodes in this cluster. As with graphs, this
is equal to the number of links involving v : d(v) = |N(v)||T | = ∑u∈V
|Tuv|
|T | = ∑u∈V muv. See Figure 4 for an
illustration.
As with graphs, the sum of the degree of all nodes in S is equal to twice the number of links in S :
∑v∈V d(v) = ∑v∈V ∑u∈V
|Tuv|
|T | = 2 ·m. In addition, if we consider a graph-equivalent stream, then the degree
of any of its nodes is equal to the degree of this node in the corresponding graph.
Matthieu Latapy, Tiphaine Viard et Clémence Magnien
a
b
c
d
0 2 4 6 8 time
a
b
c
d
0 2 4 6 8 time
FIGURE 4: Two examples of neighborhoods and degrees of nodes. We display in black the links involving the node
under concern, and in grey the other links. Left : N(a) = ([1,3]∪ [7,8])×{b}∪ [4.5,7.5]×{c} is in blue, leading to
d(a) = 310 +
3
10 = 0.6. Right : N(c) = [2,5]×{a}∪ [1,8]×{b}∪ [6,9]×{d} is in blue, leading to d(c) =
13
10 = 1.3.
Clustering coefficient
Just like in graphs, the clustering coefficient of a given node v is the density of its neighborhood :
cc(v) = δ(N(v)), leading to :
cc(v) = δ(N(v)) =
∑uw∈V⊗V |Tvu∩Tvw∩Tuw|
∑uw∈V⊗V |Tvu∩Tvw|
In other words, in a graph cc(v) is the probability that two randomly chosen neighbors of v are linked
together in G ; in a stream graph, cc(v) is the probability when one takes two random neighbors u and w of
v at time t, i.e. a random (t,uw) in T ×V ⊗V such that (t,vu) and (t,vw) are in E, that u is linked to w in S
at time t, i.e. that (t,uw) is in E.
In a graph-equivalent stream, cc(v) in the stream is equal to cc(v) in the corresponding graph.
Conclusion
In this paper, we present some key notions of the formalism we introduced in [3] for dealing directly
with the both temporal and structural nature of interactions over time. Similarly to graph theory, the
strength of our approach is to rely on very basic innovations that make it trivial to define more advanced
objects. In addition, our formalism is consistent with graph theory : if one considers a stream graph with
no dynamics (nodes are present all the time, and two nodes are either linked all the time or not at all), then
the stream graph is equivalent to a graph and its stream properties are equivalent to the properties of the
corresponding graph. As a consequence, our formalism is a generalization of graph theory, which provides
a solid ground for generalizing other graph notions.
Complementary to previous work on complexity classes in time-varying graphs [1], our formalism raises
new algorithmic challenges, like clique computations in stream graphs for instance [4, 2, 5]. It also opens
the way to better understanding of many real-world data, like network traffic, mobility traces, and fi-
nancial transactions, on which we are currently working.
*
Références
[1] Arnaud Casteigts, Paola Flocchini, Walter Quattrociocchi, and Nicola Santoro. Time-varying graphs
and dynamic networks. IJPEDS, 27(5) :387–408, 2012.
[2] Anne-Sophie Himmel, Hendrik Molter, Rolf Niedermeier, and Manuel Sorge. Adapting the bron-
kerbosch algorithm for enumerating maximal cliques in temporal graphs. Social Netw. Analys. Mining,
7(1) :35 :1–35 :16, 2017.
[3] Matthieu Latapy, Tiphaine Viard, and Clémence Magnien. Stream graphs and link streams for the
modeling of interactions over time. CoRR, abs/1710.04073, 2017.
[4] Tiphaine Viard, Matthieu Latapy, and Clémence Magnien. Computing maximal cliques in link streams.
Theor. Comput. Sci., 609 :245–252, 2016.
[5] Tiphaine Viard, Clémence Magnien, and Matthieu Latapy. Enumerating maximal cliques in link streams
with durations. CoRR, abs/1712.06970, 2017. To appear in Information Processing Letters.


Flots de liens et stream graphs pour la modélisation des interactions temporelles

https://hal.archives-ouvertes.fr/hal-01782899/file/algotel.pdf

Flots de liens et stream graphs pour la modélisation des interactions temporelles

Abstract

Similar works

Full text

Available Versions

Archive Ouverte en Sciences de l'Information et de la Communication