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Social networks: Prestige, centrality, and inuence
(Invited paper)
Agnieszka Rusinowska, Rudolf Berghammer, Harrie de Swart, Michel Grabisch
To cite this version:
Agnieszka Rusinowska, Rudolf Berghammer, Harrie de Swart, Michel Grabisch. Social networks:
Prestige, centrality, and inuence (Invited paper). de Swart. RAMICS 2011, Springer, pp.22-39,
2011, Lecture Notes in Computer Science (LNCS) 6663. �hal-00633859�
Social Networks:
Prestige, Centrality, and Influence
Agnieszka Rusinowska
1
, Rudolf Berghammer
2
, Harrie De Swart
3
, and Michel
Grabisch
1
1
Centre d’Economie de la Sorbonne, Universit´e Paris I Panth´eon-Sorbonne
106-112 Bd de l’Hˆopital, 75647 Paris Cedex 13, France
{agnieszka.rusinowska, michel.grabisch}@univ-paris1.fr
2
Institut f¨ur Informatik, Universit¨at Kiel, Olshausenstraße 40, 24098 Kiel, Germany
rub@informatik.uni-kiel.de
3
Department of Philosophy, Erasmus University Rotterdam
P.O. Box 1738, 3000 DR Rotterdam, The Netherlands; deSwart@fwb.eur.nl
Abstract. We deliver a short overview of different centrality measures
and influence concepts in social networks, and present the relation-algebraic
approach to the concepts of power and influence. First, we briefly discuss
four kinds of measures of centrality: the ones based on degree, closeness,
betweenness, and the eigenvector-related measures. We consider central-
ity of a node and of a network. Moreover, we give a classification of the
centrality measures based on a topology of network flows. Furthermore,
we present a certain model of influence in a social network and discuss
some applications of relation algebra and RelView to this model.
Keywords: social network, centrality, prestige, influence, relation alge-
bra, RelView
1 Introduction
Social networks play a central role in our activities, in social phenomena, in
economic and political life. It is therefore crucial to provide an exhaustive anal-
ysis of social network structures and to study the impact they may have on
human’s behavior. Many scholars are particularly interested in measures that
allow to compare networks. Also measures that compare nodes (representing
agents) within a network and show how a node relates to the network are of
interest. The question appears how central a node is and what its position and
prestige in a network are. The concept of centrality as applied to human commu-
nication was introduced already in the late 1940’s, and since then many different
measures of centrality have been developed. They usually capture complemen-
tary aspects of a node’s position, any hence a particular measure can be more
appropriate for some applications and less for others.
One of the aims of this paper is to deliver a brief overview of the main central-
ity measures. Four kinds of measures are presented: degree centrality, closeness
centrality, betweenness centrality, Katz prestige and Bonacich centrality. We
also briefly discuss a categorization of centrality measures based on a topology
of network flows.
2 Agnieszka Rusinowska et al.
Social networks are particularly important in studying all kinds of influence
phenomena. They are very useful for analyzing the diffusion of information and
the formation of opinions and beliefs. It is therefore not surprising that there
are numerous works in different scientific fields on the ‘network approach’ to
interaction and influence.
One of the leading dynamic models on information transmission, opinion and
consensus formation in networks is introduced by DeGroot [14]. Individuals start
with initial opinions on a subject and put some weights on the current beliefs
of other agents in forming their own beliefs for the next period. These beliefs
are updated over time. Several variations and generalizations of the DeGroot
model are presented e.g. in [15, 20, 21, 22, 36]. Surveys of models of influence
and different approaches to this phenomenon can be found e.g. in [27, 29, 36, 38].
Another framework of influence in networks is introduced in [33]. In the
original one-step model, agents have to make their acceptance-rejection decision
on a specific issue. Each agent has an inclination to say either ‘yes’ or ‘no’,
but due to possible influence of the other agents, his final decision (‘yes’ or
‘no’) may be different from his initial inclination. This framework is extensively
investigated e.g. in [24, 25, 26, 28, 29, 30, 39].
Relation algebra is used very successfully for formal problem specification,
prototyping, and algorithm development. For details on relations and relational
algebra, see e.g. [13, 16, 17, 40]. RelView is a BDD-based tool for the visu-
alization and manipulation of relations and for prototyping and relational pro-
gramming. It has been developed at Kiel University. The tool is written in the
C programming language and makes full use of the X-windows graphical user
interface. Details and applications can be found e.g. in [3, 4, 9].
Several of our works are devoted to applications of relation algebra and Rel-
View to Game Theory and Social Choice Theory. In [5] we present such an
application to coalition formation, where with the help of relation algebra and
RelView the set of all feasible stable governments is determined. A stable gov-
ernment is by definition not dominated by any other government. In [6] we deal
with the case where all governments are dominated. By using notions from rela-
tion algebra, graph theory and social choice theory, and by using RelView we
can compute a government that is as close as possible to being non-dominated.
In [7] we apply relation algebra and RelView to networks, i.e., to compute some
measures of agents’ strength in a network, like power, success, and influence. In
[8] we present relation-algebraic models of simple games and develop relational
specifications for solving some basic game-theoretic problems. We test funda-
mental properties of simple games, compute specific players and coalitions, and
apply relation algebra to determine power indices.
In this paper we also aim at presenting a relation-algebraic approach to the
concepts of influence in a social network. We recapitulate relation-algebraic spec-
ifications (presented in [7]) of the following concepts of the model of influence
([25, 33, 39]): the inclination and decision vectors, the group decision, the Hoede-
Bakker index, the inclination vectors of potential and observed influence, and
the set of followers.
Social Networks: Prestige, Centrality, and Influence 3
The paper is structured as follows. In Section 2 the basic concepts in net-
work theory are recalled. In Section 3 we discuss the main centrality measures.
Section 4 concerns the model of influence in a social network. In Section 5
the relation-algebraic preliminaries are presented. Section 6 is devoted to the
relation-algebraic approach to the concepts of influence. In Section 7 we present
some concluding remarks.
2 The basic concepts in network theory
In this section we present the preliminaries on networks. For textbooks on net-
work theory, see e.g. [23, 36, 44].
Let N = {1, 2, ..., n} be a (finite) set of nodes. By g
ij
∈ {0, 1} we denote a
relationship between nodes i and j, where
g
ij
=
1 if there is a link between i and j
0 otherwise.
(1)
In what follows we only consider undirected links, i.e., we assume that g
ij
= g
ji
.
A network g is defined as a set of nodes N with links between them. Let G
denote the collection of all possible networks on n nodes.
By N
i
(g) we denote the neighborhood (the set of neighbors) of node i in
network g, i.e., the set of nodes with which node i has a link:
N
i
(g) = {j ∈ N : g
ij
= 1}. (2)
The degree d
i
(g) of a node i in g is the number of i’s neighbors in g, i.e.,
d
i
(g) = |N
i
(g)|. (3)
A network g is said to be regular if every node has the same number of neighbors,
i.e., if for some d ∈ {0, 1, ..., n − 1}, d
i
(g) = d for each i ∈ N.
A complete network is a regular network with d = n − 1. The empty network
is a regular network with d = 0.
One of the concerns when analyzing a network is to check how one node may
be reached from another one. We distinguish between the following definitions:
- A walk is a sequence of nodes in which two nodes have a link (they are
neighbors), and a node or a link may appear more than once. Its length is
simply the number of links in the walk.
- A trail is a walk in which all links are distinct.
- A path is a trail in which all nodes are distinct.
- A cycle is a trail with at least 3 nodes in which the initial node and the end
node are the same.
- A geodesic between two nodes is a shortest path between them.
If there is a path between i and j in g, then the geodesic distance d(i, j; g)
between these two nodes i and j is therefore equal to
d(i, j; g) = the number of links in a shortest path between i and j. (4)
4 Agnieszka Rusinowska et al.
If there is no path between i and j in g, we set d(i, j; g) = ∞.
A star is a network in which there exists some node i (referred to as the
center of the star) such that every link in the network involves node i.
Two nodes belong to the same component if and only if there exists a path
between them. A network is connected if there exists a path between any pair of
nodes i, j ∈ N . Consequently, a network is connected if and only if it consists of
a single component.
The adjacency matrix G of a (undirected or directed) network g is defined
as G = [g
ij
] with g
ij
as in (1). In other words, an entry in the matrix G
corresponding to the pair {i, j} signifies the presence or absence of a link between
i and j. Let G
k
denote the kth power of G, i.e., G
k
= [g
k
ij
], where g
k
ij
measures
the number of walks of length k that exist between i and j in network g. We
have G
0
= I, where I is the n × n identity matrix.
3 Different measures of centrality in networks
The concept of centrality captures a kind of prominence of a node in a network.
The economic and sociological literature offers several such concepts. For surveys
of different notions of centrality, see e.g. [19, 23, 36]. In this paper, we recapitulate
several well-known centrality measures. The presentation is based on the three
references mentioned above.
As presented in [36], measures of centrality can be categorized into the fol-
lowing main groups:
(1) Degree centrality
(2) Closeness centrality
(3) Betweenness centrality
(4) Prestige- and eigenvector-related centrality.
3.1 Degree centrality
The degree centrality indicates how well a node is connected in terms of direct
connections, i.e., it keeps track of the degree of the node. This measure can be
seen as an index of the node’s communication activity.
The degree centrality C
d
(i; g) of node i in network g is given by
C
d
(i; g) =
d
i
(g)
n − 1
=
|N
i
(g)|
n − 1
(5)
where N
i
(g) and d
i
(g) are defined in (2) and (3). Obviously, 0 ≤ C
d
(i; g) ≤ 1.
Let i
∗
be a node which attains the highest degree centrality C
d
(i
∗
; g) in g.
The degree centrality C
d
(g) of network g is given by
C
d
(g) =
P
n
i=1
[C
d
(i
∗
; g) − C
d
(i; g)]
max
g
0
∈G
[
P
n
i=1
[C
d
(i
∗
; g
0
) − C
d
(i; g
0
)]]
. (6)
![Table 2. Flow processes and major centrality measures (see [12])](/figures/table-2-flow-processes-and-major-centrality-measures-see-12-tu9xbdkw.webp)
![Table 1. Topology of flow processes (see [12])](/figures/table-1-topology-of-flow-processes-see-12-1bhub4vw.webp)