A Set of Measures of Centrality Based on
Betweenness
Linton C. Freeman, 1977
Advisor : Professor Frank Y. S. Lin
Presented by: Tuan-Chun Chen
Presentation date: Mar. 13, 2012
Agenda
 Introduction
 Measuring point centrality
 Measuring graph centrality
 Applications
Agenda
 Introduction
 Measuring point centrality
 Measuring graph centrality
 Applications
Introduction
 Betweenness : A point in a communication network is
central to the extent that it falls on the shortest path
between pairs of other points. (Bavelas, 1948)
 Another viewpoint by Shimbel (1953): If we count all of the
minimum paths which pass through a site, then we have a
measure of the ‘stress’ which the site must undergo during
the activity of the network.
Agenda
 Introduction
 Measuring point centrality
 Measuring graph centrality
 Applications
Measuring Point Centrality
 Shaw (1954)
 Unordered pair of points {pi, pj}
 {pi, pj} are Unreachable or there are one or more paths
between them.
pk
pi
pj
Measuring Point Centrality
 A point falling between two others can facilitate, block,
distort or falsify communication between the two.
 But if it falls on some but not the shortest path connecting a
pair of points , its potential for control is more limited.
pk
pi
pj
Measuring Point Centrality
 Define “partial betweenness”
 If pi and pj are not reachable from each other, pk is not
between them. let
bij ( p k )  0
 If pi and pj are reachable.
bij ( p k )  (
1
g ij
)( g ij ( p k ))
gij
The number of geodesics linking pi and pj.
gij(pk)
The number of geodesics linking pi and pj that contain pk.
Measuring Point Centrality
 p2 and p4 each have a probability of ½ of falling between p1
and p3.
 b13 ( p 2 )  ( 1 )(1)  1
2
2
p3
p2
p4
p1
Measuring Point Centrality
 Determine overall centrality of a point:
n
cB ( pk ) 
n
b
ij
( pk )
i j i j
CB(pk)
An index of the over all partial betweenness of point pk.
n
The number of points in the graph.
Measuring Point Centrality
 Its magnitude depends upon two factors:
 1) the arrangement of edges in the graph that define the
location of pk with respect to geodesics linking pairs of points.
 2) the number of points in the graph.
Measuring Point Centrality
 Problem ! ?
 Example:
 A graph containing 5 points, CB(pi)=6.
A graph containing 25 points, CB(pj)=6.
 They have the same potential for control in absolute terms,
but differ markedly in their relative potential for control.
Measuring Point Centrality
 Maximum Value:
C m ax 
[ n ( n  1)]
2
pk
n
pi
pj
ph
n  3n  2
2
 [ n  1] 
2
The number of points in the graph.
Measuring Point Centrality
 The relative centrality of any point in a graph, expressed as a
ratio :
C 'B ( p k ) 
2C B ( pk )
n  3n  2
2
, 0  C 'B ( p k )  1
 When C’B(pk)=1, the graph is a star or a wheel.
Agenda
 Introduction
 Measuring point centrality
 Measuring graph centrality
 Applications
Measuring Graph Centrality
 A network is central to the degree that a single point can
control its communication.(Measures of graph centrality
based upon the dominance of one point.)
n

C 'B 
 [C
B
'( p k *)  C B '( p i )]
i 1
n 1
, 0  C 'B  1
C’B(pk*)
The largest centrality value associated with any point in the graph.
C’B(pi)
The centrality value of pi
n
The number of points in the graph.
Agenda
 Introduction
 Measuring point centrality
 Measuring graph centrality
 Applications
Applications
 Original application was in the study of communication in
small groups. Speed, activity and efficiency in solving
problems and personal satisfaction and leadership in small
group setting(Leavitt 1951).
 Study of the diffusion of a technological innovation in the
steel industry(Czepiel 1974)
 Examined the impact of centrality on urban growth(Pitts
1965).
 Discussing the design of organization(Beauchamp
1965)(Mackenzie 1966)
Applications
 Consider the relationship between point centrality and
personal satisfaction in Leavitt’s(1951) study of small group
problem solving.
 Each participant had a piece of information necessary for the
solution of a problem. Each could communicate only with
designated others.
 Leavitt measured point centrality as a function of the lengths
of paths or the distance between points.
Applications
Thanks for your attention！