Page 328 - Becoming Metric Wise

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Networks

N 2 1
0 , making c (i)a
0
reason closeness is usually defined as c ðiÞ 5 X
d G ði; tÞ
tAV
direct measure of centrality. In an unconnected network the closeness
centrality of any vertex is zero. Yet, a solution for this unfortunate state of
affairs is to define closeness centrality for nodes in disconnected graphs as
X 1
c}ðiÞ 5 ðN 2 1Þ tAV; (Opsahl et al., 2010). Of course, this has
t 6¼ i d G ði; tÞ
the disadvantage of introducing two types of closeness.
Graph centrality (Hage & Harary, 1995) is defined as
1
C G ðvÞ 5 . This measure too is zero in an unconnected
max tAV ðd G ðv; tÞÞ
network.
Let σ st denote the number of shortest paths between s and t, where
σ ss is set equal to 1. In an undirected, unweighted network we always
have σ st 5 σ ts but this equality does not have to hold in general. Let σ st (v)
be the number of shortest paths between s and t, passing through v. Next
we consider a centrality measure that makes use of the σ-function.

Betweenness Centrality
Betweenness centrality may be defined loosely as the number of times a
node needs a given node to reach another node. Stated otherwise: it is
the number of shortest paths that pass through a given node. As a mathe-
matical formula the betweenness centrality of node v (Anthonisse, 1971;
X σ st ðvÞ
Bavelas, 1948; Freeman, 1977) is defined as bðvÞ 5 s 6¼ v; .If v is a
t 6¼ v σ st
singleton then its betweenness centrality is equal to zero. According to
Borgatti (1995), the purpose of measuring betweenness is to provide a
weighting system so that node i is given a full centrality credit only when
it lies along all shortest paths between s and t. Betweenness gauges the
extent to which a node facilitates the network flow. It can be shown that
2
for an N-node network the maximum value for b(v)is(N 2 3N 1 2)/2.
Hence standardized betweenness centrality is:

2bðvÞ
b S ðvÞ 5 (10.5)
2
N 2 3N 1 2
Leydesdorff (2007) used centrality measures when studying interdisci-
plinarity. He claims that betweenness can be used as a valid local measure
for interdisciplinarity.