Page 332 - Becoming Metric Wise

324 Becoming Metric-Wise

We assume that we have a network N 5 (V, E), consisting of a set V of
nodes or vertices and a set E of links or edges. If a network consists of
two groups G (consisting of m nodes) and H (consisting of n nodes), then
the gefura measure of node a is defined as:

X σ g;h ðaÞ
ΓðaÞ 5 (10.6)
σ g;h
gAG
hAH
Definition: the gefura measure for a finite number of groups in an undi-
rected network
Guns and Rousseau (2009b, 2015) expanded the definition for two
groups to directed networks with any finite number of groups and
showed that one can define a global, a local and an external variant. We
restrict ourselves to the definition of the global gefura measure in an
undirected network.
Assume that there are S groups (2 # S ,1 N): G 1 ; .. .; G S ; the num-
ber of nodes in group G i is denoted as m i . The global gefura measure of
node a is then defined as:
0 1
X
X B
σ g;h ðaÞC
Γ G ðaÞ 5 B C (10.7)
@ σ g;h A
k;l gAG k
hAG l
where k takes all values from 1 to S and l takes all values strictly larger
than k. Clearly, if S 5 2, formula (10.7) reduces to formula (10.6).
An example.
Consider the example in Fig. 10.2. The network consists of three
node groups a 1 ; a 2 ; a 3 g; b 1 ; b 2 ; b 3 g; and c 1 ; c 2 g. To illustrate the procedure
f
f
f
we calculate the (unnormalized) gefura measure for node b 1 . This node is
part of the following shortest paths between different groups:
• Between a 1 and b 2 : a 1 2 b 1 2 b 2
• Between a 2 and b 2 : a 2 2 b 1 2 b 2
• Between b 2 and c 2 . In this case there are two possible shortest paths,
namely b 2 2 b 1 2 a 2 2 c 2 and b 2 2 a 3 2 c 1 2 c 2 .
Hence, we have Γ(b 1 ) 5 1 1 1 1 0.5 5 2.5. Using similar reasoning we
find the following results: Γ(a 1 ) 5 0.5; Γ(a 2 ) 5 1.5; Γ(a 3 ) 5 2; Γ(b 2 ) 5 0;
Γ(b 3 ) 5 0; Γ(c 1 ) 5 4.5; Γ(c 2 ) 5 5. This result shows that node c 2 has the
highest gefura-value, which means that it has the most important bridging
function in this network.