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paths (the shortest path between two nodes) that pass through a node
k
. To calculate
the betweenness centrality, we take every pair of the network and count how many
times a node can interrupt the geodesic distance between the two nodes of the pair.
For standardization, the denominator is
(
−
)(
−
)
/
n
1
n
2
2.
Table 17.3
Betweenness
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1 —- 1-2 1-2-3
1-4
1-4-5
1-4-6
1-4-7
1-4-5-8
1-9
2 —- —-
2-3
2-3-4 2-3-4-5 2-3-4-6 2-3-4-6-7 2-3-4-5-8 2-3-4-6-7-9
—- —-
—-
2-3-4-5-8-9
3 —- —-
—-
3-4
3-4-5
3-4-6
3-4-6-7
3-4-5-8
3-4-6-7-9
3-4-5-8-9
4 —- —-
—-
—-
4-5
4-6
4-6-7
4-5-8
4-6-7-9
4-5-8-9
5 —- —-
—-
—-
—-
—-
—-
5-8
5-8-9
6 —- —-
—-
—-
—-
—-
—-
6-8
6-7-9
7 —- —-
—-
—-
—-
—-
—-
—-
7-9
8 —- —-
—-
—-
—-
—-
—-
—-
8-9
9 —- —-
—-
—-
—-
—-
—-
—-
—-
For example, the betweenness centrality for the node 4 will be:
Betweenness centrality(4)= fraction_paths_broken
(1, 2)= 0
+
(2, 3)= 0
+
(3, 4)= 1
+
(4, 5)= 1
+
+
(1, 3)= 0
+
(2, 4)= 1
+
(3, 5)= 1
+
(4, 6)= 1
+
+
(1, 4)= 1
+
(2, 5)= 1
+
(3, 6)= 1
+
(4, 7)= 1
+
+
(1, 5)= 1
+
(2, 6)= 1
+
(3, 7)= 1
+
(4, 8)= 1
+
+
(1, 6)= 1
+
(2, 7)= 1
+
(3, 8)= 1
+
(4, 9)= 2/2=1 +
+
(1, 7)= 1
+
(2, 8)= 1
+
(3, 9)= 2/2=1 +
+
(1, 8)= 1
+
(2, 9)= 2/2=1 +
+
(1, 9)= 0
+
+
(5, 8)= 0
+
(6, 7)= 0
+
(7, 9)= 0
+
(8, 9)= 0
+ (5, 9)= 0 + (6, 9)= 0 + N
= 0+0+1+1+1+1+1+0+0+1+1+1+1+1+1+1+1
+1+1+1+1+1+1+1+1+1+0+0+0+0+0+0
= 21/32
= 0,656
