Environmental Engineering Reference
In-Depth Information
∑
d
i
,
j
i
,
j
AGD
=
6.14
2
n
The network in Figure 6.5 shows the geodesic diameter of undirected network of five nodes.
The total number of shortest paths between any node pair
i
and
j
,
P
i,j
= 1, in this case. There
are total ten unique pairs of nodes and consequently ten shortest paths, namely:
n1n2
,
n1n3
,
n1n3n4
,
n1n3n5
,
n2n3
,
n2n3n4
,
n2n3n5
,
n3n4
,
n3n5
,
n4n3n5
. Five of these paths have one
link while the other five are composed of two links. Hence, the value of
AGD
= 1.5, and
MGD
= 2.
Figure 6.5
Geodesic distances and diameter
If two nodes cannot be connected, their geodesic distance is conventionally assumed to be
infinite.
6.4.4 Betweenness Centrality
The betweenness centrality (BC) of a particular node analyses in how many shortest paths the
node is present, being a conveyor of (important) information from/to other nodes. For node
k
,
the
BC
k
will be calculated as:
P
=
∑
i
,
k
,
j
BC
;
i
≠
j
≠
k
6.15
k
P
i
,
j
i
,
j
In the network in Figure 6.5, five of the ten shortest paths pass though node
n3
. As already
mentioned, all ten pairs of nodes have one shortest path each, and also
P
i,k,j
= 1 for
k
being
node
n3
. Thus,
BC
n3
= 5. No other node is on any other shortest path; hence, their
betweenness centrality will equal 0 (also in case of nodes
n1
and
n2
, because
n2n1n3
and
n1n2n3
are not the shortest paths between
n2
and
n3
, and
n1
and
n3
, respectively).
6.4.5 Closeness Centrality
The closeness centrality,
Cc
i
, of particular node
i
is a measure of its average distance (along
the shortest path) to all other nodes. Hence:
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