Impacts of Node Connectivity on Reliability of Water Distribution Networks - Pattern Recognition for Reliability Assessment of Water Distribution Networks - page 143

n − 1

Cc = 6.16

∑

i

d

i , j

j

The results for the network in Figure 6.5 are shown in Figure 6.6. ∑d i,j indicates the total
number of links on the shortest path between node i and all other nodes.

n1 n2 n3 n4 n5 ∑d i,j Cc ,i

n1 0 1 1 0 0 6 0.67

n2 1 0 1 0 0 6 0.67

n3 1 1 0 1 1 4 1.0

n4 0 0 1 0 0 7 0.57

n5 0 0 1 0 0 7 0.57

Figure 6.6 Node connectivity matrix and closeness centrality

6.4.6 Clustering Coefficient

The clustering coefficient, Cp i , measures how the neighboring nodes of node i are connected
amongst themselves. It is expressed as the ratio of the actual number of their interconnecting
links and the total possible number of these links that can be calculated in the same was as in
Equation 6.12. Hence, for k i neighboring nodes of node i , which are mutually connected with
m k links in an undirected network:

2 m

6.17

Cp =

k

i

k ( k − 1

i i

The results for the network in Figure 6.6 are shown in Figure 6.7. Node n3 has four
neighboring nodes mutually connected with only connection, whilst the maximum number of
their connections is six.

n1 n2 n3 n4 n5 k i m k Cp i

n1 0 1 1 0 0 2 1 1.0

n2 1 0 1 0 0 2 1 1.0

n3 1 1 0 1 1 4 1 0.17

n4 0 0 1 0 0 1 0 0

n5 0 0 1 0 0 1 0 0

Figure 6.7 Node connectivity matrix and clustering coefficient

In fact, the clustering coefficient defines the probability that two neighbouring nodes of node
i are connected to each other, which enhances the connectivity in case the node i is removed
from the system. This is related to the concept of cut-points and bridges. A node is a cut-point
if its removal disintegrates the network i.e. creates sub-networks without mutual connection.