Impacts of Node Connectivity on Reliability of Water Distribution Networks - Pattern Recognition for Reliability Assessment of Water Distribution Networks - page 138

the nodes is not affected by the fact that these loops are not adjacent because they share the
same connecting node. Theoretical limit of max con for the worst connected network will be
n - 1. In that case, for l adjacent loops, l - 1 nodes will have three connections, two nodes
will have two connections, one node will have max con connections and ( n - ( l - 1) - 2 - 1)
node will be with single connection. Hence:

6.6

max con = 2 m − ( l − 1 × 3 − 2 × 2 − ( n − l + 1 − 3 × 1 = n − 1

All serial networks have two nodes with single connection and the rest with two connections,
leading to the NCF value calculated as in Equation 6.7.

1 2

[ ( ) ]

6.7 NCF = 2 + 2 n − 2 = 2 −

n n

The lowest possible NCF = 1, for two nodes and one link, will be increasing for increased
number of nodes/pipes, actually resembling the increase of reliability, too.

Branched networks will have the value of max con ≥ 2 while its theoretical maximum value
is also n - 1, as for the looped networks. The values of NCF for branched networks will
regularly be lower than 2, as is the case with serial networks. Looped and combined networks
will have the NCF value regularly higher than 2, except for the extreme cases of too high
value of max con not typical in practice.

Taking the average value of NCF as:

2 m

6.8 NCF avg =

n

makes possible to express the network connectivity index (NCI) within the range of 0 to 1, by
combining Equations 6.5 and 6.8:

NCF 1

( )

NCI = = i n + i n 6.9

max, 1 max, 1 max, 2 max, 2

NCF 2 m

avg

Equation 6.9 calculates the degree of connectivity for fixed m and n yielding the value of 1
for optimal connectivity. For instance, by comparing Equations 6.7 and 6.8, NCF will equal
NCF avg , giving the NCI = 1 for all serial networks; this result is justified because there is only
one possible connectivity scheme. Fully looped networks will also have the NCI value of 1,
indicating the optimal connectivity, while branched and combined networks will have the
value between 0 and 1, based on the value of max con and selected connectivity scheme.

Finally, combining the NGI and NCI , the network shape index (NSI) expresses the network
connectivity taking into consideration combined effect of loops and branches:

6.10 NSI = NGI × NCI

Based on the assumed NGI = 0, the NSI value for serial and branched networks will be 0 in
all cases and will be approaching the value of 1 for increasingly complex looped networks.