Impacts of Node Connectivity on Reliability of Water Distribution Networks - Pattern Recognition for Reliability Assessment of Water Distribution Networks - page 141

NCF avg in Equation 6.8. In normal situations, all nodes are connected and the minimum value
of Deg i is 1 while the maximum one is n - 1 (i.e. equal to max con in Equation 6.6). Nodes
can however have the values of Deg In,i and Deg Out,i eqaul to 0. Finally, the degree centrality
of node i , C D,i , will be calculated as:

Deg

6.11

C = n

i

D , i

− 1

Figure 6.4 shows a simple network and the corresponding values of above-discussed
parameters. The average node degree of this network is 2.

n1 n2 n3 n4 n5 Deg i Deg In,i Deg Out,i C D,i

n1 0 1 1 0 0 2 0 2 0.5

n2 1 0 1 0 0 2 1 1 0.5

n3 1 1 0 1 1 4 3 1 1.0

n4 0 0 1 0 0 1 1 0 0.25

n5 0 0 1 0 0 1 0 1 0.25

Figure 6.4 Node connectivity matrix, degree and degree centrality

6.4.2 Graph Density

In an undirected network of n nodes that are all connected to each other, the (maximum)
number of links, m max will be calculated as:

n ( n − 1

6.12 m =

max

2

The graph density, GD , as the ratio between the actual number- and maximum number of
links, indicates the level of interconnectivity amongst the nodes. Hence:

2 m

GD =

6.13

n ( n − 1

For the network from Figure 6.4, m max equals 10 and consequently, GD = 0.5. The value of
m max for directed network will double, in which case GD = 0.25.

6.4.3 Geodesic Distance and Diameter

The geodesic distance, d i,j , between any two nodes i and j is determined by the number of
links in a shortest possible path connecting these nodes. The pair of nodes with maximum
geodesic distance (MGD) defines the geodesic diameter . The average geodesic distance
(AGD) for n nodes is determined as: