Environmental Engineering Reference
In-Depth Information
Table 6.2
Connectivity of network layouts from Figure 6.2
No. conn.
1
2
3
4
2m
n
L1
2
3
6
0
26
11
L2
1
5
5
0
26
11
L3
2
4
4
1
26
11
L4
0
7
4
0
26
11
The table shows the dominant number of node connections to be two and three; the number
of corresponding nodes is shown in the shaded cells. At the same time, the average
connectivity of all four layouts is 2.36 connections per node (2
m
/
n
). As both the figure and
the table show, the fully looped network (L4) will consist only of the nodes with two or three
connections, the average value of 2.36 suggesting more of those with two than of those with
three. Reconnecting the nodes in a way that would create nodes with less than two or more
than three connections will inevitably lead to a configuration combined of loops and
branches, as is the case of L1 to L3.
6.3.2 Measures of Network Connectivity
Very rarely in reality, the nodal connectivity will result in the max
con
value above 4. It can
therefore be assumed that the network connectivity is sufficiently reflected by analysing the
two most dominant connectivity categories. Based on this hypothesis, the measure named the
network connectivity factor
(NCF) is formulated to arrive at the description of network shape:
1
(
)
NCF
=
i
n
+
i
n
6.5
max,
1
max,
1
max,
2
max,
2
n
There are three possible ways to look at the values of the two connectivity categories
represented by indices
max,1
and
max,2
in Equation 6.5:
1.
by taking into consideration the two largest numbers of nodes
n
with connectivity
i
. In
case of two categories with equal number of nodes, being the second largest, the one with
higher
i
would qualify. This approach bases the value of
NCF
1
on dominant connectivity
categories in terms of maximum number of nodes per category.
2.
by taking into consideration the largest product between the number of nodes
n
and their
corresponding connectivity
i
; this approach bases the value of
NCF
2
on dominant
connectivity categories in terms of maximum number of connections per category.
3.
by taking into consideration the number of nodes
n
in the two connectivity categories
i
closest to the average network connectivity (2
m
/
n
); this approach bases the value of
NCF
3
on dominant connectivity categories in terms of average number of connections.
Independent from the approach, the results of calculations for the layouts L1 to L4 will be the
same; they are shown in Table 6.3. The obtained
NCF
values show the same ranking of the
networks coming from the visual perception of their reliability.
Table 6.3
NCF values for network layouts from Figure 6.2
No. conn.
1
2
3
4
NCF
L1
0.18
0.55
1.64
0.00
2.18
L2
0.09
0.91
1.36
0.00
2.27
L3
0.18
0.73
1.09
0.36
1.82
L4
0.00
1.27
1.09
0.00
2.36
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