Explanation through a Case Study

To get an insight into the characteristics of the NPI and its applications, a sample road network is selected with nodes and links as shown in Figure 2.6. The selection of such a simple and semi symmetrical road network layout is solely for its ease in understanding. The network is

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networks are encircled for prominence. The shortest paths and paths-lengths between all nodes/

possible ODs are computed using Floyd-Warshall algorithm written in R-language (package e1071).

Part-I Part-II

Figure 2.6 Layout of sample road network

At first, the NPI of all links is calculated by using equation (2.1), as presented in Table 2.4.

Then as a second step, the links connected to all nodes are listed and adjacent reference node is selected to identify the most important links for all nodes individually by calculating the contribution of link in accessibility of node. The adjacent reference node of a link may be any one node of its either side. The critical one out of these two adjacent nodes will be the one for which removal of that link results into higher increase in sum of shortest path lengths from that node to the rest of nodes in the network. Once the links are listed along with their reference adjacent nodes, the NPI of all links with respect to its adjacent critical node is calculated again by using equation (2.1), as presented in Table 2.4. The number of destinations got affected in terms of revised shortest paths after a link removal are also listed.

By looking at the results, it is quite clear that the overall NPI of all links as well as the NPI of links with respect to their adjacent reference node is greater than 1.00, except for the case of isolating link GH. For the sample network, maximum value of the NPI is found for link EF and minimum for links CF and JO. The link EF possesses maximum value due to the reason as

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removal of this link causes increase in path-lengths from/to node E for majority of the nodes in the network. Whereas, minimum value of the NPI for link CF is due to presence of the link CG which is providing an alternate access to almost all of the nodes in case of removal of the link CF.

Table 2.4 Summary of Network Path-length Index for all links of sample road network

Link/s Length

* Total number of destination nodes from any origin node in sample road network = 19

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For the isolating link GH the NPI calculated based on all possible ODs path lengths is 0.355.

The NPI value of this link with reference to both of its adjacent nodes G and H is also found less than 1.00. This value is indicative of reduced number of destinations available, meaning that the accessibility of attached nodes is reduced. To avoid such situations, it is advisable to connect each node in a road network to at least two links to provide an alternate route in the case when one of the two links is removed. For instance, for the network shown in Figure 2.6, addition of a new link FH can avoid division into two sub-networks, as well as can reduce the sum of shortest path lengths from/to some nodes in the network.

The NPI value with reference to attached node can be taken as a measure of link contribution in the accessibility of that node. The link EF (E) has the highest value of the NPI of 1.73 and sixteen out of nineteen (84%) possible destinations from the node E are affected by removal of this link. (reference node of a link is written in parenthesis). On the other hand, the NPI of link CF (C), which apparently seems to be almost at similar position as that of link EF within a sub-network, possesses lowest value of 1.01 with only two destinations affected. Does the difference lie in position of a link within network? Of course it does as is clear in terms of affected destinations. But the major difference here is that of the connectivity of nodes. The node C is connected to three links whereas node E is connected to two links. The third link of node C, link CG, provides a better alternate access to the major sub-network, thereby causing least impact of removal of the link CF on accessibility of node C. Similarly, the links BC and DE have same link lengths and even the same number of affected destinations and apparently seem to have similar locations within the minor sub-network. But, the NPI of two links is still different from each other. This result clearly indicates that the NPI is dependent on link position in the network as a whole rather than its position in some local/sub-network and affected number of destinations.

Another important aspect can be observed by looking at NPI of link FG (F). Although the link has the lowest link length of 3 km only, yet its NPI value of 1.38 with 74% affected destinations is considerably higher compared to many links in the network with much higher link lengths.

This indicates that NPI of a link is independent of its length as well.

It is pertinent to observe from the results that when the link importance is to be evaluated based on impact of its removal on nodes on both sides, the results are a mix, varying from differential and non-uniform impact to a uniform one. The NPI of link EF(E) is the highest 1.73 (84% affected destinations) but the NPI of the same link with respect to the node F is 1.08 only (11% affected destinations). The reason lies here in connectivity of a node and its location within

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the network. The node E is a peripheral node and connected to network through two links only compared to node F which is relatively central and connected through four links (twice than E).

Similar is the case for link JM, where the NPI is abnormally variable with reference to adjacent nodes J, M. Contrary to this, the values of NPI and affected number of destinations for link IJ with reference to nodes I and J are more or less uniform (1.30 and 1.27 respectively). Although, the difference between the connectivity of two nodes is same as that of between the nodes E and F for link EF, yet the NPI is uniform with respect to adjacent nodes. The effect of position within a network is again the reason for uniform values of NPI. The nodes I and J have almost equal accessible portions of the whole network which is not the case for nodes E, F for link EF.

To investigate any relationship between the value of NPI of a link with reference to adjacent node and the affected number of shortest paths from that node, a graph is plotted between the two parameters as shown in Figure 2.7. Although a general trend of an increase in the NPI value with the increase in number of affected destinations/ path-lengths is observed, yet no any fixed pattern can be established. There are links with different NPI values but same number of affected destinations. Secondly, to investigate the effect of connectivity on the NPI, a graph between the NPI and number of links attached to reference node is also plotted as in Figure 2.8. Different values of the NPI for the same number of links attached to a node are observed. The plots presented in Figures 2.7 and 2.8 conclude that the NPI of a link with reference to an attached node is independent of both the affected number of destinations and the connectivity of that node measured in terms of number of links connected.

0.00

Figure 2.7 The NPI and affected number of destinations

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Figure 2.8 The NPI and number of links attached