The closeness centrality of Ahmedabad calculated using a primal approach is shown in Figure 28. Node centrality scores are calculated for the primal weighted graph, where edge lengths are used for the weighting factors (Porta et al. 2006). Figure 28a shows the node closeness , calculated for the whole network. Figure 28b shows the local closeness where is calculated for a sub-network of nodes at distance metres from each node. Figure 28c shows the local closeness where is calculated for a sub-network of nodes at distance metres from each node. For primal graphs, the closeness centrality is dominated by the so-called ‘border effect’, i.e., higher closeness scores consistently group around the geometric centre of the image. To some extent less evident in less dense cases, the border effect is overwhelming in denser urban fabrics such as those of Ahmedabad. However, in all cases the border effect affects the spatial flow of enough to prevent the emergence both of central routes and of focal spots in the city fabric (Porta et al. 2006).
C
Figure 28: Closeness centrality (C ) in Ahmedabad, India using a primal approach showing (a) global closeness, (b) local closeness (d < 400 m), and (c) local closeness (d < 200 m) where d is the distance between nodes (Porta et al. 2006). The colour red indicates a high level of closeness, whereas blue indicates a low level of closeness.
7.4.2. Characterisation of entire cities Centrality metrics only describe sub-properties of networks and are strictly relative to individual networks. A wider set of metrics are needed to characterise the networks themselves to allow normalisation and comparison between different networks (Porta et al. 2010). Comparisons between networks can be made using the Minimum Spanning Tree (MST) graph and the Greedy Triangulation (GT) graph. If a real graph and the positions of its nodes in a two-dimensional plane are given, the MST graph is the planar graph (a graph where none of the edges cross over each other) with the minimum number of edges in order to ensure connectedness, while the GT graph is the graph with the maximum number of non-planar edges. In short, MST and GT graphs represent the planar graphs with the lowest and the highest possible cost, respectively. MST and GT will serve as the two extreme cases to normalise the values of the structural measures to be computed, namely efficiency and cost. The cost is defined as the sum of the length of edges in a network:
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