## 4 Application Studies
### Criticality Assessment  
The proposed criticality indexes yield values in a range of 0-1. For a better comparison of pipe criticality, the index values are then ranked within a network. Given the values of the four indexes, finally, the OVERALL criticality of the individual pipes are estimated. The OVERALL criticality is calculated by summing the rank of each index, and then the summed values are ranked. Note the OVERALL criticality assumes the same weight for each critical index. Visualizations of these results for each network are presented in Figures 3-6 using a percentile ranking. The four colors represent the quantile of the pipe criticality, while the grey color indicates zero criticality of the pipe (blue: 0%-25%; green: 25%-50%; yellow: 50%-75%; red: 75%-100%). Note that a higher percentile indicates a higher criticality of the pipe. Figure 2: Layout of the networks. Figure 4: Critical pipe assessment for (**a**) supply shortage (\(SS\)), (**b**) economic value loss (\(EVL\)), (**c**) pressure decline (\(PD\)), (**d**) water age degradation (\(WAD\)), and (**e**) OVERALL of Network 2. Figure 3: Critical pipe assessment for (**a**) supply shortage (\(SS\)), (**b**) economic value loss (\(EVL\)), (**c**) pressure decline (\(PD\)), (**d**) water age degradation (\(WAD\)), and (**e**) OVERALL of Network 1. For all the networks, the pipes that are closer to the source tend to have a higher OVERALL criticality value compared to other pipes in the network. Failure of pipes sourcing from the reservoir would result in great loss, as the effect will reverberate downstream, thus affecting more users. Tracing down along the main lines from the reservoir, the model is capable of deciding how far the critical part of the main line is and which direction and branch are more critical. This response can be seen in Network 1 (Figure 3) as the critical main lines cease at the 6 th pipe from the reservoir, and the right-hand-side branch from the first pipe is deemed less critical compared to the left-hand-side branch. Critical pipes located further from the source are primarily seen on paths that do not have a loop or  
Figure 5: Critical pipe assessment for (**a**) supply shortage (_SS_), (**b**) economic value loss (_EVL_), (**c**) pressure decline (_PD_), (**d**) water age degradation (_WAD_), and (**e**) OVERALL of Network 3. Figure 6: Critical pipe assessment for (**a**) supply shortage (_SS_), (**b**) economic value loss (_EVL_), (**c**) pressure decline (_PD_), (**d**) water age degradation (_WAD_), and (**e**) OVERALL of Network 4.  
around the nodes with higher demand. A clear example of this response can be seen in Network 4 (Figure 6), as the pipes in loops are deemed critical because they are servicing high-demanded nodes. Each of the indexes has different responses, which is more prominently seen in Network 1 (Figure 3). The \(SS\) and \(EVL\) indexes for some of the pipes in Network 1 are colored grey, indicating that the failure of the relevant pipes does not have any effect on the social and economic aspects; however, there are clear responses in the hydraulic and water quality aspects from those failures. The distinct response from each index shows that the proposed indexes can quantify the criticality of individual components for different aspects of losses. Another interesting result can be seen from Network 3 (Figure 5), in which the \(SS\), \(EVL\), and \(PD\) index values are quite similar to each other, while the \(WAD\) results are more variable than other indexes. By taking into consideration all of the indexes as a sum, the OVERALL criticality may provide more accurate and balanced results for identification of the critical pipes. Network 1 features a densely looped network (\(DoN_{sys}=2.59\)), while Network 2 is a branch-loop hybrid network with \(DoN_{sys}\) of 2.31. Considering the different characteristic of both networks, they are chosen as representatives for further analyses. Figure 7 shows the pressure distribution on both networks to provide more details.