3.1. Hydraulic Simulations

After calculating the percentage pressure drop, the total number of nodes experiencing a pressure reduction greater than 5%, 10%, and 15% compared to normal network conditions (failure-free) is summed for each pipe break simulation. Each pipe is then assigned a value corresponding to the total number of instances where any node experienced a pressure reduction exceeding the specified percentage during the 24 h simulation period.

3.2. Complex Network Theory Application

Given the common approach to hydraulic modeling of water networks, where nodes are connected by links, these systems can easily be represented as graphs. Mathematically, a graph G is a set of vertices ($v\in V$), which are connected by edges ($e\in E$) and expressed as $G(V,E)$. The edges’ direction can be defined according to specific features of the problem, such as the flow of the network during a certain operation time. In this work, the graphs are directed and weighted based on the flow at the maximum consumption time, representing the most critical scenario to pressure reduction in the system. In a matrix notation, a graph can be defined by an adjacency matrix A, where each element $a_{ij}$ represents the connection between two nodes i and j.

Pipe breaks and leakages are modeled in this work based on the weighted and directed graph built on scenarios without anomalies. For pipe bursts, failure scenarios are created by disconnecting the edge corresponding to the broken pipe in the weighted adjacency matrix. The value of the element $a_{ij}$, which was previously equal to the flow rate of the broken pipe, indicating a connection between nodes i and j, is set to 0, which means that those nodes are no longer connected. It is important to note that i and j correspond, respectively, to the upstream and downstream nodes of the pipe based on the initial flow direction of the network. An analysis is then conducted on the increase or decrease in the shortest paths, as well as the disconnection of nodes from their sources when the distance between them becomes infinite. CNT metrics are applied to identify the most important or most connected vertices that influence a larger area. The vertices of the graphs are labeled as water sources and demand nodes. For each scenario, the shortest path between water sources and demand nodes is calculated, forming a matrix, with the sum of the edge weights belonging to the shortest path. Regarding leaks, this action is not performed differently from the rupture scenario where the edge corresponding to the broken pipe is removed from the graph. During the network characterization by CNT, the graph remains with all its edges and vertices.

3.3. Map of Normalized Criticality and Similarity Analysis

The Pearson correlation coefficient helps explore the linear relationship between normalized betweenness centrality values and hydraulic parameters, focusing on nodes influenced by pressure drops during pipe failures. This approach highlights how each pipe failure affects system performance, emphasizing its hydraulic significance. The analysis assesses the degree to which high centrality values align with notable hydraulic impacts. A strong positive correlation would demonstrate that CNT metrics effectively capture hydraulic criticality, suggesting that network theory could serve as a reliable alternative to hydraulic simulations.

Dynamic time warping, in turn, investigates the similarity of the two approaches by comparing sequential patterns of their criticality rankings. While the Pearson correlation focuses on a linear relationship, DTW will capture the shapes of series, shifts, variations, and time lags. For this very reason, DTW becomes so effective at finding non-linear correspondences and examining disagreements between hydraulic- and CNT-based ranking. DTW aligns the ranking sequences of the critical pipes. In addition, it gives the “cost” necessary to transform one sequence into another. A lower DTW cost means a closer similarity; hence, CNT metrics are close to hydraulic outcomes. The higher the cost, the higher the divergence, and one needs to investigate further to determine why the differences exist.

Together, these indicators provide a comprehensive evaluation of how well CNT-based metrics emulate the criticality insights derived from hydraulic simulations, offering both validation and a pathway for deeper analysis of mismatches.

3.4. TOPSIS for Final Ranking Formalization

TOPSIS is herein suggested to assess and rank various alternatives based on multiple criteria. This method is recommended because it accounts for the best- and worst-case scenarios, offering a clear and balanced comparison of the analyzed alternatives. The methodological steps for implementing TOPSIS are highlighted below.

• Building the input decision matrix X, whose elements $x_{ij}$ represent the performance values of each alternative i under each criterion j, which can be either quantitative or qualitative. This matrix is the foundation for the subsequent calculations.

• Computing the normalized weighted matrix U, whose elements $u_{ij}$ represent the normalized evaluations $z_{ij}$ of alternative i under criterion j multiplied by the weight of the correspondent criterion $w_j$. Criteria weights are numerical values assigned to each criterion to reflect its relative importance in the decision-making process, meaning that criteria that are considered more important are attributed higher weights. In comparison, less significant criteria receive lower weights, with the sum of all weights normalized to 1. Specifically, we first normalize the elements of the input decision matrix X as follows:

  $$z_{ij}={\displaystyle \frac{x_{ij}}{\sqrt{{\sum }_{i=1}^m{x}_{ij}^2}}}$$

  (5)

  We consequently apply the weights to the normalized values, reflecting the importance of each criterion. The weighted normalized values $u_{ij}$ of matrix U are computed as follows:

  $$u_{ij}=w_j\times z_{ij},\forall i,\forall j$$

  (6)

• Obtaining the ideal positive solution $A^{*}$ and the ideal negative solution $A^{-}$ from the normalized weighted matrix U. Specifically, the ideal positive solution $A^{*}$ includes the highest values for criteria to be maximized and the lowest values for criteria to be minimized. For example, in evaluating alternatives, criteria representing benefits such as performance or reliability should be maximized, while cost or processing times should be typically minimized. Conversely, the ideal negative solution $A^{-}$ contains the lowest values for criteria to be maximized and the highest values for criteria to be minimized. Mathematically, they are defined as follows:

  $$A^{*}=j\in I^{\prime })$$

  (7)

  $$A^{-}=j\in I^{\prime })$$

  (8)

  Here, $I^{\prime }$ refers to the set of criteria that need to be maximized, while $I^{″}$ includes criteria to be minimized. These two ideal solutions are reference points to assess how close or far each alternative is from the optimal and worst possible scenarios, as explained in the following step.

• Computing the distances between each alternative in the dataset and each of the ideal solutions previously obtained. The distances $S^{*}$ and $S^{-}$ represent how close or far each alternative is from the ideal positive solution $A^{*}$ and the ideal negative solution $A^{-}$, respectively. These distances are calculated for each alternative i using the following formulas:

  $$S^{*}=\sqrt{\sum _{j=1}^n{(u_{ij}-u_{ij}^{*})}^2}$$

  (9)

  $$S^{-}=\sqrt{\sum _{j=1}^n{(u_{ij}-u_{ij}^{-})}^2}$$

  (10)

  Here, $u_{ij}^{*}$ represents the ideal positive solution for each criterion j, and $u_{ij}^{-}$ represents the ideal negative solution for each criterion j. $S^{*}$ indicates how far an alternative is from achieving the ideal scenario across all criteria, while $S^{-}$ quantifies its closeness to the worst-case scenario. Alternatives with a smaller $S^{*}$ and a larger $S^{-}$ are considered more favorable, since they are closer to the ideal positive solution and further from the ideal negative solution.

• Calculating the closeness coefficient $C{C}_i$ for each alternative i as follows:

  $$C{C}_i={\displaystyle \frac{S^{-}}{S^{-}+S^{*}}},0<C{C}_i<1,\forall i.$$

  (11)

  The closeness coefficient quantifies how close each alternative is to the ideal positive solution $A^{*}$, relative to the ideal negative solution $A^{-}$. A higher value of $C{C}_i$ indicates that the alternative is closer to the ideal positive solution and further from the ideal negative solution, making it a more desirable option.

• Ranking alternatives based on their closeness coefficient, with the highest $C{C}_i$ values associated to the best alternatives.

The TOPSIS algorithm integrates failure scenarios and ranks the pipes in the system, offering a systematic method for guiding maintenance decisions. The ranking relies on CNT results, which serve as criteria for multi-criteria decision-making. Combining metrics from both leakage and pipe break scenarios, the algorithm assesses each pipe’s criticality in terms of its impact on network connectivity, hydraulic performance, and overall system resilience. This method ensures a well-rounded evaluation, helping decision-makers prioritize maintenance efforts effectively.

4.1. Ranking Pipes for Maintenance Strategy

The TOPSIS algorithm is applied to the Modena network, chosen as the case study due to its realistic yet medium size and relatively simple topology, which facilitates interpretation and makes the results more understandable. Applying TOPSIS to this network enables the assessment of pipe criticality based on results obtained from complex network theory for both leakage and pipe break scenarios. The Modena network’s topology provides a suitable framework to evaluate how multi-criteria decision-making integrates information from various failure scenarios into a final ranking for pipe maintenance.

The algorithm is implemented using equal weighting for both leakage and pipe break scenarios as a baseline. In addition, the analysis explores the impact of varying the scenario weights to address uncertainties and test the sensitivity of the results to the weighting scheme. Specifically, the following three configurations are evaluated:

Following the criticality of pipe 335 and pipes interconnected with 335, the analysis shifts focus to another reservoir, which is the second most significant water source in the Modena system. The pipe connecting this reservoir to the network and the pipes directly linked to this primary source appear as critical components. These elements are instrumental in the distribution of water from the reservoir to downstream sections of the network. The strategic importance of these pipes plays again a key role in maintaining the functionality of the system, particularly during periods of high demand or in the event of service interruption to the primary reservoir.

4.2. Discussion of Results

The case studies presented in this paper demonstrate the application of hydraulic simulations and CNT to assess criticality in WDNs. The Modena network provides contrasting scenarios in terms of network complexity and scale, allowing for a robust evaluation of the methodologies. In the case study, hydraulic simulations identified critical pipes based on flow rates and the extent of service disruption resulting from pipe breaks or leaks. The simulations showed that pipes near reservoirs or those carrying higher flow rates were more critical, a pattern echoed by the CNT analysis. A comparison of criticality maps revealed a strong alignment between hydraulic simulations and CNT results, particularly in identifying key elements such as trunk pipes. The consistency between these two methods underlines CNT’s potential as a fast, reliable alternative for criticality assessment, particularly in large and complex networks where computational efficiency is paramount.

In certain instances, CNT overestimated the criticality of pipes that hydraulic simulations did not identify as high-risk, largely due to differences in how reverse flows and alternative paths were handled. This underscores the need to refine CNT models to better account for specific hydraulic behaviors, such as flow reversals, which are not fully captured by topological metrics alone. Both case studies emphasized the importance of integrating hydrauli- and network-based approaches to achieve a more comprehensive risk assessment. The use of CNT demonstrated significant advantages, particularly in reducing processing time, making it feasible to model criticality across entire systems rather than being limited to smaller network sections, as is often the case with purely hydraulic approaches. Representing the network as a graph structure enables CNT to compute critical elements rapidly, facilitating full-scale analysis of the entire system. This efficiency broadens the scope of the analysis, allowing for exploring more failure scenarios in less time. Consequently, decision-makers can better understand system vulnerabilities without being constrained by time or computational resources. The methodology outlined in this work leverages CNT to prioritize inspections, repairs, and preventive actions according to criticality, making it compatible with existing maintenance scheduling techniques. Its computational efficiency allows for quick identification of high-risk areas, complementing hydraulic models as an initial screening tool. This combined approach promotes more effective resource allocation and strategic planning, improving system resilience and aligning with contemporary asset management strategies.