## Abstract

The resilience of water distribution networks (WDNs) should be proactively evaluated to reduce the potential impacts of disruptive events. This study proposes a novel hydraulically-inspired complex network approach (HCNA) to assess and enhance WDN resilience in the case of single-pipe failure. Unlike conventional hydraulic-based models, HCNA requires no hydraulic simulations for resilience analysis. Instead, it quantifies the failure consequences of edges (pipes) on the WDN graph by incorporating topological attributes with flow redistribution triggered by failures. This HCNA procedure leads to the identification of critical edges (pipes), as well as impacted ones, representing edges more susceptible to the failure of others. The impacted edges are then systematically resized by integrating HCNA with a graph-based design approach, obtaining a wide range of resilience enhancement solutions. A comparative study between HCNA and a hydraulic-based model for three WDNs confirms HCNA's effectiveness in identifying the most critical pipes in various network sizes. Furthermore, HCNA provides comparable resilience enhancement solutions with a hydraulic-based evolutionary optimization but with significantly lower computational effort (1,400 times faster). Thus, it can efficiently be used for resilience enhancement of large-scale WDNs, where the application of conventional optimizations is limited due to the intensive computational workload.

## HIGHLIGHTS

A novel graph theory-based approach for evaluating and enhancing resilience.

Utilizing topological metrics to reproduce the hydraulic behavior.

Offering optimal resilience enhancement solutions without hydraulic simulation.

Requiring significantly less computational effort than evolutionary optimization.

## INTRODUCTION

Water distribution networks (WDNs) are dynamic and complex systems serving as an integral component of urban infrastructures (Hajibabaei *et al.* 2019). They are subject to various spectrums of natural and human-made disasters, such as earthquakes, floods, and cyber-attacks, threatening their functionality (Diao *et al.* 2016; Assad *et al.* 2019). Any disruption in the functionality of WDNs could trigger cascading events, impacting inhabitants' safety and public health (Shuang *et al.* 2014; Marlim *et al.* 2019; Zhang *et al.* 2020). Consequently, WDN performance needs to be proactively investigated under possible adverse conditions to ensure that they can minimize the negative impacts of disruptions. This is where ‘resilience assessment’ is emerging as a significant consideration in the planning and management of water infrastructures (Meng *et al.* 2018). The term ‘resilience’ can be described as ‘the capability of a system to minimize the magnitude and duration of service failure over its lifetime when facing disruptive events’ (Herrera *et al.* 2016; Butler *et al.* 2017). Common approaches for assessing WDN resilience can be classified into two categories, namely (1) hydraulic-based (also known as performance-based) and (2) property-based (Yazdani & Jeffrey 2012; Pagano *et al.* 2022).

Hydraulic-based approaches quantify the response of WDNs to different failure scenarios through hydraulic simulations. For instance, in global resilience analysis (Diao *et al.* 2016), a WDN is subjected to a system malfunction (e.g., pipe failures), and response curves are generated, wherein the magnitude or duration of water supply failure is plotted as a function of the quantity of failed pipes. The area under the response curves can be used as an indicator representing how resilient a WDN is to a specified system malfunction. Although hydraulic-based approaches have been successfully employed in the literature (Diao *et al.* 2016; Chu-Ketterer *et al.* 2023), certain limitations are associated with them. For instance, the resilience assessment can become very time-prohibitive for large-scale WDNs, specifically when the analysis is conducted for multiple failure modes (Ulusoy *et al.* 2018). Additionally, they require detailed input data depending on failure scenarios, while in many water utilities, network information is often incomplete or inaccurate, and hydraulic models are even unavailable (Chen *et al.* 2021).

Property-based approaches focus on particular structural and topological properties of WDNs that could affect the hydraulic operation and, ultimately, system resilience (Pagano *et al.* 2022). These properties are often assessed based on complex network analysis (CNA), where WDNs are described mathematically as graph objects with vertices (e.g., demand nodes) and edges (e.g., pipes). Various graph metrics can be utilized within the CNA framework to quantify the structural and topological attributes of WDNs. For example, ‘link density’ (ratio of existing edges in a WDN graph to the maximum possible ones) is one of the representative graph metrics for the topological attribute called ‘connectivity’ (Yazdani *et al.* 2011).

In contrast to the hydraulic-based approaches, utilizing CNA demands less detailed data and computational efforts for resilience analysis (Di Nardo *et al.* 2018; Hajibabaei *et al.* 2022). However, most CNA approaches lack a thorough representation of the real-world hydraulics of WDNs. Thus, the present study seeks to advance the current implementation of CNA in WDN resilience assessment and enhancement by developing an efficient, hydraulically inspired approach. To provide a better overview of CNA, this paper offers a detailed research background in section 1.1, followed by the problem statement outlined in section 1.2.

### Research background

Several studies have employed conventional/classic graph metrics in CNA to identify crucial components of WDNs under unfavorable conditions (Porse & Lund 2016; Agathokleous *et al.* 2017; Pagano *et al.* 2019; Chen *et al.* 2021). A particular objective of these studies is to assess or enhance resilience, as graph metrics reflect topological attributes correlating with system resilience to a certain extent (Meng *et al.* 2018). However, it is not entirely clear to what extent conventional graph metrics can capture the performance (i.e., hydraulics) and resilience of WDNs, despite the fact that they provide useful overviews of network structure. To investigate this issue, Meng *et al.* (2018) conducted a comprehensive study on the interplay between sets of topological attributes (e.g., connectivity, efficiency, etc.) and different aspects of resilience (e.g., magnitude and duration of failures). The study revealed that while topology itself can influence WDN performance, only certain classical graph metrics are appropriate as surrogate indicators for resilience analysis. However, those so-called appropriate metrics mainly offer simplified and general insights into resilience rather than a thorough assessment. This is due to the inherent constraint of conventional metrics, which focus solely on structural characteristics and do not account for hydraulic behavior. This point was also demonstrated by Chen *et al.* (2021), who found a disparity between the hydraulic performance of pipes and the outcomes of conventional graph metrics.

Researchers have recently focused on developing ‘customized graph metrics’ that can better align CNA with the actual hydraulic behavior of WDNs. Such metrics have been utilized for various purposes, including analyzing water quality (Sitzenfrei 2021), reducing optimization workload (Diao *et al.* 2022), designing optimal diameter (Sitzenfrei *et al.* 2020; Hajibabaei *et al.* 2023), capturing hydraulic characteristics (Giustolisi *et al.* 2019; Simone *et al.* 2020), and, especially, assessing and enhancing resilience (Pagano *et al.* 2019; Lorenz & Pelz 2020). For instance, Herrera *et al.* (2016) proposed a graph-based resilience index that incorporates network constants (e.g., pipe length and diameter) as a surrogate for potential energy loss in water transport routes. Since Herrera's index mainly relies on structural attributes, Lorenz & Pelz (2020) modified it by including nodal demands as a hydraulic feature. They employed the modified index as an objective function in a cost–benefit optimization problem for resilience enhancement. Li *et al.* (2023) proposed a customized metric based on ‘edge betweenness centrality’ for seismic resilience evaluation, which can determine the seismic functionality of pipelines after an earthquake. In another study, Ulusoy *et al.* (2018) introduced a metric of pipe criticality for resilience analysis using a random walk between a couple of sources (e.g., reservoir) and demand nodes. This metric facilitated the pre-selection of critical pipes and can be further complemented by detailed hydraulic modeling to achieve accurate results.

Some other studies have utilized graph metrics alongside the structural self-similarity observed in WDNs, known as fractality, to conduct resilience and vulnerability analyses. For example, Di Nardo *et al.* (2018) employed classic fractal and graph metrics and introduced a resilience analysis approach to identify specific critical pipes. This approach was developed based on the relationship between the resilience and ‘average path length’, specifically for single-pipe failures that do not disconnect the WDN graph. Giudicianni *et al.* (2021) assessed WDN vulnerability by randomly and progressively removing pipes until the disconnection of networks. They utilized fractal and graph metrics in their evaluation, leading to the development of a vulnerability index integrating both fractal and topological aspects.

The other body of literature has specifically focused on CNA-based methods for pipe/node ranking. For example, Pagano *et al.* (2019) proposed a graph-based resilience analysis framework that targets the potential effect of single-pipe failure on the connection between the nodes and sources, ultimately ranking critical pipes. Yazdani & Jeffrey (2012) presented a technique incorporating pipe sizes and flow direction into topological metrics to identify and rank critical nodes. In another study, a pipe ranking method was proposed by Pagano *et al.* (2022) using several graph metrics correlated with WDN resilience. In this approach, the responses of the metrics to pipe failures are aggregated through the Bayesian Belief Network to rank critical pipes.

### Problem statement

The above-mentioned studies have established a valuable foundation for the CNA-based resilience assessment. However, there are still certain limitations that need to be addressed, especially regarding resilience analysis under pipe failures. These limitations can be summarized as follows: firstly, most proposed metrics/approaches have predominantly focused on the connectivity features (i.e., a connection between sources and nodes) and overlooked the impacts of water flow and its redistribution in case of failures. Secondly, they are unable to quantify the failure propagation (i.e., spatial impact of failures on network components), which is important for resilience improvement. Lastly, they cannot be used individually for optimal resilience enhancement, particularly for complex and large WDNs.

To address these limitations, the current study aims to develop a hydraulically-inspired complex network approach (HCNA), which can efficiently assess and enhance WDN resilience under single-pipe failure. HCNA integrates topological attributes (connectivity, redundancy, and efficiency) with hydraulic features (nodal demands and water flow) to imitate the real-world hydraulic behavior of WDNs. It utilizes tailored graph metrics enriched with dynamically adjusted weights to track the failure propagation resulting from pipe failures. Accordingly, HCNA can quantify the spatial impact of failures and identify critical pipes. HCNA is also integrated with a graph-based design approach, providing optimal/near-optimal solutions for WDN resilience enhancement without hydraulic simulation. The effectiveness of HCNA in identifying critical pipes and enhancing resilience is validated by comparing its results with those obtained through a conventional hydraulic-based approach.

## METHODS AND MATERIALS

*et al.*2020) to resize particular pipes for resilience enhancement. Lastly, the results of HCNA are compared and validated with those derived from the hydraulic-based approach. Detailed explanations of each part are provided in the following sections.

### Hydraulic-based approach

WDN resilience can be described through different metrics from the hydraulic perspective (Meng *et al.* 2018). To better elaborate, consider the ‘Quality of Serviceability’ plot illustrated in Figure 1(a). In case of an adverse event at time *t*_{0}, a loss of serviceability appears from time *t*_{0} to *t*_{1}, which could occur immediately (e.g., due to an earthquake) or gradually (e.g., due to a single-pipe failure). Then, a progressive recovery of serviceability is observed from time *t*_{1} to *t*_{2} as a result of recovery attempts. The shaded area between time *t*_{0} and *t*_{2} can be described as ‘resilience loss’ (Bruneau & Reinhorn 2007). Thus, from the hydraulic point of view, WDN resilience can be enhanced either by 1) decreasing the level/magnitude of serviceability loss or/and 2) increasing the recovery rate of serviceability (Pudasaini & Shahandashti 2020). This study focuses solely on the first category; therefore, the service failure magnitude is chosen as a measure to assess and enhance resilience in sections 2.1.1 and 2.1.2.

#### Resilience assessment and pipe ranking

*SFM*).’ The

*SFM*resulting from the failure of pipe

*j*is calculated as follows:where

*t*denotes time steps () during the simulation period

*T*,

*n*denotes the number of junctions, denotes the required demand of node

*i*at time

*t*(), denotes the supplied demand (outflow) of node

*i*at time

*t*() in the case of failure of pipe

*j*.

*et al.*1988) is employed for pressure-driven analysis as follows:where denotes the pressure of

*i*at time

*t*, denotes the minimum pressure, denotes the pressure required to fulfill the entire demand, and

*γ*is the pressure exponent. In this study, the values of and , and

*γ*are considered as 0 m, 30 m, and 0.5, respectively (Önorm 2018; Gorev

*et al.*2021).

The *SFM* resulting from the failure of each pipe is determined by setting its status to ‘closed’ for 24 h in EPANET 2.2. Consequently, important pipes are identified and ranked based on their corresponding *SFM*. This procedure assumes that each pipe can be isolated individually by isolation valves. This means that each pipe is considered as one segment. However, the proposed method also works if there is more than one pipe in each segment.

#### Resilience enhancement based on optimization

*ΙΙ*(non-dominated sorting genetic algorithm) (Deb

*et al.*2002) is used as a multi-objective optimization engine, which can self-adaptively handle the constraints (Minaei

*et al.*2020). The current pipe resizing problem is formulated with two conflicting objectives as follows:

In Equation (3), denotes the number of resized pipes, denotes the unit cost of pipe as the function of the diameter and the road type , and denotes the pipe length. In Equation (4), denotes the number of total pipes and is the average *SFM* of a WDN calculated only at the peak hour to capture the maximum impact of pipe failures.

In Equation (5), the optimization problem is constrained by and , which are physical and practical constraints, respectively. Since hydraulic simulation is performed by EPANET 2.2, the physical constraints, i.e., the conservation of mass and energy, are automatically satisfied. According to the practical constraint, every solution is feasible as long as it keeps the piezometric pressure of nodes above the minimum pressure (). The mathematical representation of these constraints is provided in the supplementary information (SI) file. It is worth mentioning that the constraints are handled within the optimization process in a self-adaptive manner using the tournament selection method, as detailed in Minaei *et al.* (2020).

In Equation (6), the optimization decision variable for a pipe includes a set of available diameter classes in the investigated WDNs. In this equation, represents the number of available commercial diameters. Additionally, 0 stands for a ‘do nothing’ option, which is considered to give the possibility of not replacing the pipes. Hence, the search space size of the pipe resizing optimization problem for a WDN with pipes encompasses possible solutions.

The optimization parameters (e.g., population size and generation number) are chosen regarding the characteristics of investigated WDNs, described in the case study section.

### Hydraulically-inspired complex network approach

#### Graph metrics used for HCNA

The foundation of HCNA is laid on graph theory. Accordingly, a WDN is mathematically described as a graph *G(N,E)*, where *N* is a set of nodes representing reservoirs, tanks, and junctions, and *E* is a set of edges representing pipes, valves, and pumps. The number of elements in the sets *N* and *E* is denoted as *#N* and *#E*, respectively. In addition, a subset *D* (*D ⊆ N*) is defined in the WDN graph, which encompasses demand nodes (sinks) that are supplied by a set of source nodes *S (S ⊆ N)*. To better reflect hydraulic behavior, network information can be incorporated into a WDN graph as node/edge weights. For example, the ratio of pipe length to diameter (Herrera *et al.* 2016; Pagano *et al.* 2019, 2022) or the inverse of pipe diameter (Di Nardo *et al.* 2018) can be used as edge weights, serving as an indicator of flow resistance in WDN resilience analysis.

The weights assigned to the edges of a WDN graph can be divided into two categories – ‘static’ and ‘dynamic.’ The static approach has been commonly used in the literature, wherein edge weights remain constant throughout the analysis. However, a more recent approach, ‘dynamic weights’, allows for continuous modification of edge weights during analysis (Sitzenfrei *et al.* 2020; Hajibabaei *et al.* 2023). While various traditional graph metrics have been proposed to assess WDN resilience, HCNA utilizes only two graph metrics based on dynamic weights.

*Shortest path (SP)*’, which determines a route between two graph nodes whose corresponding cost is minimal. In this context, the term ‘cost’ can be interpreted as the sum of all (positive) edge weights connecting those two nodes (Dijkstra 1959). In a WDN graph, the

*SP*between sources and demand nodes are of interest, especially those with the least hydraulic resistance, as they represent efficient paths for flowing water. Therefore, the hydraulic resistance can be included in the weighting function of

*SP*to find the most efficient routes in WDNs. The resistance of an edge

*e*() with the diameter and the length can be estimated as follows (Herrera

*et al.*2016; Lorenz & Pelz 2020):where denotes the friction factor of edge

*e*calculated regarding roughness and diameter by assuming the turbulent flow in pipes (Lorenz & Pelz 2020):

*Demand edge betweenness centrality (EBC*is selected as the second graph metric, which is a customized version of the traditional edge betweenness centrality metric (Sitzenfrei

^{Q})*et al.*2020).

*EBC*can be particularly utilized for flow estimation in WDN graphs, where nodal demands are the primary driving force for water flow. In addition, it can be computed directly for the weighted graph of WDN with minimal computational effort. The

^{Q}*EBC*method identifies the

^{Q}*SP*connecting a source node

*S*and every demand node

*i*and adds the corresponding demand of each demand node

*i*() to the

*EBC*values located in

^{Q}*SP*. Accordingly, the of edge

_{S,i}*e*is formulated as follows:

*SP*(see Figure 2(b))

_{N4,S}*.*This demand increases friction losses along the edges located in the

*SP*. Consequently, inspired by the energy balance concept in a looped WDN, the next larger demand in Figure 2(c) (i.e., 3 L/s) should be routed to the alternative route (the left branch). To accomplish that, in the second iteration, the weights (resistance) of the edges along the

_{N4,S}*SP*are modified with a factor that is the function of the nodal demand () and maximum demand in the network (), calculated by . By performing the mentioned process for all the demand nodes and modifying the weights dynamically, the final values are calculated and assigned to the graph edges (see Figure 2(d)). It is worth mentioning that the inspiration behind using this dynamic weighting function comes from the quadratic correlation between water flow and friction losses observed in the Darcy–Weisbach equation (Rossman 2000). This dynamic approach provides a more realistic estimation of water flow and prevents the values from being concentrated in only a few particular edges (shortest-path tree). This is especially significant when dealing with large demands that need to be efficiently redistributed throughout looped networks under pipe failures. In such a situation, the dynamic approach endeavors to emulate real-world behavior by utilizing multiple available pathways for flow redistribution, aligning with the concept of maintaining energy balance in networks. Further details on various dynamic weights are provided in Hajibabaei

_{N4,S}*et al.*(2023).

#### HCNA for assessing and enhancing resilience

In HCNA, the first objective is to identify and rank critical edges in the case of single-edge failure, thereby assessing the resilience of WDN in such scenarios. The second objective is to detect edges with the potential for resizing, aiming to enhance WDN resilience and reduce the adverse effects of edge failures. To achieve these two objectives, the impact of each edge failure on the WDN graph is assessed considering two aspects: (1) connectivity between the source and the nodes associated with the failed edge and (2) the flow redistribution triggered by the failure. The main idea here is to track the (possible) redistributing paths systematically (using ) and investigate if the network has sufficient connectivity and capacity to handle this redistribution. The outcome of this systematic investigation for every single edge is summarized in the form of a matrix called ‘failure matrix’. This innovative matrix serves a dual purpose: firstly, it identifies edges that would have a more substantial impact in the event of failure (i.e., critical edges); secondly, it detects edges that would be more vulnerable to the failure of others (i.e., highly impacted/overloaded edges), highlighting their potential for resizing. The HCNA framework and its components are explained in detail as follows.

*F*consisting of #

*E*×#

*E*elements. The element in

*F*represents the failure consequence of edge

*n*on edge

*k*. Next, of edges are calculated for the ordinary situation without any failures, referred to as ‘normal condition’ (as shown in Figure 3(a)). Following, a pipe failure scenario is considered by removing an edge

*n*from

*G*, referred to as an abnormal condition. In this situation, two possible scenarios can occur:

In the first scenario, an edge failure leads to the disconnection of at least one demand node from the source (see *e1* and *e5* in Figure 3(b)). Under these circumstances, the total unfulfilled demand is equivalent to the that has been routed through the removed (failed) edge *n* under normal conditions (i.e., ). Thus, to represent the effect of the failed edge in the failure matrix, is allocated to (i.e., = 20.8, = 3.0), while the remaining elements are set to zero ( = 0, *k**n*). Accordingly, the non-zero values along the diagonal of *F* (Figure 3(d)) are associated with edges such as *e1* and *e5,* whose removal disconnects a portion of the network from the source node.

*e*2 fails (

*n*

*=*2 in Figure 3(c)), its corresponding load () must be redistributed by existing edges. However, only particular edges are responsible for this redistribution. These edges (blue-colored ones in Figure 3(c)) can be determined by calculating the extra load , which represents the differences between with . Next, it is assumed that edge

*k*is only impacted (or overloaded) by the failure of edge

*n*when its exceeds its maximum capacity (see Figure 3(c)). Accordingly, the failure consequence of edge

*n*on edge

*k*is formulated as follows:

Here denotes the extra load on *k* due to the failure of *n* (L/s), denotes the maximum capacity of *k* (L/s), and denotes the overload coefficient of *k* (–).

is calculated based on the maximum acceptable velocity (2.5–3.5 m/s (Baur *et al.* 2019)) by . Besides, converts the overloaded values () from the maximum scale to the optimal scale, assuming that most pipes are designed optimally. Thus, is calculated by , where as the optimal velocity is obtained from the recommended values listed in Table 1.

D (mm) | 80 | 100 | 125 | 150 | 200 | 250 | 300 | 350 | 400 | 500 | 600 | 700 |

V_{opt} (m/s) | 0.80 | 0.80 | 0.80 | 0.85 | 0.90 | 0.95 | 1.00 | 1.05 | 1.10 | 1.20 | 1.30 | 1.40 |

D (mm) | 80 | 100 | 125 | 150 | 200 | 250 | 300 | 350 | 400 | 500 | 600 | 700 |

V_{opt} (m/s) | 0.80 | 0.80 | 0.80 | 0.85 | 0.90 | 0.95 | 1.00 | 1.05 | 1.10 | 1.20 | 1.30 | 1.40 |

*F*is completed. Thereafter, the ranking of failed edges is determined using their graph-based failure magnitude (

*GFM*) calculated by Equation (11). Accordingly, each column of

*F*is summed and then normalized to the total demand value (i.e., the sum of nodal demands) indicated by

*TD*. For instance, the percentage of

*GFM*for

*e*1 and

*e*3 in Figure 3 is 100% (20.8/20.8) and 14% (3/20.8), respectively. After ranking the edges based on their

*GFM*, we investigate the correlations and trends between their corresponding

*SFM*(determined using the hydraulic approach) for all pipes in three WDNs. This analysis provides insights for classifying the criticality of edges based on their

*GFM*.

*F*(excluding the main diagonal), resulting in overloaded magnitude (

*OM*) for the impacted edges (Equation (12)).

*OM*are then used for edge resizing. For this purpose, the graph-based design approach proposed by Sitzenfrei

*et al.*(2020) is employed, where is used as the representative of water flow in the volumetric flow equation (i.e., ). In this study, we enlarge overloaded edges (i.e., impacted edges with

*OM >*0) by adding their corresponding

*OM*to their as an estimation for water flow. Thus, using Equation (13), new diameters for overloaded edges can be estimated as follows:

Here, denotes design velocity, ranging from 0.5 to 2.5 m/s (Baur *et al.* 2019). By varying with 0.01 m/s time steps, 201 various solutions can be obtained. For instance, by setting for *e*6 and *e*7 (Figure 3(f)), new diameters are determined, which is considered as one solution for resilience enhancement. Note that if the new diameter is equal to or smaller than the existing one, it is assumed that no replacement will happen. Once all possible solutions are determined, the WDN graph of each solution is converted to EPANET 2.2 files. Subsequently, the corresponding *SFM* and costs of every solution are calculated and compared with those obtained with the hydraulic-based optimization.

## CASE STUDIES

The third case study (Figure 4(c)) is a real-world large WDN in Austria comprising 4,021 pipes, 3,558 junctions, and a total length of 211 km. More information on case studies can be found in SI.

All of the WDNs are selected for resilience enhancement scenarios through pipe resizing. Optimizing D-Town and the large WDN using the evolutionary algorithm and hydraulic simulations is challenging due to the high number of evaluations required in the optimization model. This process demands intensive computational efforts, which cannot be completed within a reasonable timeframe. Thus, the hydraulic-based resilience enhancement is conducted for the small WDN (Figure 4(a)), and its results are used to validate HCNA outcomes. After the validation, HCNA is employed for the resilience enhancement of the D-Town and the large WDN.

For pipe resizing optimization of the small WDN, six diameter classes from 76.2 to 254 mm (3–10 inches) with unit pipe costs from 8 to 32 $/m were considered. The optimization was solved with 2,000 generations and a population size of 100, which were tuned after multiple initial trials and sensitivity analysis. The solutions converged after 53,600,000 function evaluations (generation number × population size × number of pipes), fulfilling the convergence condition (i.e., stopping after 10,000 function evaluations without any further improvements in the solutions). For the D-Town WDN, pipe resizing with HCNA was conducted based on the pipe options and related costs suggested in the literature (Marchi *et al.* 2014). For the large WDN, pipe resizing based on HCNA was performed using 15 pipe options varying from 76.2 to 914.4 mm (3–36 inches), with the unit costs ranging from 8 to 1,200 $/m (Sitzenfrei *et al.* 2020).

## RESULTS AND DISCUSSION

The results are presented in two main sections. Section 4.1 involves a comparison of pipe ranking between the hydraulic-based approach and HCNA. This study explores pipe rankings from two aspects. Firstly, it investigates the correlations and trends between the hydraulic-based ranking (based on *SFM*) and HCNA ranking (based on *GFM*) for all pipes in each case study. For this purpose, Spearman's rank correlation coefficient (Spearman 1904) is employed, a metric that ranges from −1 to +1 and measures the relationship between the rankings of two variables. Secondly, this study conducts a more in-depth comparison, specifically focusing on the critical pipes. In this paper, a pipe is considered ‘critical’ from the hydraulic perspective if its corresponding *SFM* is greater than or equal to 1%. To facilitate a meaningful comparison between the critical pipes of the two approaches, we look at the top-ranked pipes with *SFM* ≥ 1% and compare them with an equal number of top-ranked edges obtained from HCNA. This comparative analysis enables us to evaluate how effectively HCNA can replicate real-world hydraulic behavior by identifying critical pipes where *SFM* ≥ 1%. Finally, we provide recommendations for classifying the criticality of edges solely based on their *GFM* values, drawing upon all the aforementioned comparisons.

After identifying critical pipes in section 4.1, section 4.2 focuses on enhancing the resilience of WDNs through pipe resizing.

### Pipe ranking comparison

*SFM*(sorted in descending order) compared to the corresponding distribution of

*GFM*for the small WDN with 268 pipes. In Figure 5(b), the

*x*-axis is represented on a logarithmic scale to better highlight the comparison among the critical pipes.

*SFM*= 0. Nearly all of these pipes, except for the 66th one, also have

*GFM*values of zero. Shifting the focus to critical pipes, the hydraulic-based model identifies 46 pipes as critical (i.e., those with

*SFM*≥ 1%). HCNA can successfully recognize 45 of them among its 46 top rankings, accounting for 98%. Figure 6 visually compares the 46 highest-ranked pipes between the hydraulic-based approach and HCNA. In this figure, pipes (edges) are color-coded and sized based on their ranking. Accordingly, HCNA provides highly similar rankings to the hydraulic-based model for most pipes. The only pipe that HCNA overlooks (P495 in Figure 6(a)) has a very low

*SFM*(i.e., 1%), ranked last among the critical pipes. Additional details on the pipe ranking for this case study can be found in the SI file.

*SFM*and their corresponding

*GFM*. This figure reveals notable similarities in the trend between these metrics, particularly among pipes with higher

*SFM*, as illustrated in Figure 7(b). The Spearman correlation index computed for this WDN yields a value of 0.78, signifying a strong positive relationship between the two metrics. According to the hydraulic-based approach, 228 pipes in this network are identified as critical (

*SFM*≥ 1%), with 85% among the top 228 ranked edges in HCNA.

*et al.*(2019) for comparing a graph-based ranking against the hydraulic-based ranking, focusing on the top 40 high-ranked pipes. To extend this comparison, we have also evaluated our HCNA for these 40 highly-ranked pipes. Figure 8 provides a visual comparison between the hydraulic-based approach and HCNA for these pipes. Detailed results of the graph-based method proposed by Pagano

*et al.*(2019) are presented in the SI file. Accordingly, HCNA provides better results than the one in the literature, especially regarding the order of ranking. The difference in the order of rankings between HCNA and the hydraulic-based approach is more than five for only six pipes highlighted in Figure 8(a). On the other hand, the proposed graph-based method in the literature creates two subsets for pipe ranking, wherein most rankings differ from the ones obtained by the hydraulic-based approach (shown in Table S3 in SI). This is because it relies solely on topological features and neglects the effects of water flow and edge capacity on resilience assessment.

*SFM*and

*GFM*metrics for the third case study (large WDN) in Figure 9 reveals that the similarities in their trend are less prominent than those observed for the previous WDNs. The Spearman correlation index, computed at 0.68, confirms this observation but still indicates a good positive correlation between these metrics. Hydraulic simulations of this WDN show that only 82 out of 4,021 pipes have

*SFM*≥ 1% under single-pipe failure. A total of 78 out of the 82 critical pipes are located in loops, meaning their failure only alters the connections between the reservoir and demand nodes and does not cut off a portion of the WDN. HCNA can identify only 63% of these critical pipes among the top 82 ranked edges. This is primarily attributed to the more complex structure of this WDN than the previous case studies, comprising over 4,000 pipes distributed across multiple loops, making it inherently difficult for HCNA to identify all critical pipes. Nevertheless, when focusing on the most critical pipes with

*SFM*≥ 10%, HCNA shows promising results by identifying 91% of them. According to the hydraulic-based model, the top 32 ranked pipes have

*SFM*≥ 10%, and the first 29 of these are recognized by HCNA (see Figure 9(b) and Table S4 in SI).

Apart from comparing the two methods, it is essential to note that some users may exclusively rely on HCNA and require a threshold to classify the criticality of edges based on *GFM* values. According to the results from Figures 5, 7, and 9, a similar threshold to *SFM* can be adopted for small to medium-sized networks, such as the first and second case studies, where *GFM* demonstrates high accuracy. This implies edges can be classified as critical from the HCNA perspective in such WDNs when their *GFM* ≥ 1%. However, for large and complex networks (e.g., third case study) where *GFM* accuracy decreases, a more conservative threshold of 10% is suggested to focus only on the most critical pipes. It is worth mentioning that the novel approach introduced in this study can also be employed in a hybrid manner. For instance, one can pre-select a set of pipes using HCNA (e.g., those with *GFM* ≥ 1%) and then accurately rank them based on the hydraulic-based model.

### Resilience enhancement

*SFM*of the networks (

*SFM*

_{avg}) through pipe resizing. For this purpose,

*SFM*

_{avg}reduction for each solution can be determined as . Figure 10 shows the obtained solutions for the resilience enhancement of the WDNs. In this figure, ‘optimization’ refers to the solutions obtained with the evolutionary optimizations and the hydraulic-based model. Besides, HCNA (colored dots) represents the solutions derived from the failure matrix and the graph-based design approach, where each dot corresponds to one design velocity (see Equation (13)). Note that for obtaining the HCNA solutions, no hydraulic run is required. We only converted all the HCNA solutions to EPANET 2.2 files and ran the hydraulic simulations to compare their corresponding costs and

*SFM*

_{avg}with those obtained with the optimization. In addition, ‘#critical pipes’ in Figure 10 denotes the number of pipes in each solution with

*SFM*≥ 1%.

Figure 10(a) shows that HCNA can offer optimal solutions for resilience enhancement. Even a few HCNA solutions can outperform those obtained with the optimization. This figure indicates that a 38% reduction in the at the cost of 10,000$ leads to a decrease in the number of critical pipes from 46 to 37. However, resizing additional pipes beyond this point does not further decrease the number of critical pipes. This is because, beyond this point, almost all the critical pipes of the solutions (36 out of 37) are not located in loops, meaning their failure only isolates the network from the source. Therefore, adding redundant capacity by resizing the pipes in loops does not have any additional impact on the and only increases the costs.

One can argue that HCNA provides fewer solutions than optimization. However, it accurately predicts failure propagations and optimally resizes the overloaded edges (i.e., impacted edges with *OM* > 0) without hydraulic simulations. Moreover, it is significantly efficient regarding computational time. The execution time for optimizing the small WDN with 2,000 generations took around 240 h (10 days) on a desktop PC (Intel^{®} Core™ i7–8,700 CPU @ 3.2 GHz). In comparison, HCNA only required 14 s to obtain the solutions and 10 min to convert them to the EPANET files and calculate their corresponding . This implies that for this specific WDN, HCNA can provide optimal solutions around 1,400 times faster than traditional optimization. Detailed information on the computational time of approaches is provided in Table S5. Note that when employing the traditional optimization approach, we assume that we do not have prior knowledge about the impacted/overloaded edges (obtained by HCNA) in the optimization problem. Therefore, the optimization search space size is immensely large, and the search is conducted considering all existing pipes in the WDN.

Figure 10(b) illustrates the resilience enhancement solutions for D-Town based on HCNA. The results exhibit a similar trend to the first case study, demonstrating that resizing the edges up to a specific point corresponding to an investment of $240,000 can reduce the average *SFM* in the existing WDN by approximately 30.5%. Beyond this investment threshold, the most critical pipes in the new solutions are those whose failures would disconnect them from the source. Consequently, by adding extra capacity to the pipes within loops, the resilience enhancement solutions stagnate.

The results of HCNA for resilience enhancement of the large WDN are presented in Figure 10(c). The findings reveal that HCNA can reduce the by up to 92%, only by resizing the overloaded edges (i.e., impacted edges with *OM* >0) derived from the failure matrix. Going further into the details, 368 pipes are considered as ‘overloaded edges’ in the failure matrix. Resizing 86 of them leads to a 65% reduction in , decreasing the number of critical pipes from 82 to 40. Moreover, resizing 234 out of 386 overloaded edges at the cost of 1.53 M$ results in a 92% reduction in , decreasing the number of critical pipes from 82 to 10. Further resizing beyond this point only increases costs without contributing to resilience enhancement.

As previously mentioned, due to the computational burden of the evolutionary optimization process, only HCNA is used for this network. To better illustrate this issue, consider integrating the hydraulic model with the optimization engine for this WDN with only 100 generations and 100 populations. In this scenario, the optimization would require 40,210,000 evaluations (100 × 100 × 4,021), with each evaluation in the optimization model taking approximately 15 min (i.e., closing pipes one by one in EPANT 2.2 and finding the optimal combination). Hence, the process would necessitate about 10 M h to acquire the Pareto-optimal solutions. In comparison, HCNA only takes 3 h to obtain the 142 solutions presented in Figure 10(c) and 35 h to calculate their associated (for detailed information, see Table S5 in the SI file). Consequently, the novel approach proposed in this study can significantly reduce the execution time for obtaining resilience enhancement results, providing decision-makers with cost-effective and highly efficient solutions for WDN rehabilitation.

It can be argued that HCNA's accuracy could diminish with increasing WDN complexity (as depicted in Figure 9), potentially impacting the reliability of resilience enhancement results. However, it is crucial to note that despite variations in rankings between the two approaches in Figure 9, HCNA correctly pinpoints the highly impactful pipes while effectively identifying over 3,000 ones with minimal impacts (*SFM* < 1%). This demonstrates HCNA's ability to distinguish highly impacted/overloaded edges requiring resizing and insignificant edges that should be left untouched. Consequently, HCNA contributes to enhancing resilience by offering solutions, although they may not be optimal due to disparities in top rankings compared to the hydraulic-based approach. To better illustrate the effectiveness of HCNA solutions in improving the resilience of the large WDN, we provided Fig. S4 in the SI file. This figure compares the *SFM* distribution of two HCNA solutions against the *SFM* distribution of the existing WDN. It proves how these solutions can significantly reduce the failure magnitude of critical pipes, ensuring that no pipes have *SFM* values exceeding 10% (see Fig. S4).

## SUMMARY AND CONCLUSION

This study proposes an innovative HCNA, which can simultaneously assess and enhance WDN resilience in the case of single-pipe failure. HCNA attempts to reproduce WDN hydraulics by combining topological characteristics like connectivity, efficiency, and redundancy with hydraulic features such as flow, friction loss, and demand. HCNA can identify critical pipes, predict failure propagation, and provide (near) optimal solutions for resilience enhancement under pipe failures. A comparative analysis of HCNA with a hydraulic-based model on three WDNs with various sizes and complexities confirmed the efficiency and accuracy of the newly developed technique. The results revealed that HCNA could significantly alleviate the computational workload, especially for optimal resilience enhancement, by being approximately 1,400 times faster than the evolutionary optimization for the small WDN. This high efficiency is particularly beneficial for large-scale and real-world WDNs, where conventional optimization methods are excessively computationally intensive. Therefore, HCNA can be a valuable tool for decision-makers, particularly advantageous for our ongoing ‘RESIST’ project, where the focus is on evaluating and improving the resilience of an Alpine network using graph theory.

While HCNA offers several benefits, it also has some shortcomings and limitations. For instance, HCNA's accuracy still requires further improvement, particularly when employed for accurate pipe ranking in complex and large WDNs characterized by numerous loops. Besides, the graph metric used in HCNA (demand edge betweenness centrality) lacks topographical information (elevation data) that could potentially impact the hydraulics. Additionally, HCNA results may be impacted by the selection of velocities (optimal and maximum) used to estimate failure consequences. Thus, it is crucial to determine them accurately based on the characteristics of case studies. Furthermore, HCNA relies on specific network data (e.g., pipe diameter, length), which can pose challenges in cases where this information is unavailable, especially for aged or poorly documented WDNs. However, it is important to note that HCNA eliminates the need for complex and time-consuming hydraulic models that demand extensive data calibration, often at a substantial cost.

For future work, HCNA can be expanded to other failure modes, including multiple-pipe failure and excess demand. In addition, it can be of interest to integrate HCNA into the evolutionary optimization approaches to reduce the runtime and search space for rehabilitation problems. The other potential of HCNA relates to the failure matrix concept. The proposed failure matrix in this study was exclusively employed to recognize the overloaded edges under pipe failure, representing ‘negative propagation.’ Nonetheless, this innovative concept can also be utilized to explore ‘positive propagation’ by implementing relevant strategies (e.g., adding redundant paths).

## ETHICAL APPROVAL

Ethical responsibilities of authors are approved.

## CONSENT TO PARTICIPATE

All the authors agreed with the content and participation.

## CONSENT TO PUBLISH

All the authors agreed with submission and publishing.

## FUNDING

The project ‘RESIST’ is funded by the Austrian security research programme KIRAS of the Federal Ministry of Finance (BMF). This study was also part of a research stay at the University of Texas at Arlington, which was funded by the Austrian Marshall Plan Foundation, and the National Science Foundation (NSF): 1926792.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

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