https://orcid.org/0000-0001-8603-6941
ABSTRACT
In this study, the spatial relationship between critical pipes identified using edge-based centrality measures and pipes with higher failure probability-based on selected vulnerability indicators were analysed in sanitary sewer networks. By analysing two sub-networks, one residential and the other a central network, significant spatial associations between pipes with high centrality values and those exhibiting adverse conditions (poor CCTV grades, previous blockages, and low self-cleaning capabilities) were identified. Path-based centrality measures, particularly edge betweenness and K-path edge centrality were less influenced by weights when identifying critical pipes. In contrast, non-path-based measures like nearest neighbour edge centrality could identify more localised critical pipes within the sewer networks investigated. The results showed that the spatial patterns between critical pipes and pipes in adverse conditions were not random and could support proactive maintenance planning and the development of more resilient networks. Additionally, the impact of network structure, connectivity, and differences in the composition of pipe attributes could contribute to variations in the strength of observable spatial associations.
HIGHLIGHTS
Significant spatial clustering was identified between sewer pipes investigated with higher centrality values and those in adverse conditions.
These spatial associations identified have practical implications for vulnerability assessment and maintenance planning.
Factors such as network assortativity, topology, and physical characteristics of pipes can affect the interpretation of spatial associations.
INTRODUCTION
The ability of sewer networks to withstand disturbances is crucial for operation and maintenance (Barthélemy 2011a). Vulnerability in the context of this study refers to the criticality or importance of a pipe, such that the occurrence of an operational disturbance such as a blockage can disproportionately impact larger portions of the sewer network (Zhang et al. 2017). Common methods that have been used for assessing sewer network vulnerability include risk assessment tools (Cardoso et al. 2007; Marlow et al. 2011), condition assessment using CCTV inspections (Caradot et al. 2020), and hydraulic and hydrodynamic modelling (Möderl et al. 2009). Some strengths of using CCTV inspections for vulnerability assessment in sewers include enhanced safety, image storage and examination, precise defect identification, cost-effectiveness, non-destructive nature, versatility, and the potential for automation and increased efficiency (Romanova et al. 2013; Meijer et al. 2019; Caradot et al. 2021). Hydraulic and hydrodynamic modelling and simulations facilitate the analysis of various scenarios at a relatively low cost and within a limited time frame, providing valuable information on the vulnerability of sewer networks and individual pipes (Bijnen et al. 2017; Shende & Chau 2019). However, these methods have limitations, for example, risk assessment tools often lack adequate network data (Ugarelli & Sægrov 2022). Condition class is typically derived from CCTV observations and is regarded to have no direct correlation with measurable characteristics of pipes, such as material properties, geometry, and hydraulic capacity. While the CCTV condition class is considered a structured and standardised approach to aggregate expert opinions, its simplicity limits its ability to provide a deeper understanding of the underlying reasons and methodologies behind rehabilitation decisions (Tscheikner-Gratl et al. 2020). Some drawbacks of hydraulic and hydrodynamic modelling for vulnerability assessments include output uncertainties due to data availability, quality limitations, and calibration issues (Bijnen et al. 2017). Furthermore, risk assessments, CCTV condition class, and hydraulic modelling do not typically consider the influential role of the sewer network's topological properties on vulnerability. Studies such as Simone et al. (2022) argue that to better understand and analyse sewers in terms of vulnerability, a tailored complex network theory approach, such as edge-based centrality measures, can provide valuable insights.
The topology of sewer networks can play an important role in the occurrence and impact of operational failures, such as blockages (Reyes-Silva et al. 2020), suggesting that the topological properties of sewers are linked to vulnerability (Ganesan et al. 2020). Most sewer networks are generally considered to have branched topologies, however meshed or looped topologies are considered to be more topologically resilient (Zhang et al. 2017). The topology of sewer networks may also facilitate the occurrence of cascading failures, i.e. failures in one part of the network that can affect the functionality of other interconnected parts (Dong et al. 2020). For example, bridging pipes connecting different network parts is critical in analysing such cascading failures (Dong et al. 2020). Graph theory centrality measures have been used to identify such critical bridging pipes and other topologically important pipes or manholes in urban drainage networks (Demšar et al. 2008; Meijer et al. 2018), and aid vulnerability assessments (Ganesan et al. 2020; Simone 2023). Despite the extensive development of numerous centrality metrics over the past decade, only a few have been used and accepted in practice (Wan et al. 2021). While centrality measures such as betweenness, closeness, degree, and eigenvector are commonly utilised for node centrality analysis, relatively few accepted metrics are available for measuring the centrality of individual edges (Ficara et al. 2022). The few edge centrality measures that do exist typically centre around the concept of betweenness centrality, and the concept of bridging, or are based on the spectrum of the network's Laplacian (Bröhl & Lehnertz 2022). The applicability of these edge centrality measures has not yet been adequately evaluated in the context of sanitary sewer networks. Combining risk assessment tools, condition assessment methods, and hydraulic and hydrodynamic modelling with edge-specific graph centrality measures may provide a more comprehensive assessment of vulnerabilities and failure propensity in sewer networks.
The objective of this paper was to examine the potential spatial relationships between operational failures and the topological characteristics of sanitary sewer networks (SSNs) and their implications for maintenance planning. Specifically, by examining the spatial relationship between the criticality of sewer pipes (determined using edge-based graph centrality measures) and vulnerability indicators (CCTV condition grades, previous incidents of blockages and low self-cleaning potential of pipes). The suitability of edge centrality measures and their associated weights in the context of SSNs was also compared and analysed.
METHODOLOGY
Flowchart illustrating the methodology for investigating edge-based centrality measures and spatial analysis to support vulnerability assessment in sanitary sewer networks.
Flowchart illustrating the methodology for investigating edge-based centrality measures and spatial analysis to support vulnerability assessment in sanitary sewer networks.
The network cross K-function was then used to examine how pipes with higher centrality values were spatially related to pipes with a higher probability of failure determined by vulnerability indicators. These indicators included CCTV condition grades, previous incidents of blockage, and low self-cleaning potential. Following the Swedish classification system (Svenskt Vatten (SWWA) 2006), Closed-Circuit Television (CCTV) condition grades for pipes were categorised from 1 to 4. Pipes with grades 3 and 4 were considered in adverse condition due to various defects, such as cracks, misaligned joints, sediment deposition, root intrusion, and other issues. Self-cleaning capabilities of pipes in the sewer networks were assessed based on how the pipe slope compared to the inverse of its diameter. If the slope was flatter than the inverse of the diameter, it indicated a low self-cleaning potential (Ackers et al. 2001). Furthermore, a two-sided Maximum Absolute Deviation (MAD) test was also used to directly compare spatial patterns of the various centrality measures, weights, and vulnerability indicators (Baddeley et al. 2014). The spatial associations identified were also compared to historical maintenance data, specifically the frequency of maintenance actions and their temporal distribution (2014–2023) for residential and central networks. The municipality compiled maintenance data from which data for this study was sourced.
The methodology was applied to two sanitary sewer networks: one in a residential area and the other in the central part of the city. The subsequent subsections provide descriptions of the centrality measures investigated along with associated weights and the network cross K-function used to assess spatial associations within the sewer networks and identifiability analysis.
Graph theory centrality measures and topological properties of SSNs
Centrality measures investigated in this study are tailored towards identifying critical or important edges (i.e. pipes) that are most or least influential in comprehensively understanding the structural and functional characteristics within a given network topology (Wan et al. 2021). The centrality measures used are classified into path-based and non-path-based methods. The analysed path-based included betweenness edge centrality and K-path edge centrality measures. Non-path-based methods included Nearest neighbour edge centrality and Shannon entropy edge centrality measures. These measures (Table 1) were selected because of their ease of interpretation and reduced computational requirement.
Centrality measures used in the analyses, including their description, formula, and relevant references
| Edge centrality measure | Description | Formula | References |
|---|---|---|---|
| Path-based centrality measures | |||
| Edge betweenness (EB) | Measures the importance of a pipe in a graph based on the number of shortest paths that pass through it. EBC calculates the fraction of shortest paths between all pairs of nodes that contain a particular pipe. The steps involved include:
|  Variables:
| Simone et al. (2020); Ganesan et al. (2020) |
| K-path (WERW K-path; KEC) | Measures the extent to which the pipe contributes to the overall flow in the network through random K-path paths along random simple paths. The steps include:
|  Variables:
| De Meo et al. (2012) |
| Non-path-based centrality measures | |||
| Nearest neighbour (NNE) | The pipe's centrality is influenced by its weight in relation to the cumulative weights of adjacent pipes. The idea behind this centrality measure is that the weight of the pipe alone does not determine the pipe's importance but also how it compares to the pipes in its immediate surroundings. The steps include:
|  Variables:
| Bröhl & Lehnertz (2022) |
| Shannon entropy (SEC) | Shannon entropy quantifies the amount of uncertainty or randomness in a system. When applied to networks, it can help identify the diversity or unpredictability of paths in which a pipe participates. Each pipe's weight represents the importance of that pipe for ensuring flow. The relative importance of the pipe is assessed by converting the weights into probabilities (pi). Higher entropy values indicate pipes that may significantly impact the flow and are crucial for the network. Specific steps include:
|  Variables:
| Omar & Plapper (2020); Lockhart et al. (2016) |
| Edge centrality measure | Description | Formula | References |
|---|---|---|---|
| Path-based centrality measures | |||
| Edge betweenness (EB) | Measures the importance of a pipe in a graph based on the number of shortest paths that pass through it. EBC calculates the fraction of shortest paths between all pairs of nodes that contain a particular pipe. The steps involved include:
| Variables:
| Simone et al. (2020); Ganesan et al. (2020) |
| K-path (WERW K-path; KEC) | Measures the extent to which the pipe contributes to the overall flow in the network through random K-path paths along random simple paths. The steps include:
| Variables:
| De Meo et al. (2012) |
| Non-path-based centrality measures | |||
| Nearest neighbour (NNE) | The pipe's centrality is influenced by its weight in relation to the cumulative weights of adjacent pipes. The idea behind this centrality measure is that the weight of the pipe alone does not determine the pipe's importance but also how it compares to the pipes in its immediate surroundings. The steps include:
| Variables:
| Bröhl & Lehnertz (2022) |
| Shannon entropy (SEC) | Shannon entropy quantifies the amount of uncertainty or randomness in a system. When applied to networks, it can help identify the diversity or unpredictability of paths in which a pipe participates. Each pipe's weight represents the importance of that pipe for ensuring flow. The relative importance of the pipe is assessed by converting the weights into probabilities (pi). Higher entropy values indicate pipes that may significantly impact the flow and are crucial for the network. Specific steps include:
| Variables:
| Omar & Plapper (2020); Lockhart et al. (2016) |
*Consider a sewer network G (v, E) where v represents the nodes in the network (e.g. manhole, joints, pump stations), (v1 … vn) and E represent pipes (E1 … En), C denotes the centrality value for each pipe in the network G.
Weights
Pipe weights in graph centrality measures represent attributes such as relationship strength, intensity, distance, cost-associated flow or the inherent importance of the pipes within the network (Meijer et al. 2018). Weights are typically positive numerical values assigned to the pipes in the network. In a study, three main types of weights were investigated, with all four centrality measures. These weights include:
Pipe location: In gravity sewers, pipes within the sewer network can possess inherent relevance independent of the network's topology (Giustolisi Ridolfi & Simone 2020). For instance, certain pipes may represent crucial locations, such as sewer pipes connected to important facilities (e.g. hospitals, water, and treatment plants). The Strahler hierarchical stream order methodology (Gleyzer et al. 2004) was used to order the pipes in the sub-networks investigated in this study. The Strahler order is a numerical measure of the network's branching complexity, beginning with the peripheral parts of the network assigned relevance starting from 1 at a confluence of two pipes. The weight increases by n + 1 downstream of the network until the treatment plant (Simone et al. 2020). The Strahler order values were considered pipe weights.
Pipe diameter: Certain pipes within the sewer network, such as pipes with larger diameters, may play a more crucial role in conveying flow to the wastewater treatment plants. Diameter can also simplistically reflect higher population density areas connected to the network, amplifying the consequences of operational disturbances such as blockages. Therefore, the pipe diameter was considered another type of pipe weight and interpreted as pipes with higher diameters being more critical for the function of the sewer network (Hesarkazzazi et al. 2020; Reyes-Silva et al. 2020).
Pipe age is critical in the deterioration modelling of sewer pipes (Malek Mohammadi et al. 2020; Caradot et al. 2021). Studies such as van Riel et al. (2014) reported age as one of the most relevant parameters for the renewal planning of sewers. Hence, the age of sewer pipes was used as a structural weight. Older pipes were considered more critical based on a higher propensity to structural failures compared to newly installed ones.
Network cross K-function analysis
In this study, the network cross K-function was used to test the null hypothesis that the relationship between pipes with high centrality values and those with high failure probability within a network are spatially independent or random. The null hypothesis is based on complete spatial randomness (CSR) theory (Yamada & Okabe 2001). The network cross K-function quantifies the spatial interrelationships between pipes with high centrality values and those with high failure probability by considering them as two types of points constrained to a network. Monte Carlo simulation comparisons between the distribution of the two sets of points, events or locations being investigated, and random points/events or locations are generated based on the binomial point process. The analysis results were then presented as a curve and a simulation envelope. The curve indicates the spatial pattern of the two sets of points, events or locations investigated at a significance level of 0.05, referred to as the observed network cross-K-function curve. The simulation envelope represents the acceptance of the null hypothesis, i.e. a random spatial pattern. More specifically, the spatial pattern is considered random when the observed curve lies within the simulation envelope (Kunene & Scientiae 2020). When the observed curve lies above the simulation envelope, the relationship between sets of points is considered clustered, implying a significant, influential relationship between the sets of points (Yamada & Okabe 2001). When it is located below the simulation envelope, the pattern is considered dispersed, implying a weak association between the sets of points.
The Maximum Absolute Deviation (MAD) test was used in the Monte Carlo simulations to measure the difference between the observed curve and the simulation envelope for clustered relationships. The MAD values were calculated as the maximum absolute difference between the functions over a specified range. A smaller MAD value suggested that the observed spatial patterns were closely aligned with the null hypothesis, i.e. showing a random pattern. A larger MAD value indicated a more significant deviation from the null hypothesis, i.e. a clustered or dispersed pattern.
Data
In this study, the methodology outlined in Figure 1 was applied to two subnetworks – a residential and a central subnetwork of a Swedish municipality. Data on network structures in shapefiles and maintenance records in spreadsheets were collected from 2014 to 2023. Due to confidentiality agreements, the specific name of the municipality is not disclosed. The topological parameters described in Meijer et al. (2022) and Barthélemy (2011b) for both (residential and central) networks are presented in Table 2. The distribution of pipe attributes, weight types, and vulnerability indicators for both networks are also detailed in Table 3.
Topology parameters of the residential and central networks
| Category | Topological parameters | Residential Network | Central Network |
|---|---|---|---|
| Type (Buhl et al., 2006) | Meshness level | Higher levels of meshness within the network (Meshness coefficient M = 0.2) | Lower levels of meshness within the network (Meshness coefficient M = 0.03) |
| Size | Number of edges, i.e. pipes | 787 | 1,248 |
| Number of nodes, i.e. manholes, pump stations, joints | 1,154 | 1,186 | |
| Structure (Yuan et al. 2021) | Assortativity coefficient: location-weighted | 0.7 | 0.8 |
| Assortativity coefficient: diameter weighted | 0.5 | 0.8 | |
| Assortativity coefficient: age weighted | 0.6 | 0.3 | |
| Connectivity (Telesford et al. 2011) (Broido & Clauset 2019) | Node degree distribution of residential and central networks vs. a random graph: Deviations from the random graph's degree distribution indicate structural characteristics, such as the small-world effect, clustering, or hubs with highly connected nodes. Scale-free networks may exhibit a degree distribution of many nodes with low degrees and a few with extremely high degrees. |  Shows a deviation from the random graph indicating both small-world effect and scale-free network |  It shows more of a deviation from the random graph, indicating an increased small-world effect and scale-free network compared to the residential network. |
| Category | Topological parameters | Residential Network | Central Network |
|---|---|---|---|
| Type (Buhl et al., 2006) | Meshness level | Higher levels of meshness within the network (Meshness coefficient M = 0.2) | Lower levels of meshness within the network (Meshness coefficient M = 0.03) |
| Size | Number of edges, i.e. pipes | 787 | 1,248 |
| Number of nodes, i.e. manholes, pump stations, joints | 1,154 | 1,186 | |
| Structure (Yuan et al. 2021) | Assortativity coefficient: location-weighted | 0.7 | 0.8 |
| Assortativity coefficient: diameter weighted | 0.5 | 0.8 | |
| Assortativity coefficient: age weighted | 0.6 | 0.3 | |
| Connectivity (Telesford et al. 2011) (Broido & Clauset 2019) | Node degree distribution of residential and central networks vs. a random graph: Deviations from the random graph's degree distribution indicate structural characteristics, such as the small-world effect, clustering, or hubs with highly connected nodes. Scale-free networks may exhibit a degree distribution of many nodes with low degrees and a few with extremely high degrees. | Shows a deviation from the random graph indicating both small-world effect and scale-free network | It shows more of a deviation from the random graph, indicating an increased small-world effect and scale-free network compared to the residential network. |
Pipe attributes, weights, and vulnerability indicators for the residential and central network
| Category | Characteristics | Residential Network | Central Network |
|---|---|---|---|
| Pipe attributes | Total pipe length (km) | 15.8 | 32.3 |
| Pipe material type |  |  | |
| Weight type | Pipe age |  |  |
| Diameters range (mm) |  |  | |
| Vulnerability indicators | CCTV condition grades |  |  |
| Previous incidents of blockages | 28 | 91 | |
| Pipes with potential self-cleaning problems (True) if the slope is less than the inverse of diameter, otherwise False |  |  |
| Category | Characteristics | Residential Network | Central Network |
|---|---|---|---|
| Pipe attributes | Total pipe length (km) | 15.8 | 32.3 |
| Pipe material type | | | |
| Weight type | Pipe age | | |
| Diameters range (mm) | | | |
| Vulnerability indicators | CCTV condition grades | | |
| Previous incidents of blockages | 28 | 91 | |
| Pipes with potential self-cleaning problems (True) if the slope is less than the inverse of diameter, otherwise False | | |
RESULTS AND DISCUSSION
Pipe centrality distribution and associated weights
Distribution of centrality values for pipes in residential and central networks for the four centrality measures betweenness (a), K-path (b), Nearest neighbour (c), and Shannon entropy (d), and three weight types (location, diameter and age) investigated.
Distribution of centrality values for pipes in residential and central networks for the four centrality measures betweenness (a), K-path (b), Nearest neighbour (c), and Shannon entropy (d), and three weight types (location, diameter and age) investigated.
Since the distribution of centrality values for path-based measures, appear to not be influenced by weights, they may be more practical for identifying critical pipes when accurate weight values are complex, uncertain, or unavailable. This observed distribution aligned with findings from Reyes-Silva et al. (2020), who observed that edge weights did not substantially influence edge betweenness centrality in assessing eight various sub-networks, which comprised between 427 and 1,358 edges, and a total sewer length ranging from 60 to 185 km. However, another study aimed at identifying preferential paths in a stormwater network to improve redundancy and mitigate overflows found edge betweenness centrality analysis to be sensitive to the weight type used (ratio between length and volumetric flow rate based on Manning equation) (Hesarkazzazi et al. 2020). This suggests that further research is needed to increase the understanding of the sensitivity of weights in path-based centrality measures with various network types and sizes. Additionally, most of the critical pipes identified by path-based centrality measures tended to be located on the main sewer path of the investigated sub-networks. However, the pipes directly connected to the outlet of the studied networks typically had lower centrality values, i.e. were less critical. This was in alignment with Reyes-Silva et al. (2020) and Simone et al. (2022), who indicated that pipes near the wastewater treatment plants might have lower centrality values due to fewer possible shortest paths, especially when the wastewater treatment plants are located at the outskirts. This suggests the need to investigate the impact of the outlet location on the centrality values and various weighting schemes to optimise the use of edge-based centrality measures.
For the non-path-based centrality methods, the pipe centrality values tended to vary for each sub-network and weight type since the edge weights were more integral in the calculations. For Nearest neighbour edge centrality (Figure 2(c)), the median of the distribution of centrality values was slightly below 0.25 for the pipes in both networks when the weight type ‘location’ was considered. The IQR was relatively narrow, with outliers on the higher end. This distribution suggested that pipe weights were relatively uniformly distributed or that the significance of neighbouring pipes often balanced the significance of a pipe. However, the higher values (the outliers) indicated that these pipes may play a dominant role in their local regions in both networks.
When diameter was considered the weight type, a slightly higher median value was observed compared to the location-weighted analysis, with outliers at both the upper and lower ends of the distribution. This suggested that most pipes in the network did not significantly differ in weight from their immediate neighbours. Pipes with higher centrality values on the upper end could be considered crucial and may play important roles in their local regions, with their removal potentially causing intense, localised impacts. Conversely, outliers on the lower end may locally indicate weaker or less critical pipes. These outliers were considered essential for pointing out potential vulnerabilities in the network, prioritising network integrity or connectivity, and strategically planning maintenance and resource allocation.
Regarding the weight type ‘pipe age’, the distribution of centrality values for the residential network showed a wide IQR, with only one outlier on the higher end. In contrast, a narrow IQR with a median around 0 and one outlier at the higher end was also observed for the central network. This indicated a considerable variability in the significance of pipes within the residential network. While some pipes closely match the weight of their neighbouring pipes (in terms of pipe age), others vary significantly. The narrow IQR in the central sub-network may imply that most of the network might have been developed or refurbished around the same time, simplifying maintenance planning, as most pipes might have similar deterioration patterns. Previous findings, such as Bröhl & Lehnertz (2022), have highlighted that Nearest neighbour edge centrality is particularly suitable for highlighting local network bottlenecks. Moreover, it enhances the characterisation of the path structure within complex networks. Therefore, incorporating this measure alongside path-based centrality can yield a more comprehensive understanding of network dynamics and behaviour.
In the context of Shannon entropy centrality (Figure 2(d)), the values for pipes in the central network showed a wide IQR with no outliers for all three weight types investigated, with a median value of around 0.25. This distribution suggested that the pipes generally had a moderate level of unpredictability in their path involvements, i.e. while the network was designed to handle a variety of flows, no specific pipe stood out as an extreme hub or bottleneck based on the metrics used. The distribution of centrality values for the residential network showed a wide IQR with a median of around 0.25 when weighted by pipe location and age. However, the median was about 0 when weighted by diameter. When considering location and age as weights in the residential network, the pipes showed a similar unpredictable role as in the central network. This could indicate that older pipes or pipes in specific locations played different roles in the connectivity and flow patterns in the network. However, when the diameter was used as the weight, most pipes had a very predictable role, suggesting that the size of the pipes (diameter) is more uniform in this network.
Spatial association between edge centrality measures and vulnerability indicators and implications for maintenance planning
The network cross-k-function analysis was conducted at a 95% confidence level (Okabe and Sugihara, 2012; Kunene & Scientiae 2020). For the residential network, the results indicated a non-random spatial association between pipes with higher centrality values and pipes with a higher probability of failure based on vulnerability indicators, i.e., pipes with CCTV condition grades 3 and 4, pipes with previous blockage incidents, and low self-cleaning potential (Table 4). Ganesan et al. (2020) and Simone (2023) reported similar results in their studies.
Spatial relationship between critical pipes identified by centrality measures >0.7 and pipes with adverse conditions in the residential and central networks
 |
| |
MAD (maximum absolute deviation) alpha values in Bold indicate a significant degree of spatial association between weight types and across centrality measures and vulnerability indicators (P < 0.05). Cells shaded grey represent instances where two types of spatial relationships were identified. The MAD values in these instances are related to only the more pronounced relationship; the less prominent relationship is indicated in parentheses.
+Clustered pattern, –Random pattern, *Dispersed pattern.
aThe differences in MAD between weight types were negligible. This can also be observed with edge weights in Figure 1(a) and 1(b), which are not influenced by edge weights.
Among the centrality measures, edge betweenness and K-path edge centrality showed the most significant spatial association and displayed strongly clustered patterns. Particularly, pipes with high betweenness centrality demonstrated a significant spatial clustering, in relation to pipes with CCTV condition grades 3 and 4 and previous blockage incidents, as emphasised by MAD values. On the other hand, pipes with higher centrality based on K-path edge centrality showed stronger clustered patterns associated with pipes with low self-cleaning potential. For detailed network cross K-function plots for the residential network, see supplementary data I. These results could imply that pipes with higher centrality values may be more likely to cause network disruptions due to structural failures and blockage events. Additionally, these results suggested a plausible interaction between structural failures and blockage occurrence; for example, the occurrence of blockages may be due to structural-related problems. Increased structural failures such as cracks may also permit higher levels of infiltration into the pipes, which aid in reducing sediment deposition (Beheshti & Sægrov 2018) and, thus, lower blockage occurrences around pipes with higher centrality values.
For the central sub-network, analogous to the residential, it was observed that pipes with higher centrality values, determined by path-based centrality measures, exhibited a stronger degree of spatial association with pipes in adverse conditions (Table 4). For detailed network cross K-function plots for the central network, see supplementary data II. The strength of the spatial associations was also observed to be stronger in the residential network compared to the central network. However, the strength of these spatial associations may be affected by other network attributes not evaluated in this study.
The non-path-based centrality measures, Nearest neighbour edge and Shannon entropy exhibited clustered spatial associations with adverse CCTV condition grades when weighted by pipe age and diameter. However, these non-path-based measures showed a random spatial association in relation to pipes with previous incidents of blockages (Table 4). Both investigated non-path-based methods showed a clustered pattern in relation to pipes exhibiting low self-cleaning potential.
In contrast to the residential network, the non-path-based centrality measures displayed a less distinct spatial pattern in connection to pipes with adverse conditions in the central network. For example, pipes with adverse CCTV grades tended to cluster around pipes with higher Shannon entropy centrality in the residential network. In contrast, in the central network, a combined (slightly clustered/dispersed) pattern was observed when weighted by pipe diameter and age. The above indicates that while some pipes with adverse CCTV grades tend to be located in close proximity to pipes with high centrality, there were also instances of dispersion, i.e. grouped away from pipes with high centrality. In the case of the Nearest neighbour centrality, a predominantly random spatial pattern was observed in the central network vs. a more clustered pattern in the residential network in relation to pipes with adverse CCTV grades when weighed with location and diameter. Among the non-path-based centrality measures used, only the Shannon entropy centrality measure in the central network showed a significant spatial association, i.e. slightly clustered and dispersed pattern with pipes that had previous incidents of blockages across all weight types investigated.
The difference in the degree of spatial associations between sub-networks may be influenced by topological factors such as structure and connectivity. For example, assortativity tended to be more pronounced in the central network compared to the residential network. This was especially true when pipe diameter and location were considered weights. These suggest that lower assortativity networks may have more pronounced spatial relationships. The central network also showed more evidence of being scale-free. Scale-free networks may be more vulnerable to targeted failures (Amaral et al. 2000; Aarstad et al. 2013). Meng et al. (2018) also suggested that network structure, topology, and connectivity significantly influenced the occurrence and magnitude of failures in separate sewers. Another observation in this study was the differences in pipe age and material composition between networks, which may have also contributed to the distinctive spatial patterns between centrality values and adverse conditions in pipes. Specifically, a less diverse composition in pipe material may show more variable spatial patterns. For example, the residential network consisted of predominantly concrete pipes, while the central network consisted of a mix of concrete, clay, plastic pipe materials and others. The uncertainty in data of pipe conditions (CCTV grades) may vary across different networks (Fugledalen et al. 2023), and the implications of such variations on the variability of spatial association need to be further investigated.
Frequency of maintenance actions in the Residential and Central Networks compiled by the municipality from where data for this study was sourced. Each code represents one type of maintenance action.
Frequency of maintenance actions in the Residential and Central Networks compiled by the municipality from where data for this study was sourced. Each code represents one type of maintenance action.
Temporal comparative analysis of operational disturbances between Residential and Central Networks from 2014 to 2023 compiled by the municipality from where data for this study was sourced. Numbers indicate the total number of operational disturbances that occurred.
Temporal comparative analysis of operational disturbances between Residential and Central Networks from 2014 to 2023 compiled by the municipality from where data for this study was sourced. Numbers indicate the total number of operational disturbances that occurred.
The strength of spatial associations may also be used as an indicator to identify temporal variability in operational problems or maintenance needs. For example, significant deviations in entropy centrality values over time can highlight anomalies. Pipes with high centrality values showing a significant spatial association with vulnerability indicators may be suitable locations for sewer monitoring (Simone et al., 2022), such as installing sensors or in-depth CCTV inspections to aid proactive measures. Such pipes may also be in locations where redundancy or redesigns may be beneficial. For example, in locations where blockages tend to cluster around pipes with higher centrality values, it may be cost-effective to redesign critical pipes to increase flow redundancy (Hesarkazzazi et al. 2020) compared to conventional approaches like regular pipe flushing.
CONCLUSIONS
The results showed significant spatial clustering between pipes with higher centrality values and pipes with adverse CCTV condition grades, previous blockage incidents, and low self-cleaning in the two sewer networks investigated. These findings suggest a level of spatial aggregation and colocation between high centrality pipes and those prone to failure than could be expected under random conditions. Path-based centrality measures, such as edge betweenness and K-path edge centrality, were more practical measures due to being less influenced by weights in identifying critical pipes in sewer networks, especially when accurate weight values are difficult to obtain. Combined with spatial approaches, such as the network cross K-function, edge-based centrality measures provided realistic preliminary analyses that could be useful in analysing complex sewer networks where there is a lack of data, high data uncertainty, or modelling problems. Factors such as assortativity, small-world and scale-free network effects, and pipe physical characteristics (such as pipe age) may also influence spatial association. This complexity requires further research to interpret spatial association for sewer maintenance.
Spatial clustering between pipes with higher centrality and vulnerability indicators in sewer networks can form the basis for developing data-driven maintenance strategies. Insights from their use can aid in more targeted maintenance strategies for SSNs, leading to enhanced proactive maintenance planning, optimization, and remedial actions such as redesigning sewers to include more redundancies for failure prevention. Additionally, the distinct topological differences between residential and central networks necessitate tailored maintenance approaches, with residential networks showing the need for more diverse strategies. Such nuanced understanding may enhance the efficiency and reliability of sewer maintenance planning.
Integrating different methodologies, such as those investigated in this study, provided an approach to identifying critical points and potential weaknesses in sanitary sewer networks. The synergistic effect of an integrated approach may enable a more systematic and robust evaluation of sewer network vulnerabilities. This study presented an initial step towards a methodology that uses edge-based centrality measures to develop asset management of sanitary sewer networks. However, more investigations are needed for other sewer networks to validate the findings from this study.
ACKNOWLEDGMENTS
This work was supported by the projects “Stormwater & Sewers” funded by Swedish Water . I would also like to acknowledge Youen Pericault his support in carrying out this work.
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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