Water distribution system vulnerability analysis using weighted and directed network models


Corresponding author: P. Jeffrey, Cranfield Water Science Institute, Cranfield University, College Road, Cranfield, MK43 0AL, United Kingdom. (p.j.jeffrey@cranfield.ac.uk)


[1] The reliability and robustness against failures of networked water distribution systems are central tenets of water supply system design and operation. The ability of such networks to continue to supply water when components are damaged or fail is dependent on the connectivity of the network and the role and location of the individual components. This paper employs a set of advanced network analysis techniques to study the connectivity of water distribution systems, its relationship with system robustness, and susceptibility to damage. Water distribution systems are modeled as weighted and directed networks by using the physical and hydraulic attributes of system components. A selection of descriptive measurements is utilized to quantify the structural properties of benchmark systems at both local (component) and global (network) scales. Moreover, a novel measure of component criticality, the demand-adjusted entropic degree, is proposed to support identification of critical nodes and their ranking according to failure impacts. The application and value of this metric is demonstrated through two case study networks in the USA and UK. Discussion focuses on the potential for gradual evolution of abstract graph-based tools and techniques to more practical network analysis methods, where a theoretical framework for the analysis of robustness and vulnerability of water distribution networks to better support planning and management decisions is presented.

1. Introduction

[2] Critical physical infrastructure networks, such as water, energy, and transport systems share common physical features important to the study and analysis of their structure and function. The supply-demand (source-sink) and physical nature of these systems influences their spatial evolution over time, imposing significant restrictions on network connectivity, efficiency, and robustness under component failure and targeted attack. In addition, interdependencies between different infrastructure systems make them increasingly vulnerable to systemic risk and cascading failures. The reliability of physical infrastructure systems and the services they provide is threatened by several phenomena including random failures, asset deterioration, and catastrophic events. For water infrastructure systems in particular, the quality and quantity of delivered water can be compromised by a variety of problems, such as reductions in system capacity and performance due to pipe tuberculation, leakage, corrosion, and increasing frequency of bursts and breakage correlated with deterioration and aging [Berardi et al., 2008; Walski et al., 2003, pp. 6 and 151].

[3] Pressure loss, high water age, and high energy consumption in centralized systems can be precipitated by long transport distances from source and treatment works to end users [National Research Council, 2006]. Such challenges can be intensified in frequency and impact by drought, flooding, and extreme weather conditions. Furthermore, it has been proposed that the damage caused by water systems failures to other infrastructure systems through interdependency is much greater than the damage water systems sustain from other infrastructure system failures [Zimmerman and Restrepo, 2006]. Consequently, improving the robustness and resilience of water infrastructure systems is an increasingly prominent concern, not only for water system engineers and utility managers, but also for the critical infrastructure protection community concerned with the sustainable design and operation of infrastructure systems.

[4] Two rather different (but potentially complementary) approaches to investigating the susceptibility of networks to damage are: (i) simulation-based probabilistic models, known as reliability theory and (ii) deterministic measurements based on network graph topology, known as structural (in)vulnerability analysis. Despite a well developed methodology and the degree of precision offered, the first approach is regarded as highly data dependent and computationally costly for studying large complex systems. The structural vulnerability approach, meanwhile, presents a less computationally intensive “first approximation” to the analysis of complex water distribution system robustness. A combination of the two approaches may arguably provide adequate information on identifying weak points and at risk components and detecting structural flaws which make the system vulnerable to failures. Such information may subsequently be used to develop sustainable design and planning strategies with the goal of reducing failure likelihood and consequences, and reducing the time of system recovery and overcoming perturbations. In this work it is the latter of the two approaches that is of interest. In other words, without attempting to fully characterize the multidimensional vulnerability of complex water distribution systems, this work seeks to propose an analytical framework and present some necessary tools to establish fundamental relationships between network topology and susceptibility to service loss.

[5] Study of the interplay between the structure and function of complex networks [Strogatz, 2001; Albert et al., 2000; Cohen et al., 2000; Holme et al., 2002; Boccaletti et al., 2006; Estrada, 2006], in particular of the component failure and attack resilience of infrastructure systems such as the Internet, telecommunication networks, and power grids [Latora and Marchiori, 2005; Holmgren, 2006; Rosas-Casals et al., 2007; Kim and Motter, 2008; Bompard et al., 2009; Dueñas-Osorio and Vemuru, 2009], transport systems [Barthelemy and Flammini, 2008; Berche et al., 2009; He et al., 2010], gas and water supply systems [Carvalho et al., 2009; Yazdani and Jeffrey, 2011a, 2011b] has grown rapidly in recent years. Structurally based assessments of network and service delivery robustness seek to establish relationships between changes in network connectivity patterns, resulting from random or cascading component failures, and the system's ability to maintain functionality as measured by the global loss of connectivity (e.g., reduction in size of the largest connected subnetwork) or reduction of system efficiency (e.g., increase in reachability between components) [Albert et al., 2000; Latora and Marchiori, 2005]. Among notable findings in this area is the notion of the fragility of scale-free networks to targeted removal of highly connected nodes, also known as hubs [Albert et al., 2000]. This is characterized by an avalanche breakdown of network connectivity following the removal of a small percentage of critical nodes giving rise to cascading failures and broad disruptions to network performance.

[6] While the existence of highly connected nodes and centrally located critical components with large failure consequences is frequently observed in many social, information, and technological networks, the connectivity of subsystem components in spatially organized urban infrastructure systems including municipal water supply networks is largely restricted by physical and geographical constraints [Carvalho et al., 2009; Newman, 2003; Yazdani and Jeffrey, 2011a]. This is to say that little variation is observed in the connectivity patterns of the nodes in water distribution networks, no hubs are present in the network as most of the nodes have very low degree (two or three and mostly less than five) and any differences are virtually indistinguishable from a purely topological point of view (as represented by the abstract graphs' node degree). Such networks are also equally sensitive to random or malicious failures [Barthelemy and Flammini, 2008, Yazdani and Jeffrey, 2011a].

[7] The above implies that purely topological connectivity metrics, which are based on abstract graph node degree metrics, may not adequately characterize the sensitivity of a system to perturbations as they fail to capture important component and subsystem level information related to the role and characteristics of components and how the failure of individual components may affect system performance and cause disruptions both locally (e.g., as a result of the failure of smaller distribution pipes or unavailability of local sites) and at broader scales (e.g., unavailability of the source and storage tanks or breakage of a single trunk main adjacent to the source). Consequently, additional network characteristics including the physical attributes of infrastructure system components and information on network flow dynamics (i.e., hydraulics) and governing equations are required for a more realistic analysis of vulnerability and an in-depth understanding of the interplay between system topology and performance.

[8] A water distribution system may be represented as a spatially organized network of multiple interconnected components. In an abstract modeling context, a mathematical graph may be used to show the relationships between collections of nodes (e.g., junctions such as pipe intersections, reservoirs, and consumers) that are joined by links (e.g., pipes and sometimes valves). Notable previous works on the application of graph theoretic methods to studying water distribution systems are concerned with optimization of redundancy (e.g., construction of alternative supply paths) and least cost reliable network design [Jacobs and Goulter, 1989; Kessler et al., 1990], segmentations according to the location of isolation valves [Walski, 1993], system skeletonization, aggregation, and connectivity analysis [Walters and Lohbeck, 1993; Yang et al., 1996; Ostfeld, 2005], network graph decomposition models for efficient system control and identifying connected components [Deuerlein, 2008], and equivalent matrix methods for the detection of topological changes following failures [Perelman and Ostfeld, 2011; Giustolisi et al., 2008]. However, there have been few systematic studies to date of water distribution system connectivity, vulnerability, and robustness by using advanced and emerging network theory measurements.

[9] As discussed above, a purely topological graph representation fails to fully describe many operational and hydraulic characteristics of water distribution systems in the context of vulnerability to failures, as it does not account for network component attributes and system-specific information. Subsequently, a topological connectivity analysis may be viewed as a fundamental and necessary yet insufficient basis for water distribution systems reliability and vulnerability analysis. Previous studies [Yazdani and Jeffrey, 2011a, 2011b; Yazdani et al., 2011] have deployed graph theory derived analyses to emphasize the significant role of topology in system performance while also discussing some limitations of the approach due to the restricted access to system component data and, in particular, ignoring the direction of flow and the relative significance of nodes.

[10] However, as will be demonstrated in this work, using weighted and directed network models and techniques based on the available information on component attributes and system hydraulics may considerably improve such connectivity analysis through strengthening the analogy between network graphs and real systems, and by providing support for monitoring, management, and understanding of the interaction of network components. This study utilizes a set of simple network component attributes to model water distribution systems as weighted and directed networks where multiple nodes (reservoirs, tanks, and junctions) are joined by links (pipes, valves), and where physical (pipe size and flow direction) and hydraulic (nodal demands) information is used to develop measurements of component criticality to serve the analysis of structural vulnerability. In particular, as an alternative to established graph theory measures of node centrality which are typically limited to a consideration of the node degree, a combined topological and physical metric, the demand-adjusted entropic degree or generalized connectivity metric, has been developed to quantify (rank) the level of centrality (criticality) of water distribution network nodes based on a joint evaluation of node degree, weight of adjacent links, and the nominal demand.

[11] Moving beyond the purely topological notion of connectivity as an indicator of component criticality and system performance, generalized connectivity metrics are applied to benchmark water distribution systems which incorporate both physical and hydraulic attributes, and which identify and rank critical network components including cut sets, bottlenecks, and large flow passage routes whose failures may result in broad disruptions to water supply. Importantly, this particular stable of metrics is extended by proposing a novel measure of node criticality which offers a more targeted assessment of network robustness. Through strengthening the similarity between network graphs and real water distribution systems and exposing the contrast between global (network level) and local (component level) network measurements, this work demonstrates the potential for gradual evolution of abstract graph-based tools and techniques so that they are capable of providing credible advice to practitioners about the robustness and vulnerability of water distribution networks.

2. Network Topology and Measurements

[12] Water distribution systems may be represented as networks of nodes (e.g., reservoirs, tanks, pumps, and fittings) connected by links (e.g., pipes, valves) [Walski et al., 2003] where physical and hydraulic attributes (e.g., pipe size, nodal head, or demand) may be used to inform the role and characteristics of system components. Network theory provides an invaluable framework for representation and quantitative study of water infrastructure systems, through analysis of subsystem connections, quantifying topological redundancy (e.g., the existence of alternative supply paths) and identifying critical components whose failure may impact the overall performance of the system.

[13] Here, a network is modeled as a graph inline image in which inline image is the set of inline image graph nodes (vertices) and inline image is the set of inline image graph links (edges). In a nondirected and nonweighted network model, the topology of the network and the relationships between components are studied and quantified by using connectivity measurements. One of the fundamental measures of connectivity, describing the centrality of a given node inline image is the node degree. The node degree is the number of links adjacent to a node inline image given by inline image, where inline image is the number of lines between nodes inline image and inline image. The nonincreasing set of graph node degrees is called the degree sequence with its frequency histogram known as the degree distribution. The shape of degree distribution, in addition to the minimum, maximum, and average values of degree sequence (also known as average node degree given by inline image [Newman, 2010, p. 134]) provide useful information about the network's overall connectivity and its tolerance to failures and targeted attacks [Albert et al., 2000]. A detailed analysis of degree distribution and connectivity for a sample of benchmark water distribution systems along with the relevant interpretations on redundancy in networks and structural robustness is presented by Yazdani and Jeffrey [2011a, 2011b].

[14] Among other measurements of network topology is the geodesic distance (also referred to as shortest path length), the length of shortest geodesic path between two nodes which is the smallest number of links traversed to reach one node from another [Newman, 2010, p. 139]. This is, in general terms, a topological measure of efficiency when dispatching flow across a network. The average (respectively, maximum) value of geodesic distance between all pairs of nodes is called average path length denoted by inline image (respectively, graph diameter denoted by inline image) with its value indicating the structural complexity of the network. Changes in average path length and graph diameter following component failures may be used to assess the intensity of the damage and network vulnerability to failures. The average path length and average value of geographical distances of all network nodes may to a limited extent be interpreted as the surrogates of network efficiency [Yazdani and Jeffrey, 2011a].

[15] Two important metrics useful for quantifying the connectivity and structural robustness of networks are meshedness coefficient [Buhl et al., 2006] and algebraic connectivity [Fiedler, 1973], respectively. In an abstract graphical model of water distribution networks, the number of independent loops is given by inline image for single-source networks and by inline image for multiple-source systems [Larock, 2000]. Without a major loss of generality, it can be assumed that water distribution networks are nearly planar graphs (although not strictly so due to the presence of crossovers) where only a tiny percentage of the link intersections do not match to their end nodes [Yazdani and Jeffrey, 2011b]. The maximum number of independent loops in planar graphs is given by inline image. The meshedness coefficient for quantifying the density of loops in the network structure is defined as inline image, which is regarded as a global (network level) indicator of redundancy through the existence of loops and alternative supply paths [Yazdani and Jeffrey, 2011a, 2011b; Buhl et al., 2006].

[16] Algebraic connectivity inline image is the second smallest eigenvalue of the normalized Laplacian matrix of a network [Fiedler, 1973]. The Laplacian matrix of a graph inline image with inline image nodes is a inline image matrix inline image, where inline image with inline image being the degree of node inline image, and inline image is the adjacency matrix of inline image, where inline image if there is a link joining nodes inline image and inline image otherwise. The algebraic connectivity of a graph is a nonnegative number whose magnitude represents structural robustness against efforts to decouple parts of the network. The algebraic connectivity may be used as a strong indicator of connectivity at network level which enables the comparison of structural robustness of different network layouts. However, due to its property of representing a global average only, this metric provides little information on specific points of weakness and structural flaws, and is not capable of pinpointing critical locations such as bridges and cut sets. Applications of graph theory techniques to identify bridges and their location in water distribution systems (and more generally the partition of networks into blocks, forests, and bridges) may be found by Deuerlein [2008].

3. Weighted and Directed Networks

[17] In weighted networks (where each edge has a weight or value), the concept of node degree may be generalized to node strength. The strength of a given node inline image is given by inline image, where inline image is the weight of the line joining nodes inline image and inline image. The selection of edge weights is not straightforward, as there are multiple physical and operational parameters which could be selected. In real physical flow circulation networks, important parameters including nodal demand, geographical distances between nodes (specially the distance between the source node and consumers), and the associated cost of construction and maintenance (of sites and links) need to be considered for the purpose of developing indicators of network efficiency, reliability, and robustness. In particular, combining information on the weight of the links and the direction of flow may reveal useful information regarding the importance and role of nodes in a network. From the point of view of water distribution system reliability analysis, purely topological (abstract graph theory) connectivity metrics often fail to distinguish between different components (which sometimes have multiple functions) in a network, for they do not capture important physical and performance-related information related to network components. This is demonstrated in Figure 1, where a hypothetical water distribution network is represented as (a) nonweighted and nondirected, and (b) pipe-length-weighted, flow-directed, and nodal-demand labeled.

[18] If viewed as nonweighted and nondirected, purely topological measurements (described above) would characterize the two networks as identical ( inline image, inline image, inline image, inline image, inline image, inline image, inline image, and inline image). However, the weighted and directed network model provides more insight into the differences between these networks and reveals vital information on the role, physical attributes, and hydraulic properties of network components. By way of example, consider that junctions N2, N5, and N8 in Figure 1 have node degree measures of two which makes them topologically indistinguishable. However, while N2 is a transmission-only node, N5 is simultaneously a transmission and demand node (nominal demand of 200 L s−1) and N8 is a demand-only node (nominal demand of 300 L s−1). Moreover, due to the adjacency of node N2 to the source, the consequences of its failure are far greater than those of the two other nodes. Furthermore, the shortest possible topological distance from source to N8 is five, which means that the source can be reached by N8 in no less than five steps (for example by going through the sequence {source, N2, N3, N4, N5, N8} or {source, N2, N3, N6, N7, N8}). However, looking only at total pipe length in each selected path, ceteris paribus, significant differences would be expected in terms of total headloss and (pumping) energy consumption, hydraulic balance of the system, cost of construction and maintenance, and operational consequences of the likely failure of the intermediate components. Consequently, a quantitative analysis of vulnerability and derivation of a component criticality index requires the selection of weights to be associated with network links and nodes.

Figure 1.

A topological representation of (a) a hypothetical water distribution system and (b) the pipe-length weighted flow-directed version of the same network with the nodal base-demand values shown in blue.

[19] In such a quantitative robustness analysis of water distribution systems using network models, the weight of the links may be specified by, inter alia, the given pipe size (i.e., diameter, length) or physical capacity. This provides an indication of the cost of building and maintaining the network and, as will be demonstrated in section 5, may be used for ranking network nodes based on their level of centrality and connectivity, important in terms of studying the operational consequences of failures. Subject to data availability, other types of data such as volumetric flow rate or total headloss may be used to weight network water mains, while nodal demand, hydraulic head, or the number of consumers served may be used as the weighting associated with network nodes. However, in general there are always limitations in terms restricted access to systems performance data, technical difficulties of quantifying different cause and effects of water supply disruption, and other issues which may hinder the full characterization of system vulnerability. These limitations are to some extent applicable to any quantitative analysis of infrastructure system robustness seeking to rank the importance of network components and the impacts of their failure.

[20] In a flow network, any quantitative measure of the importance of network nodes based on their level of centrality should not only be determined by node degree, but also by the flow passing through each adjacent pipe, and the distribution of link weights [Bompard et al., 2009]. For example, as seen in Figure 1, the impact of failure of the node N2 on the performance of the network is much greater than the impact of failure of the node N7, despite N7 having a larger number of pipe connections. This is due to the fact that the relationship between topology (and its changes) and the system's operational response is determined by multiple other important parameters and not only by the topological connectivity in isolation.

[21] Therefore, a nontopological measure of connectivity is required to incorporate the notion of degree, with the value and distribution of weights among the links adjacent to one node. One useful such measure is the “entropic degree” [Bompard et al., 2009] defined for a node inline image by inline image, where inline image is the normalized weight of the link between nodes inline image and inline image, with the property of inline image. The entropic degree is a quantitative measure of node importance by which network nodes may be ranked according to their centrality and failure induced impact on network performance. This metric will be used in section 4 as the basis for derivation of a more generalized measure of connectivity which uses specific water distribution system information to identify the critical nodes in the network.

4. Demand-Adjusted Entropic Degree

[22] The criticality of nodes in a water distribution network is partially a function of the node type and associated hydraulic attributes such as outflow, total demand, and hydraulic head. A simple classification by function of different types of water distribution system nodes would include: boundary source nodes with defined elevation grade (e.g., reservoirs and storage tanks), transmission only nodes (e.g., T-junctions, pumps), and nodes with specified positive base demand (end users). While the physical, geographical, and hydraulic attributes of the source nodes are usually known very well, it is crucial to distinguish between other node types in a quantitative modeling context. To this end, an adjusted measure of node criticality is proposed here as a “demand-adjusted entropic degree,” defined for a node inline image by inline image, where inline image is the entropic degree and inline image is the base (nominal) demand for water at node inline image (in liters per second), and inline image is the maximum of all such base demand values of the network nodes.

[23] The proposed demand-adjusted entropic degree uses a dimensionless weighting factor to incorporate the nodal base demand for water into the definition of entropic degree (for nodes other than sources or reservoirs). This factor adjusts the entropic degree by weighting it by the relative demand for water at each node. In general, it is conceivable to use nodal base demand (or other indicators of node criticality) as an independent parameter in the initial definition of entropic degree. However, in this work, due to the emphasis placed on the physical capacity of the pipes adjacent to each node and due to measurement unit differences, the nominal nodal demand is only used as a supplementary weighting factor in quantifying node importance.

[24] Deriving system-specific (rather than generic network) measurements such as the above specified demand-adjusted entropic degree is central to being able to describe detailed properties of water distribution systems and evaluate system robustness more realistically. The important contrasts between this refined local-scale metric and the ones introduced in the previous sections (that are generally viewed as global metrics of redundancy, efficiency, or robustness which quantify average network structure characteristics) needs to be highlighted. Overall, the global (network level) metrics only provide a first approximation for network structure as they are generally inadequate in characterizing the real system and they usually fail to describe the relationship between structure and function. This is partly due to the definition of these metrics which makes them mostly applicable to quantifying the structure of nonweighted and nondirected networks.

[25] However, system specific measurements provide a more detailed and meaningful view of system vulnerability through studying component specifications and the impact of local and global failures. In particular, the demand-adjusted entropic degree provides a measurement of node centrality refined and suited to the analysis of spatially organized infrastructure networks with limited connectivity. In the absence of hubs and highly connected nodes, which are responsible for a great percentage of network connectivity and regarded as the points of vulnerability to targeted attacks in scale-free networks [Albert et al., 2000], the notion of demand-adjusted entropic degree may be viewed as an alternative generalized measure of connectivity which quantifies the importance of each node among other nodes toward operation of the network, based on the specific role of each node, its known demand for water and the physical (or potentially hydraulic) attributes of the adjacent pipes.

5. Case Studies and Numerical Analysis

[26] In order to compare the use and value of these global and local metrics we turn to two model water distribution networks which provide useful case studies (Figure 2). Employing such recurrently studied test-bench networks enables the analysis (particularly that which utilizes the new demand-adjusted entropic degree metric) to be easily contrasted with the results of previous work in this area. The purpose of conducting the analysis is to both demonstrate the practical application of the proposed metric and explore its value in the context of water distribution network planning and management.

Figure 2.

Graph representation of studied networks: (left) Richmond and (right) Colorado Springs.

[27] The “Richmond” distribution system, an example from the Yorkshire Water service area in the UK reported by van Zyl et al. [2004], is a medium size network ( inline image) with high link density in the urban center and a sparse configuration at suburban areas and transmission levels (where the reservoirs and water supply sources are located). This specific layout may have important consequences in terms of large-scale disruptions as a result of failure of pipes or the outage of system component at upstream network locations. Contrastingly, the “Colorado Springs Utilities” system reported by Lippai [2005] is a large network ( inline image) with mesh-like structure at the central distribution levels. The existence of structural holes and cut sets (regions where failures of a small number of network components may disconnect large segments of the network) make this network structurally vulnerable in certain areas. The somewhat locally regular structure suggests the formation of this network as a result of some construction planning with its evolution and expansion over a shorter period of time as compared with Richmond.

[28] By definition, the degree of each node in the studied water distribution networks is given by the number of pipes adjacent to it. However, due to spatial organization and the physical constraints imposing restrictions on patterns of connectivity, little variation is observed in the graph degree sequence of water distribution networks; in both the networks presented in Figure 2, the maximum value of graph degree sequence is four, the minimum value is one, and the largest percentage of nodes have degree three with 39.5% for Richmond and 48.2% for Colorado Springs. Moreover, both degree distributions are relatively random with their simple shapes indicating topological homogeneity of structures, and the average node degrees are found to be very low (as compared to other types of networks [Newman, 2003, p. 182]) with inline image for Richmond and inline image for Colorado Springs [Yazdani and Jeffrey, 2011a].

[29] These observations prompt the use of nontopological metrics and specific physical system information (on components attributes and their functions) in order to distinguish between otherwise “typically the same” nodes. Table 1 provides an overview of some of the global topological measurements of the studied networks. A comparison of the obtained values suggests that the structure of the Colorado Springs network provides, on average, a more optimal connectivity, more redundancy (and hence greater reliability), higher efficiency, and larger structural robustness as compared to the structure of the Richmond network. More details on the hydraulic and structural properties of these water distribution systems may be found by Yazdani and Jeffrey [2011a, 2011b], van Zyl et al. [2004], and Lippai [2005].

Table 1. Global Structural Properties of Richmond and Colorado Springsa
Networknmkldr inline image
  • a

    n, nodes; m, links; 〈k〉, average degree; l, average path length; d, graph diameter; r, meshedness; inline image, algebraic connectivity.

Colorado Springs178619942.2325.9469 inline image inline image
Richmond8729572.1951.44135 inline image inline image

[30] In order to evaluate the node importance by using demand-adjusted entropic degree, each pipe adjacent to a node has been weighted by the physical capacity of the pipe (in cubic meters). The physical capacity of a pipe between nodes inline image and inline image is given by inline image, where inline image and inline image are the pipe length and diameter, respectively. While other choices of weighting (such as pipe length or a combination of different pipe attributes) are possible subject to availability of data, the physical capacity may be viewed as a relatively simple to evaluate yet fundamental indicator of the pipe size, the cost of operation (e.g., pumping, major headloss), the cost of network construction and maintenance, and an estimator of the average (respectively, maximum) volume of flow passing through water mains during normal demand (respectively, peak demand) circumstances. Moreover, the specified nodal base demand (in liters per second) has been utilized to calculate the demand-adjusted entropic degree. Other methods such as the evaluation of the total inflow minus total outflow of a node may be used to approximate the nodal demand in directed flow networks.

[31] Upon applying the demand-adjusted entropic degree to the benchmark water distribution networks (weighted and directed) and comparing the obtained results on node criticality with those obtained from a purely topological definition of node degree (nonweighted and nondirected), the two methods yield very different results. In the case of the Richmond network, the highest node degree is four with a majority of the nodes having degree three while there is a much greater variation observed in the values of demand-adjusted entropic degree (Figure 3). Moreover, it is observed that the most important nodes identified using the demand-adjusted entropic degree are not necessarily those identified using the purely topological degree, and vice versa (Table 2). For example, none of the nodes with topological degree of four in the Richmond network are among the top ten most important nodes identified and ranked according to the magnitude of demand-adjusted entropic degree.

Figure 3.

Comparison of topological degree and demand-adjusted entropic degree for Richmond. The limited range of the node degree values (integers from 1 to 4) and variations in the demand-adjusted entropic degree for the same nodes (going up to 94.37), indicate a total difference of discriminatory power from the first metric to the second.

Table 2. Richmond's Top Ten Critical Nodes
NodeNode DegreeBase Demand (L s−1)Demand-Adjusted Entropic DegreeNormalized Demand-Adjusted Entropic Degree

[32] This approach to identifying critical nodes may also be applied to Colorado Springs, as illustrated in Figure 4 and listed in Table 3. In Colorado Springs, similar to Richmond, the ranking of node criticality by demand-adjusted entropic degree has been largely determined by the physical capacity and only partially by topological degree or nodal base demand. Consequently, it is observed that the two approaches (i.e., purely topological and weighted) yield different results, and that demand-adjusted entropic degree enables discrimination between nodes with typically low connectivity (i.e., low degree) thereby providing a more realistic assessment of node criticality according to their specified demand, their number of adjacent links, and the magnitude and distribution of the weight among those links.

Figure 4.

Comparison of topological degree and demand-adjusted entropic degree for Colorado Springs. The limited range of the node degree values (integers from 1 to 4) and variations in the demand-adjusted entropic degree for the same nodes (going up to 542.28), indicate a total difference of discriminatory power from the first metric to the second.

Table 3. Colorado Springs' Top Ten Critical Nodes
NodeNode DegreeBase Demand (L s−1)Demand-Adjusted Entropic DegreeNormalized Demand-Adjusted Entropic Degree

[33] As discussed previously, studying the patterns of connectivity among network components (e.g., via studying cumulative degree distribution) and comparing the structure of the network against models of randomly structured or scale-free networks may reveal useful information on global network tolerance against random failure of nodes and targeted removal of the most important nodes [Albert et al., 2000]. Therefore, it would be useful to study the distribution of demand-adjusted entropic degree, derived as the generalized measure of connectivity in this work. To this end, the obtained values of demand-adjusted entropic degree are normalized to a scale of zero to one by dividing each value by the maximum demand-adjusted entropic degree in each network. This enables a direct comparison of the two networks' generalized connectivity distributions through quantifying the proportion of critical nodes that constitute each network structure.

[34] The cumulative distribution of the normalized demand-adjusted entropic degree for the two studied networks is shown in Figure 5 for comparison. In the Colorado Springs network, approximately 80% of nodes have normalized demand-adjusted entropic degree of less than 0.1, with around 87% of nodes having normalized demand-adjusted entropic degree of less than 0.2. By comparison, in the Richmond network, only around 57% of nodes have normalized demand-adjusted entropic degree of less than 0.1, with only 62% having normalized demand-adjusted entropic degree of less than 0.2. These results support the observation that Colorado Springs enjoys a globally more distributed and uniform structure than Richmond. Among other factors, the uniformity of the network structure and shorter transport distances (both topological and physical) are linked to properties of more equalized distribution of flow and pressure in the network and its overall higher operational and hydraulic efficiency [Yazdani and Jeffrey, 2011a]. In addition, it is observed that Richmond, despite being the smaller network, has a greater percentage of its nodes with generalized connectivity value of more than 0.2. This indicates the presence of a greater number of highly connected critical nodes whose failures may result in broad disruptions to water supply. Overall, the observation that the generalized connectivity distribution curve of Richmond is almost everywhere above the generalized connectivity distribution curve of Colorado Springs, implies a greater uniformity and invulnerability in the structure of Colorado Springs.

Figure 5.

The cumulative distribution (pr{>f}) of normalized demand-adjusted entropic degree for weighted and directed networks: (crosses) Richmond and (triangles) Colorado Springs. The accumulation a relatively high percentage of nodes with low values of normalized demand-adjusted entropic degree suggests higher topological uniformity and greater invulnerability to node failures in Colorado Springs than in Richmond.

[35] As discussed previously, the representation and analysis of degree distribution in scale-free networks reveals crucial information on what degree of network connectivity is controlled by what percentage of nodes, and how resilient the network topology is against random failures and targeted attacks on network hubs [Albert et al., 2000]. The analysis of demand-adjusted entropic degree and generalized connectivity distribution reveals similar information on node centrality and the impact of component failures on network operation through offering a refined measure of connectivity which is suitable for water distribution systems where node degree values are typically low due to spatial constraints. A more detailed quantitative analysis of the failure consequences of most important nodes is beyond the scope of this work and further details of the possible approaches to such analysis may be found by Albert et al. [2000], Cohen et al. [2000], Holme et al. [2002], Boccaletti et al. [2006], Latora and Marchiori [2005], Bompard et al. [2009], and Berche et al. [2009].

6. Discussion and Conclusions

[36] This paper presents an extended view of the connectivity analysis of water distribution systems and elaborates on employing weighted and directed network models and using measurements to quantitatively assess system topology, vulnerability, and robustness against likely failure of components. In particular, it proposes a technique (or more specifically a combined topologically and physically based metric) for identifying and ranking critical components (e.g., nodes) in water distribution systems, by making use of some of the fundamental physical and hydraulic attributes of network components. The proposed technique may well be modified to generate similar combined metrics to serve the analysis of vulnerability to failures in water distribution systems. Additionally, in this paper a direct comparison of structural vulnerability of the studied water distribution networks has been presented based on the undertaken weighted and directed network measurements. In general, common topological graph techniques only provide a limited overview of the relationship between system connectivity and performance, mostly reporting average values at a global scale. On the one hand, the failure of different components with similar topological properties (e.g., failure of two nodes with identical degree) may result in totally different operational consequences (e.g., total duration of disruption and the amount of lost water). On the other hand, for actual indicators of system performance such as pipe discharge, flow direction, or the consumed energy at pumping stations, the likely changes of system topology due to failures may require different measurements which may lead to different assessments of the failure and its consequences. Consequently, employing the extended weighted and directed network models and utilizing tailored measurements to address specific local scale problems may improve analyses of structural vulnerability of water distributions systems. In other words, by using the information on system components (e.g., their attributes and function) the existing analogy between abstract link-node graphs and real water distribution systems may be strengthened. This, in turn, leads to a more credible assessment of susceptibility to damage in water distribution networks through a more accurate quantification of system topology and its relationship with the likelihood and impacts of failures, despite limited access to reliable performance/failure data and other challenges of quantifying real system properties.

[37] Importantly, this contribution has endeavored to progress the adoption of graph theoretic approaches to network analysis. In proposing the use of weighted and directed graph representations and offering an illustrative novel metric which is based on the nature of the specific network under consideration in terms of both topological and physical properties, a more widespread initiative has emerged which is driving the use of graph theoretic approaches into the hands of engineers and technicians. Such initiatives are realizing more practical value from graph theoretic techniques, both in the water sector [Deuerlein, 2008; Di Nardo and Di Natale, 2011] and in other areas of application [Laita, 2011]. This is not to argue that nondirected and nonweighted analyses are in any way faulty or unhelpful. Within the confines of their assumptions they can bring new insights, particularly with regard to global connectivity and interdependence. Early work in this area [Jacobs and Goulter, 1989; Yang et al., 1996; Todini, 2000] together with the more nuanced understandings generated in recent years [Yazdani and Jeffrey, 2011b; Ostfeld, 2005] continue to offer useful characterizations of water network connectivity and resilience. However, their use for planning and operational management is hamstrung by stark discrepancies between the way the network is represented in the analysis and its actual form and functionality. This disparity between the actual and the modeled both limits the utility and compromises the plausibility of the analysis. Closing this credibility gap is increasingly feasible given the opportunities available through well-conceived (and perhaps judicious) utilization of advanced and emerging network theory tools and techniques. Future work might usefully seek to reconcile some of the discrepancies noted above and extend the range of vulnerability metrics on offer to water network engineers and asset managers.


[38] The authors would like to thank The Leverhulme Trust for financial support and The Centre for Water Systems of the University of Exeter for providing access to benchmark network data.