Generic patterns in the evolution of urban water networks: Evidence from a large Asian city

We examine high-resolution urban infrastructure data using every pipe for the water distribution network (WDN) and sanitary sewer network (SSN) in a large Asian city (≈4 million residents) to explore the structure as well as the spatial and temporal evolution of these infrastructure networks. Network data were spatially disaggregated into multiple subnets to examine intracity topological differences for functional zones of the WDN and SSN, and time-stamped SSN data were examined to understand network evolution over several decades as the city expanded. Graphs were generated using a dual-mapping technique (Hierarchical Intersection Continuity Negotiation), which emphasizes the functional attributes of these networks. Network graphs for WDNs and SSNs are characterized by several network topological metrics, and a double Pareto (power-law) model approximates the node-degree distributions of both water infrastructure networks (WDN and SSN), across spatial and hierarchical scales relevant to urban settings, and throughout their temporal evolution over several decades. These results indicate that generic mechanisms govern the networks’ evolution, similar to those of scale-free networks found in nature. Deviations from the general topological patterns are indicative of (1) incomplete establishment of network hierarchies and functional network evolution, (2) capacity for growth (expansion) or densification (e.g., in-fill), and (3) likely network vulnerabilities. We discuss the implications of our findings for the (re-)design of urban infrastructure networks to enhance their resilience to external and internal threats.

Figure 1

Schematic of primal- versus dual-mapping approach applied in this study (top: spatial maps, bottom: network graphs). Left: primal mapping counts each pipe segment between intersections as edges ($e$), and the intersections as nodes ($n$), resulting in $e=7$ and $n=3$ in this schematic example. Right: Dual mapping creates a node from several pipe segments, which form a functional unit based on unchanged pipe diameter (flow capacity). Intersections connecting different functional pipe units form edges, resulting in $e=3$ and $n=4$.

Figure 2

Temporal evolution of sewer network 1970–2015. The 1970 network is highlighted in all time steps. Topological analysis was performed for 10 time steps with results being consistent with those of the functional subnets; data for six time steps are shown.

Figure 3

Pipe hierarchies (diameters) of the entire WDN (left) and SSN (right). Network topologies were analyzed separately for the highest pipe hierarchy and for networks with pipes added for an incrementally shrinking pipe diameter threshold. Networks shown here are the entire networks “grown” from the “backbone” (largest diameter pipes).

Figure 4

Topological network metrics for all (110) analyzed subnets, including the entire networks for various time steps, and hierarchical subnetworks. WDNs (blue circles) and SSNs (red dots): (a) network density follows a PL distribution ($y=1.76x^{-0.96},R^2=0.995$); (b) average number of neighbors (average node degree) increases for small network sizes, and converges to ≈2 for SSNs, and ≈2.3 for WDNs); (c) clustering coefficient versus average node degree; (d) centralization follows a power law ($y=2.03x^{-0.62},R^2=0.85$); (e) characteristic path length increases in the form: $y=1.38x^{0.31},R^2=0.83$); (f) network heterogeneity.

Figure 5

Characteristics of node-degree distributions: (a) NDD of SSN in 2005 (3938 dual-mapped nodes, 52,675 primal nodes) follows a (double) PL function with breaking point $k_{\mathrm{break}}=10,p(k\ge 2)=1.22{k^{-}}^{2.41}$ for the trunk, and $p\left(k>k_{\mathrm{break}}\right)=2.74k^{-2.95}$ for the tail; (b) comparison of NDD of similarly sized subnets of WDN (DZ10, circles) and SSN (2005 network, asterisks) highlights the similarity of topologies among SSN and WDN; (c) box plot showing heteroscedasticity of PL exponents for the trunks of WDN (hollow) and SSN (dashed) across the full range of subnet sizes; [mean (small squares), median (thick line), interquartile range (box), [25–75th percentile ±(1.5* Interquartile Range)] (whiskers), outliers (diamonds)]. ${\gamma }_{\mathrm{trunk}}$ converges at ${\gamma }_{\mathrm{trunk}}=2.45\pm 0.27$ for network size ≥200 dual-mapped nodes (mean of ${\gamma }_{\mathrm{trunk}}=2.53$ for WDNs and ${\gamma }_{\mathrm{trunk}}=2.41$ for SSNs), except for two WDN outliers (see text); (d) Breaking points between the two power laws of NDD for WDNs (blue circles) and SSNs (red asterisks). Outliers are discussed in the text and shown in Fig. 6, 7, 8.

Figure 6

WDN subnets along a gradient of breaking points between the power laws of trunk and tail ($k_{\mathrm{break}}$ outliers from panel (d) in Fig. 5): (a) DZ11: $k_{\mathrm{break}}=5,p{\left(k\right)}_{\mathrm{trunk}}=0.40k^{-2.49}$ and $p{\left(k\right)}_{\mathrm{tail}}=0.09k^{-1.47},N=2425$ (dual-mapped nodes); (b) DZ10: $k_{\mathrm{break}}=8,p{\left(k\right)}_{\mathrm{trunk}}=1.46k^{-2.80}$ and $p{\left(k\right)}_{\mathrm{tail}}=0.26k^{-1.88},N=3179$; (c) DZ01: $k_{\mathrm{break}}=10,p{\left(k\right)}_{\mathrm{trunk}}=1.17k^{-2.37}$ and $p{\left(k\right)}_{\mathrm{tail}}=0.138k^{-1.686},N=1497$; (d) DZ32: in this subnet power-law distributions of trunk and tail converge as the breaking point between the two power laws increases $\left(k_{\mathrm{break}}=20\right)$, and $p\left(k\right)=1.07k^{-2.27}\left(N=2271\right)$.

Figure 7

WDN graphs of selected subnetworks: (a) DZ11: tendency to contain high node-degree hubs, $\mathrm{heterogeneity}=2.34$; (b) DZ32: $h=1.58$.

Figure 8

Spatial maps of the selected water distribution subnetworks: (a) DZ11, (b) DZ32.

Figure 9

Geometric characteristics (pdfs) of the water districts (subzones of DZs): area: $y=0.47x^{-1.58},R^2=0.90$; population density: $y=14x^{-1.38},R^2=0.87$; pipe length per customer: SSN (closed circles): $y=2.63x^{-1.32},R^2=0.73$; WDN (open circles): $y=1.11x^{-1.00},R^2=0.68$.

Figure 10

Growth of population and SSN in our case study city occurs in waves, with a distinct stepwise growth function for SSN. Dashed lines are fitted models, exponential growth model for population, superpositioned logistic growth model for SSN.