# Topological Taxonomy of Water Distribution Networks

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Link Density q

#### 2.2. Average Node Degree k

#### 2.3. Diameter D

#### 2.4. Average Path Length l

#### 2.5. Spectral Radius (or Spectral Index) ${\lambda}_{1}^{A}$

#### 2.6. Spectral Gap $\Delta {\lambda}^{A}$

#### 2.7. Algebraic Connectivity ${\lambda}_{2}^{L}$

#### 2.8. Eigengap ${\Delta}^{L}$

## 3. Materials

## 4. Results and Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

WDN | Water Distribution Network |

CWDN | Complex Water Distribution Network |

DMAs | District Metered Areas |

## References

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**Figure 1.**Examples of the synthetic networks used for comparison: (

**a**) a small $3\times 3$ square grid with 9 nodes and 12 edges; and (

**b**) a small random 3-regular graph with 10 nodes and 15 edges.

**Figure 2.**Relationship between the link density q and the network size n. The data points follow a trend that can be fitted with a power law decrease $q\propto {n}^{-1.4}$. Both water networks, random regular graphs and planar square lattice show a similar $q\sim \frac{1}{n}$ scaling expected for sparse networks. The blue continuous line and the broken red line refer to random graph and square grids, respectively. The black dots represent the studied WDNs. For numerical values, please see Table 2.

**Figure 3.**Relationship between the average node degree $\overline{K}$ and the network size n compared to the case of random regular networks and planar square grids. Notice that, due to boundary effects, the square grids approach their theoretical value $\overline{K}\sim 4$ only for large network sizes n. The blue continuous line and the broken red line refer to random graph and square grids, respectively. The black dots represent the studied WDNs. For numerical values, please see Table 2.

**Figure 4.**Plot of the diameter D versus the network size n for synthetic and real water networks, planar square grids and random graphs. Notice that, similar to planar square grids, water networks also show the $D\sim {n}^{1/2}$ scaling that is expected for planar spatial networks. The blue continuous line and the broken red line refer to random graph and square grids, respectively. The black dots represent the studied WDNs. For numerical values, please see Table 2.

**Figure 5.**Plot of the average path length l versus the network size n for synthetic and real water networks, square grids and random graphs. Notice that, similar to planar square grids, water networks also show the $l\sim {n}^{1/2}$ scaling that is expected for planar spatial networks. The blue continuous line and the broken red line refer to random graph and square grids, respectively. The black dots represent the studied WDNs. For numerical values, please see Table 2.

**Figure 6.**Relationship between the inverse of the spectral index ${\lambda}_{1}^{A}$ and the network size n for water networks, random cubic networks and planar square grids. The blue continuous line and the broken red line refer to random graph and square grids, respectively. The black dots represent the studied WDNs. For numerical values, please see Table 2.

**Figure 7.**Relationship between the spectral gap $\Delta {\lambda}^{A}$ and the network size n. For the water networks, the fit of the data points is a power law $\Delta {\lambda}^{A}\sim {n}^{-0.36}$; for the random networks, thick line is fitting to an exponential approach $\sim {e}^{-n/35}$ to a plateau value of $\Delta {\lambda}^{A}$; and, for the square grids, to a power law $\Delta {\lambda}^{A}\sim {n}^{-0.93}$. The blue continuous line and the broken red line refer to random graph and square grids, respectively. The black dots represent the studied WDNs. For numerical values, please see Table 2.

**Figure 8.**Relationship between the algebraic connectivity ${\lambda}_{2}^{L}$ and the network size n. For the water networks, the fit of the data points is a power law ${\lambda}_{2}^{L}\sim {n}^{-1.26}$; for the random networks, thick line is fitting to an exponential approach $\sim {e}^{-n/35}$ to a plateau value of ${\lambda}_{2}^{L}$; and, for the square grids, to a power law ${\lambda}_{2}^{L}\sim {n}^{-0.99}$. The blue continuous line and the broken red line refer to random graph and square grids, respectively. The black dots represent the studied WDNs. For numerical values, please see Table 2.

**Figure 9.**Relationship between the optimal number of clusters ${c}_{opt}$ and the network size n. The thick brown line is a power law fit for the analyzed water networks, yielding ${c}_{opt}\sim {n}^{0.28}$, providing an indication for the optimal number of District Metered Areas (DMAs) of a water network of given size. The black dots represent the studied WDNs. For numerical values, please see Table 2.

**Table 1.**Name of network, number of nodes n, number of pipes m, number of loops r, type and data-file sources for all WDNs.

Name | Number of Nodes n | Number of Links m | Number of Loops r | Type | Source |
---|---|---|---|---|---|

Two Loop | 6 | 8 | 2 | synthetic | [38] |

Two Reservoirs | 10 | 17 | 8 | synthetic | [39] |

New York tunnel | 19 | 42 | 24 | synthetic | [40] |

Goyang | 22 | 30 | 9 | synthetic | [41] |

Anytown | 22 | 43 | 22 | synthetic | [42] |

Blacksburg | 30 | 35 | 6 | synthetic | [43] |

Hanoi | 31 | 34 | 4 | synthetic | [44] |

Bakryan | 35 | 58 | 24 | synthetic | [45] |

Fossolo | 36 | 58 | 23 | synthetic | [46] |

Richmond Skelton | 68 | 99 | 4 | synthetic | [47] |

Pescara | 41 | 44 | 32 | synthetic | [46] |

BWSN2008-1 | 126 | 168 | 43 | real | [48] |

Skiathos | 175 | 189 | 15 | real | [49] |

Parete | 184 | 282 | 101 | real | [14] |

Villaricca | 196 | 249 | 54 | real | [14] |

Monteruscello | 205 | 231 | 27 | real | [50] |

Modena | 268 | 317 | 50 | real | [46] |

Celaya | 333 | 477 | 145 | real | [51] |

Castellamare | 365 | 439 | 75 | real | GORI Spa |

D-Town | 399 | 443 | 45 | real | [52] |

Balerma Irrigation | 443 | 454 | 12 | real | [53] |

Oreto | 462 | 792 | 331 | real | [54] |

Richmond | 865 | 949 | 85 | real | [47] |

Giugliano | 994 | 1077 | 84 | real | [16] |

Matamoros | 1283 | 1651 | 369 | real | [10] |

Wolf Cordera Ranch | 1782 | 1985 | 204 | real | [55] |

San Luis Rio Colorado | 1890 | 2681 | 792 | real | [10] |

Exeter | 1891 | 3032 | 1142 | synthetic | [56] |

Exnet | 1891 | 2465 | 575 | synthetic | [56] |

Denia | 6276 | 6555 | 280 | real | Aqualia |

E-Town | 11063 | 13896 | 2834 | real | [57] |

Alcala | 11473 | 12454 | 982 | real | Aqualia |

BWSN2008-2 | 12523 | 14822 | 2300 | real | [48] |

Chihuahua | 34868 | 40330 | 5463 | real | [10] |

SG1 | 9 | 12 | 4 | synthetic | Matlab |

SG2 | 100 | 180 | 81 | synthetic | Matlab |

SG3 | 1024 | 1984 | 961 | synthetic | Matlab |

SG4 | 10000 | 19800 | 9801 | synthetic | Matlab |

SG5 | 34969 | 69564 | 34596 | synthetic | Matlab |

RG1 | 10 | 15 | 6 | synthetic | Matlab |

RG2 | 100 | 150 | 51 | synthetic | Matlab |

RG3 | 1000 | 1500 | 501 | synthetic | Matlab |

RG4 | 10000 | 15000 | 5001 | synthetic | Matlab |

RG5 | 35000 | 52500 | 17501 | synthetic | Matlab |

**Table 2.**Topological metric values calculated for all case studies: link density q, average node degree $\overline{K}$, graph diameter D, average path length l, spectral gap $\Delta {\lambda}^{A}$, algebraic connectivity ${\lambda}_{2}^{L}$, inverse spectral radius $1/{\lambda}_{1}^{A}$, optimal cluster number ${c}_{opt}$.

Name | q | $\overline{\mathit{K}}$ | D | l | $\mathbf{\Delta}{\mathit{\lambda}}^{\mathit{A}}$ | ${\mathit{\lambda}}_{2}^{\mathit{L}}$ | $1/{\mathit{\lambda}}_{1}^{\mathit{A}}$ | ${\mathit{c}}_{\mathbf{opt}}$ |
---|---|---|---|---|---|---|---|---|

Two Loop | 0.5333 | 2.67 | 4 | 1.90 | 1.2213 | 0.68862 | 0.404 | 2 |

Two Reservoirs | 0.3778 | 3.40 | 6 | 2.59 | 0.8799 | 0.37909 | 0.303 | 2 |

New York tunnel | 0.2456 | 4.42 | 9 | 4.21 | 0.5560 | 0.11799 | 0.400 | 3 |

Goyang | 0.1299 | 2.73 | 9 | 3.75 | 0.2595 | 0.09969 | 0.331 | 3 |

Anytown | 0.1861 | 3.91 | 7 | 2.94 | 0.4581 | 0.28044 | 0.218 | 2 |

Blacksburg | 0.0805 | 2.33 | 9 | 4.37 | 0.3077 | 0.08998 | 0.372 | 2 |

Hanoi | 0.0731 | 2.19 | 13 | 5.31 | 0.2739 | 0.06116 | 0.412 | 4 |

Bakryan | 0.0975 | 3.31 | 12 | 4.30 | 0.4793 | 0.07860 | 0.299 | 3 |

Fossolo | 0.0921 | 3.22 | 8 | 3.67 | 0.3516 | 0.21888 | 0.307 | 2 |

Richmond Skelton | 0.0537 | 2.15 | 24 | 9.24 | 0.0266 | 0.01091 | 0.411 | 3 |

Pescara | 0.0435 | 2.91 | 20 | 8.69 | 0.3024 | 0.00891 | 0.306 | 4 |

BWSN2008-1 | 0.0213 | 2.67 | 25 | 10.15 | 0.1004 | 0.00750 | 0.330 | 4 |

Skiathos | 0.0124 | 2.16 | 27 | 11.52 | 0.0461 | 0.00835 | 0.374 | 3 |

Parete | 0.0171 | 3.10 | 20 | 8.80 | 0.1714 | 0.02117 | 0.303 | 4 |

Villaricca | 0.0130 | 2.54 | 32 | 11.29 | 0.1194 | 0.00665 | 0.334 | 4 |

Monteruscello | 0.0110 | 2.25 | 47 | 20.24 | 0.0481 | 0.00152 | 0.352 | 5 |

Modena | 0.0089 | 2.37 | 38 | 14.04 | 0.1385 | 0.00908 | 0.334 | 6 |

Celaya | 0.0086 | 2.86 | 32 | 11.81 | 0.1915 | 0.01336 | 0.281 | 6 |

Castellamare | 0.0066 | 2.41 | 37 | 13.62 | 0.1640 | 0.00627 | 0.311 | 6 |

D-Town | 0.0056 | 2.22 | 66 | 26.32 | 0.0703 | 0.00065 | 0.350 | 5 |

Balerma Irrigation | 0.0046 | 2.05 | 60 | 23.89 | 0.0845 | 0.00069 | 0.370 | 6 |

Oreto | 0.0074 | 3.43 | 27 | 11.98 | 0.2016 | 0.00492 | 0.252 | 4 |

Richmond | 0.0025 | 2.19 | 135 | 51.44 | 0.0727 | 0.00014 | 0.345 | 8 |

Giugliano | 0.0022 | 2.17 | 51 | 21.22 | 0.1354 | 0.00243 | 0.327 | 9 |

Matamoros | 0.0020 | 2.57 | 80 | 27.76 | 0.1439 | 0.00100 | 0.291 | 8 |

Wolf Cordera Ranch | 0.0013 | 2.23 | 69 | 25.94 | 0.0612 | 0.00053 | 0.326 | 8 |

San Luis Rio Colorado | 0.0015 | 2.84 | 76 | 28.86 | 0.0063 | 0.00089 | 0.268 | 7 |

Exeter | 0.0017 | 3.21 | 54 | 20.61 | 0.6121 | 0.01021 | 0.257 | 10 |

Exnet | 0.0014 | 2.61 | 59 | 21.31 | 0.1190 | 0.00102 | 0.257 | 10 |

Denia | 0.0003 | 2.09 | 186 | 70.48 | 0.0797 | 0.00004 | 0.328 | 17 |

E-Town | 0.0002 | 2.51 | 289 | 71.13 | 0.0570 | 0.00003 | 0.281 | 13 |

Alcala | 0.0002 | 2.17 | 163 | 64.88 | 0.0957 | 0.00009 | 0.295 | 13 |

BWSN2008-2 | 0.0002 | 2.37 | 297 | 93.30 | 0.0147 | 0.00002 | 0.308 | 14 |

Chihuahua | 0.0001 | 2.31 | 368 | 186.05 | 0.0175 | 0.00001 | 0.282 | 18 |

SG1 | 0.3333 | 2.67 | 4 | 2.00 | 1.4142 | 1.00000 | 0.354 | - |

SG2 | 0.0364 | 3.60 | 18 | 6.67 | 0.2365 | 0.09790 | 0.261 | - |

SG3 | 0.0037 | 3.88 | 62 | 21.33 | 0.0271 | 0.00963 | 0.251 | - |

SG4 | 0.0004 | 3.96 | 198 | 66.67 | 0.0029 | 0.00099 | 0.250 | - |

SG5 | 0.0001 | 3.98 | 398 | 124.67 | 0.0008 | 0.00028 | 0.250 | - |

RG1 | 0.3333 | 3.00 | 3 | 1.80 | 1.3820 | 1.38200 | 0.333 | - |

RG2 | 0.0303 | 3.00 | 8 | 4.73 | 0.2614 | 0.26142 | 0.333 | - |

RG3 | 0.0030 | 3.00 | 13 | 8.04 | 0.1805 | 0.18055 | 0.333 | - |

RG4 | 0.0003 | 3.00 | 17 | 11.37 | 0.1737 | 0.17368 | 0.333 | - |

RG5 | 0.0001 | 3.00 | 19 | 13.01 | 0.1717 | 0.17170 | 0.333 | - |

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Giudicianni, C.; Di Nardo, A.; Di Natale, M.; Greco, R.; Santonastaso, G.F.; Scala, A.
Topological Taxonomy of Water Distribution Networks. *Water* **2018**, *10*, 444.
https://doi.org/10.3390/w10040444

**AMA Style**

Giudicianni C, Di Nardo A, Di Natale M, Greco R, Santonastaso GF, Scala A.
Topological Taxonomy of Water Distribution Networks. *Water*. 2018; 10(4):444.
https://doi.org/10.3390/w10040444

**Chicago/Turabian Style**

Giudicianni, Carlo, Armando Di Nardo, Michele Di Natale, Roberto Greco, Giovanni Francesco Santonastaso, and Antonio Scala.
2018. "Topological Taxonomy of Water Distribution Networks" *Water* 10, no. 4: 444.
https://doi.org/10.3390/w10040444