# Assessment of the Water Distribution Networks in the Kingdom of Saudi Arabia: A Mathematical Model

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## Abstract

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## 1. Introduction

- Vulnerability framework: This provides a vulnerability prediction measure that can be compared with other networks or with the same network under different conditions.
- Centrality metrics: These are the most widely employed indicators for identifying important nodes in a network.
- Structural features: These are based merely on the connections between nodes, which in this case represent how the pipes and the junctions are connected.
- Operational features: These are based on the operation of the system, considering the demand, the supply, and the water flows in a particular state.
- Malware profile: This is an ordered collection of network elements (nodes or links).
- Functional damage: This is measured as the fraction of unsatisfied demand determined after removing the faulty elements.
- Physical damage: This is measured by determining the fraction of removed elements in a particular stage of the failure sequence.
- Vulnerability curve: This is a graph in which the axis along the horizontal direction indicates a physical damage metric and the vertical axis a functional damage measure.
- Vulnerability prediction measure (VPM): This is derived from the area under the vulnerability curve and expresses the degree of damage using the physical damage measure.

## 2. Water Distribution Network Modelling

## 3. Vulnerability Framework

## 4. Centrality Metrics

#### 4.1. Degree Metric of Centrality

#### 4.2. Eigenvector Metric of Centrality

#### 4.3. Betweenness Metric of Centrality

#### 4.4. Closeness Metric of Centrality

#### 4.5. PageRank Metric of Centrality

#### 4.6. Katz Metric of Centrality

## 5. Representation Using Line Graphs

**A**) of the original graph are shown in Table 1. Figure 4 shows the line graph obtained after applying the proposed transformation, where each node represents a line (or pipe) in the original graph; the edges and the adjacency matrix of the transformed graph

**M**are shown in Table 2.

## 6. Vulnerability Assessment on Benchmarks

#### 6.1. Attacks on Junctions

#### 6.2. Attacks on Pipes

## 7. Vulnerability of an Urban Water Distribution System Based in KSA: A Case Study

#### 7.1. Attacks on Junctions

#### 7.2. Attacks on Pipes

## 8. Discussion

## 9. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Torres, J.; Duenas-Osorio, L.; Li, Q.; Yazdani, A. Exploring Topological Effects on Water Distribution System Performance Using Graph Theory and Statistical Models. J. Water Resour. Plan. Manag.
**2017**, 143, 04016068. [Google Scholar] [CrossRef] - Klempous, R.; Kotowski, J.; Nikodem, J.; Olesiak, M.; Ulasiewicz, J. Some models for water distribution systems. J. Comput. Appl. Math.
**1988**, 21, 257–269. [Google Scholar] [CrossRef] - Ko, M.J.; Choi, Y.H. Optimal Design of Water Distribution Systems Considering Topological Characteristics and Residual Chlorine Concentration. Mathematics
**2022**, 10, 4721. [Google Scholar] [CrossRef] - Faramondi, L.; Setola, R.; Panzieri, S.; Pascucci, F.; Oliva, G. Finding critical nodes in infrastructure networks. Int. J. Crit. Infrastruct. Prot.
**2018**, 20, 3–15. [Google Scholar] [CrossRef] - Giudicianni, C.; Di Nardo, A.; Di Natale, M.; Greco, R.; Santonastaso, G.; Scala, A. Topological taxonomy of water distribution networks. Water
**2018**, 10, 444. [Google Scholar] [CrossRef] - Meng, F.; Fu, G.; Farmani, R.; Sweetapple, C.; Butler, D. Topological attributes of network resilience: A study in water distribution systems. Water Res.
**2018**, 143, 376–386. [Google Scholar] [CrossRef] - Albarakati, A.; Bikdash, M.; Dai, X. Line-graph based modeling for assessing the vulnerability of transmission lines. In Proceedings of the SoutheastCon 2017, Concord, NC, USA, 30 March–2 April 2017; pp. 1–8. [Google Scholar]
- Gutiérrez-Pérez, J.; Herrera, M.; Pérez-García, R.; Ramos-Martínez, E. Application of graph-spectral methods in the vulnerability assessment of water supply networks. Math. Comput. Model.
**2013**, 57, 1853–1859. [Google Scholar] [CrossRef] - Nardo, A.; Giudicianni, C.; Greco, R.; Herrera, M.; Santonastaso, G. Applications of graph spectral techniques to water distribution network management. Water
**2018**, 10, 45. [Google Scholar] [CrossRef] - Albarakati, A. Evaluation of the Most Harmful Malicious Attacks in Power Systems Based on a New Set of Centralities. J. Electr. Eng. Technol.
**2021**, 16, 1929–1939. [Google Scholar] [CrossRef] - Albarakati, A.; Tassaddiq, A.; Kale, Y. Evaluation of the vulnerability in water distribution systems through targeted attacks. AQUA—Water Infrastruct. Ecosyst. Soc.
**2021**, 70, 1257–1271. [Google Scholar] [CrossRef] - Kim, A.; van der Beek, H. Holistic Assessment of the Water-for-Agriculture Dilemma in the Kingdom of Saudi Arabia; Georgetown University Qatar: Doha, Qatar, 2018. [Google Scholar]
- Gazzeh, K.; Abubakar, I. Regional disparity in access to basic public services in Saudi Arabia: A sustainability challenge. Util. Policy
**2018**, 52, 70–80. [Google Scholar] [CrossRef] - Government of Saudi Arabia Saudi Vision 2030. 2016. Available online: https://vision2030.gov.sa/en (accessed on 17 January 2023).
- Holme, P.; Kim, B.J.; Yoon, C.N.; Han, S.K. Attack vulnerability of complex networks. Phys. Rev. E
**2002**, 65, 056109. [Google Scholar] [CrossRef] [PubMed] - Sharif, M.; Farahat, A.; Al-Zahrani, M.; Islam, N.; Rodriguez, M.; Sadiq, R. Optimization of chlorination boosters in drinking water distribution network for Al-Khobar City in Saudi Arabia. Arab. J. Geosci.
**2016**, 9, 1–11. [Google Scholar] [CrossRef] - Kapelan, Z. Modelling in water distribution systems. In Simplicity Complexity And Modelling (Statistics in Practice); John Wiley & Sons: Hoboken, NJ, USA, 2011; pp. 103–124. [Google Scholar]
- Eliades, D.; Kyriakou, M.; Vrachimis, S.; Polycarpou, M. EPANET-MATLAB Toolkit: An Open-Source Software for Interfacing EPANET with MATLAB. In Proceedings of the Computing and Control for the Water Industry CCWI 2016, Amsterdam, The Netherlands, 7–9 November 2016; pp. 1–8. [Google Scholar]
- Eliades, D.; Kyriakou, M.; Vrachimis, S.; Polycarpou, M. OpenWaterAnalytics Epanet-Matlab Toolkit. MathWorks File Exchange. 2021. Available online: https://la.mathworks.com/matlabcentral/fileexchange/25100-openwateranalytics-epanet-matlab-toolkit (accessed on 17 January 2023).
- The Mathworks Inc. Graph and Network Algorithms. Matlab Documentation. 2020. Available online: https://www.mathworks.com/help/matlab/graph-and-network-algorithms.html (accessed on 17 January 2023).
- Agathokleous, A.; Christodoulou, C.; Christodoulou, S. Topological Robustness and Vulnerability Assessment of Water Distribution Networks. Water Resour. Manag.
**2017**, 31, 4007–4021. [Google Scholar] [CrossRef] - Giustolisi, O.; Ridolfi, L.; Simone, A. Tailoring Centrality Metrics for Water Distribution Networks. Water Resour. Res.
**2019**, 55, 2348–2369. [Google Scholar] [CrossRef] - Bloch, F.; Jackson, M.; Pietro, T. Centrality Measures in Networks. Soc. Choice Welf.
**2023**, 61, 413–453. [Google Scholar] [CrossRef] - Page, L.; Brin, S.; Motwani, R.; Winograd, T. The Page Rak Citation Ranking: Bringing Order to the Web. In Technical Report; 1999; pp. 1–17. [Google Scholar]
- Riquelme, F.; Gonzalez-Cantergiani, P.; Molinero, X.; Serna, M. Centrality measure in social networks based on linear threshold model. Knowl.-Based Syst.
**2018**, 140, 92–102. [Google Scholar] [CrossRef] - Katz, L. A new status index derived from sociometric analysis. Psychometrika
**1953**, 18, 39–43. [Google Scholar] [CrossRef] - Godsil, C.; Royle, G. Algebraic Graph Theory; Springer: New York, NY, USA, 2001; Volume 207. [Google Scholar]
- Yoshida, T. Weighted line graphs for overlapping community discovery. Soc. Netw. Anal. Min.
**2013**, 3, 1001–1013. [Google Scholar] [CrossRef] - United States Environmental Protection Agency Epanet Application for Modeling Drinking Water Distribution Systems. Available online: https://www.epa.gov/water-research/epanet (accessed on 17 January 2023).
- Albarakati, A.J.; Boujoudar, Y.; Azeroual, M.; Eliysaouy, L.; Kotb, H.; Aljarbouh, A.; Khalid Alkahtani, H.; Mostafa, S.M.; Tassaddiq, A.; Pupkov, A. Microgrid energy management and monitoring systems: A comprehensive review. Front. Energy Res.
**2022**, 10, 1097858. [Google Scholar] [CrossRef] - Albarakati, A.J.; Azeroual, M.; Boujoudar, Y.; EL Iysaouy, L.; Aljarbouh, A.; Tassaddiq, A.; EL Markhi, H. Multi-Agent-Based Fault Location and Cyber-Attack Detection in Distribution System. Energies
**2023**, 16, 224. [Google Scholar] [CrossRef]

**Figure 4.**Line graph corresponding to the EPANET-MSX Example Network applying the line graph conversion.

**Figure 11.**Vulnerability curves obtained from attacks on junctions with different centrality measures and attack sequences for the representative urban Saudi Arabian WDS.

**Figure 12.**Zone definition and vulnerability ranking of junctions for the representative urban Saudi Arabian WDS.

**Figure 14.**Vulnerability curves obtained from attacks on pipes with different centrality measures and attack sequences for the representative urban Saudi Arabian WDS.

From Node | To Node | Weight | Adjacency Matrix (A) | ||||
---|---|---|---|---|---|---|---|

1 | 2 | 4.0691 | 0 | 4.0691 | 7.1309 | 0 | 15.3000 |

1 | 3 | 7.1309 | 4.0691 | 0 | 0.6691 | 0 | 0 |

1 | 5 | 15.3000 | 7.1309 | 0.6691 | 0 | 2.3000 | 0 |

2 | 3 | 0.6691 | 0 | 0 | 2.3000 | 0 | 0 |

3 | 4 | 2.3000 | 15.3000 | 0 | 0 | 0 | 0 |

**Table 2.**Edges and adjacency matrix corresponding to the line graph obtained from the EPANET-MSX Example Network.

From Node | To Node | Weight | Adjacency Matrix (M) | ||||
---|---|---|---|---|---|---|---|

1 | 2 | 13.8885 | 0 | 13.8885 | 23.7508 | 0 | 0 |

1 | 3 | 23.7508 | 13.8885 | 0 | 4.3525 | 1.5449 | 0 |

2 | 3 | 4.3525 | 23.7508 | 4.3525 | 0 | 1.2582 | 7.2421 |

2 | 4 | 1.5449 | 0 | 1.5449 | 1.2582 | 0 | 0.4354 |

3 | 4 | 1.2582 | 0 | 0 | 7.2421 | 0.4354 | 0 |

3 | 5 | 7.2421 | |||||

4 | 5 | 0.4354 |

Benchmark | Junctions | Pipes | Reservoirs | Tanks | External src |
---|---|---|---|---|---|

Net1 | 11 | 13 | 1 | 1 | 0 |

Net2 | 36 | 40 | 0 | 1 | 1 |

Net3 | 97 | 119 | 2 | 3 | 0 |

Strategy | Centrality | VPM | ||
---|---|---|---|---|

Net1 | Net2 | Net3 | ||

RMCEF | Degree | 0.785 | 0.783 | 0.768 |

Eigenvector | 0.706 | 0.723 | 0.627 | |

Betweenness | 0.876 | 0.830 | 0.662 | |

Closeness | 0.876 | 0.750 | 0.612 | |

PageRank | 0.834 | 0.958 | 0.842 | |

Katz | 0.706 | 0.768 | 0.760 | |

IMCEF | Degree | 0.863 | 0.918 | 0.953 |

Eigenvector | 0.863 | 0.914 | 0.952 | |

Betweenness | 0.863 | 0.926 | 0.901 | |

Closeness | 0.863 | 0.922 | 0.875 | |

PageRank | 0.834 | 0.863 | 0.889 | |

Katz | 0.863 | 0.914 | 0.952 |

Technique | Centrality Metrics | VPM Types | ||
---|---|---|---|---|

Net1 | Net2 | Net3 | ||

RMCEF | Degree | 0.730 | 0.776 | 0.750 |

Eigenvector | 0.664 | 0.709 | 0.598 | |

Betweenness | 0.503 | 0.791 | 0.611 | |

Closeness | 0.580 | 0.682 | 0.506 | |

PageRank | 0.569 | 0.690 | 0.681 | |

Katz | 0.664 | 0.778 | 0.749 | |

IMCEF | Degree | 0.884 | 0.932 | 0.962 |

Eigenvector | 0.884 | 0.921 | 0.961 | |

Betweenness | 0.566 | 0.918 | 0.881 | |

Closeness | 0.723 | 0.894 | 0.798 | |

PageRank | 0.643 | 0.775 | 0.805 | |

Katz | 0.884 | 0.943 | 0.962 |

Centrality Metrics | VPM Types | |
---|---|---|

RMCEF | IMCEF | |

Degree | 0.745 | 0.819 |

Eigenvector | 0.452 | 0.802 |

Betweenness | 0.747 | 0.851 |

Closeness | 0.701 | 0.824 |

PageRank | 0.795 | 0.817 |

Katz | 0.762 | 0.815 |

Centrality Metrics | Kinds of VPM | |
---|---|---|

RMCEF | IMCEF | |

Degree | 0.644 | 0.613 |

Eigenvector | 0.692 | 0.684 |

Betweenness | 0.548 | 0.743 |

Closeness | 0.554 | 0.651 |

PageRank | 0.558 | 0.654 |

Katz | 0.633 | 0.617 |

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**MDPI and ACS Style**

Albarakati, A.; Tassaddiq, A.; Srivastava, R.
Assessment of the Water Distribution Networks in the Kingdom of Saudi Arabia: A Mathematical Model. *Axioms* **2023**, *12*, 1055.
https://doi.org/10.3390/axioms12111055

**AMA Style**

Albarakati A, Tassaddiq A, Srivastava R.
Assessment of the Water Distribution Networks in the Kingdom of Saudi Arabia: A Mathematical Model. *Axioms*. 2023; 12(11):1055.
https://doi.org/10.3390/axioms12111055

**Chicago/Turabian Style**

Albarakati, Aiman, Asifa Tassaddiq, and Rekha Srivastava.
2023. "Assessment of the Water Distribution Networks in the Kingdom of Saudi Arabia: A Mathematical Model" *Axioms* 12, no. 11: 1055.
https://doi.org/10.3390/axioms12111055