# Complex Networks Theory for Evaluating Scaling Laws and WDS Vulnerability for Potential Contamination Events

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Complex Networks

#### 2.2. Cayley Regular Graphs

_{t}(l), supplied by a given node that is at level l is given by:

_{t}(l) can also be expressed as:

_{t}(l’) behaves as a power law:

_{S}(l), is given by:

_{S,a}(l’), denoted N

_{S,a,cum}(l’), as:

_{S,a,cum}(l’) represents the quantity we need in order to calculate the distribution function of potentially contaminated nodes. From Equation (11), it is easily seen that N

_{S,a,cum}(l’) scales linearly with respect to N

_{t}(l’) on a log-log plot (with a negative slope). In a real system, where loops are present and the direction of flows is not easily determined, we need to perform hydraulic simulations. The following section describes the methodology adopted for calculating the distribution function for a generic WDS.

#### 2.3. Methodology for Determining the Virtual Network and Its Distribution Functions

- Extended period simulations and source tracing analyses are performed with Epanet, one for each junction as a trace node, with sufficient simulation time in order to reach steady-state (or cyclic) conditions.
- A directed functional network is built, where the adjacency matrix is characterized by having 1 at position ij (for i ≠ j) if, and only if, the contaminant injected at the i-th node reaches the j-th node, with a predefined concentration threshold (typically, 0.1%).
- The out-degree distribution of this directed, complex network is what we have previously called N
_{S,a,cum}(l’), and may be plotted on a log-log scale with respect to N_{t}(l’), the number of nodes supplied. - A stretched exponential function is fitted to the results of numerical simulations in such log-log plots. This allows the evaluation of its scaling behavior and can be compared to the power-law distribution describing the tree-like regular network. Accordingly, two separate fittings may also be obtained, namely, a power-law and an exponential function for subsequent analytical derivation.
- The analytical formula derived in Section 2.4 allows an estimate of the vulnerability of the system. Thus, the more “distant” (in a probabilistic sense) the two distributions are, the more different the behavior of the real WDS is from a purely tree-like and auto-similar network, resulting in higher values of the index.

#### 2.4. Analytical Expression of the Vulnerability Index: A Pragmatic Derivation

_{S,a,cum}and N

_{t}for a generic WDN may be approximated by a stretched exponential distribution on a log-log plot, as emerged from the results (see Section 3). However, in order to derive an analytical expression for the Vulnerability Index, the first part of the out-degree distribution is approximated by a power-law, and the second part by an exponential fit. If we denote N

_{inf}= 1 and N

_{sup}as the lower and upper limits of the range for which power-law is adopted and N

_{sup}and N

_{max}as the corresponding limits for the exponential approximation, we can say that the following expressions hold (x represents N

_{t}):

_{I}—that is, the area between the out-degree distribution of the WDN and the tree-like Cayley graph—can be written as:

_{inf}= 1, simple manipulation gives the following expression:

## 3. Results

#### 3.1. Characteristics of the Real WDSs Adopted for the Analyses

#### 3.2. Scaling Laws of Different WDSs

_{t}, they are not properly represented by the exponential fitting.

#### 3.3. Scaling Laws Obtained by Different Fittings

#### 3.4. Determination of the Vulnerability Index for the Real Networks: A Pragmatic Method

_{I}, we perform a separate approximation: power law (Figure 8) and exponential (Figure 9). Such curves allow the determination of, for each WDS, the coefficients, allowing the calculation of the V

_{I}: The results are reported in Table 4. In order to calculate V

_{I}, it has been assumed that N

_{sup}= 100 and N

_{max}= 1000.

## 4. Discussion and Concluding Remarks

_{t}), while when moving to the right of the diagram, we are virtually going “upstream” in the water distribution system, towards the reservoirs (which supply a large amount of N

_{t}).

_{t}and the exponential for higher values.

## Funding

## Conflicts of Interest

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**Figure 1.**Example of a Cayley regular graph with five levels (L = 5) and a bifurcation ratio of X = 3 (solid lines represent actual links). The symbol l represents the number of the level counted from the origin, so l = 0, 1, 2, 3, 4.

**Figure 2.**Layout of the four real systems adopted in the numerical analyses: (

**a**) G water distribution system (WDS); (

**b**) U WDS; (

**c**) T WDS; (

**d**) P WDS. The first three supply medium-sized cities, while P WDS supplies twelve small towns.

**Figure 3.**Log-log plot of cumulative out-degree distribution function for several WDSs, together with an overall exponential fitting. The tree-like auto-similar network of Figure 1 is also indicated, with its power-law distribution.

**Figure 4.**Cumulative out-degree distribution functions for G WDS, with stretched exponential fitting.

**Figure 5.**Cumulative out-degree distribution functions for T WDS, with stretched exponential fitting.

**Figure 6.**Cumulative out-degree distribution functions for U WDS, with stretched exponential fitting.

**Figure 7.**Cumulative out-degree distribution functions for P WDS, with stretched exponential fitting.

**Figure 8.**Cumulative out-degree distribution functions limited in the range 100–1000 N

_{t}, together with their exponential fitting.

**Figure 9.**Cumulative out-degree distribution functions limited in the range 1–100 N

_{t}, together with their power-law fitting.

**Table 1.**Main characteristics of the real WDSs analyzed: n, number of junctions; p, number of pipes; t, number of tanks; L, total length of pipes; Pop, population served; ρ, network density; <k>, average node degree; <ℓ>, average path length; D, network diameter; <C

_{c}>, average clustering coefficient.

WDS | n | p | t | L (km) | Pop | ρ | <k> | <ℓ> | D | <C_{c}> |
---|---|---|---|---|---|---|---|---|---|---|

G | 3958 | 4170 | 5 | 201 | 45,000 | 0.001 | 2.11 | 50.04 | 137 | <0.001 |

T | 4140 | 4601 | 20 | 326 | 85,000 | 0.001 | 2.22 | 46.82 | 130 | 0.009 |

U | 5772 | 6453 | 19 | 408 | 97,000 | <0.001 | 2.23 | 49.58 | 127 | 0.001 |

P | 4619 | 5115 | 26 | 756 | 66,000 | <0.001 | 2.21 | 60.93 | 165 | 0.020 |

**Table 2.**Main properties of the virtual networks built from extended period simulations and source tracing analyses (n, number of nodes; m, number of links). Symbols with the subscript u refer to the undirected network. ρ, network density; <k

_{out}>, average out-node degree; <ℓ>, average path length; D, network diameter; <C

_{c}>, average clustering coefficient.

WDS | n | m | ρ | ρ_{u} | <k_{out}> | <ℓ> | <ℓ_{u}> | D | D_{u} | <C_{c}> | <C_{c,u}> |
---|---|---|---|---|---|---|---|---|---|---|---|

G | 3958 | 281,185 | 0.018 | 0.0359 | 71.042 | 1.1114 | 2.0970 | 5 | 8 | 0.204 | 0.443 |

T | 4140 | 286,719 | 0.017 | 0.0335 | 69.256 | 1.5126 | 2.3657 | 8 | 13 | 0.184 | 0.433 |

U | 5772 | 343,422 | 0.010 | 0.0206 | 59.498 | 2.0941 | 3.6016 | 11 | 10 | 0.289 | 0.583 |

P | 4619 | 219,646 | 0.010 | 0.0206 | 47.553 | 1.6635 | 3.6291 | 5 | 14 | 0.270 | 0.578 |

WDS | A | B | C |
---|---|---|---|

G | 1.0139 | 0.064 | 0.550 |

T | 1.0336 | 0.078 | 0.551 |

U | 1.5449 | 0.272 | 0.370 |

P | 1.1431 | 0.229 | 0.410 |

**Table 4.**Values of power-law and exponential fitting coefficients, and resulting Vulnerability Index, V

_{I}, for the real networks analyzed.

WDS | α | β | γ | δ | V_{I} |
---|---|---|---|---|---|

G | 1.0826 | 0.181 | 0.6627 | 0.003 | 205.73 |

T | 1.0865 | 0.188 | 0.5685 | 0.003 | 182.91 |

U | 1.3901 | 0.270 | 0.5028 | 0.003 | 165.83 |

P | 1.2942 | 0.364 | 0.2830 | 0.003 | 98.22 |

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Nicolini, M.
Complex Networks Theory for Evaluating Scaling Laws and WDS Vulnerability for Potential Contamination Events. *Water* **2020**, *12*, 1296.
https://doi.org/10.3390/w12051296

**AMA Style**

Nicolini M.
Complex Networks Theory for Evaluating Scaling Laws and WDS Vulnerability for Potential Contamination Events. *Water*. 2020; 12(5):1296.
https://doi.org/10.3390/w12051296

**Chicago/Turabian Style**

Nicolini, Matteo.
2020. "Complex Networks Theory for Evaluating Scaling Laws and WDS Vulnerability for Potential Contamination Events" *Water* 12, no. 5: 1296.
https://doi.org/10.3390/w12051296