# Applications of Graph Spectral Techniques to Water Distribution Network Management

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Spectral Graph Theory

^{3}), where n is the number of vertices/nodes (it is usual to name the elements of a graph as vertices and edges and the elements of a network as nodes and links; we make this distinction throughout the paper.) in the associated graph/network. From the 1990s, graph spectra have been used for several important applications in many fields [31]; such as expanders and combinatorial optimization, complex networks and the internet topology, data mining, computer vision and pattern recognition, internet search, load balancing and multiprocessor interconnection networks, anti-virus protection, knowledge spread, statistical databases and social networks, quantum computing, bioinformatics, coding theory, control theory, and computer sciences.

#### 2.1. Graph Matrices

**A**, and the Laplacian matrix,

**L**, are widely used in graph analysis. Another matrices such as the Modularity matrix, the Similarity matrix, and the sign-less Laplacian are omitted from the current GST tool-set. Using them will make a wider GST mathematical framework but require a further investigation that falls out of the scope of this proposal. The following items synthetically describe a number of graph matrices that are related to

**A**and

**L**, whose properties are introduced and developed in this paper.

- Adjacency Matrix
**A**: let G = (V, E) be an undirected graph with n-vertices set V and m-edges set E. A common way to represent a graph is to define its Adjacency matrix**A**, whose elements a_{ij}= a_{ji}= 1 if nodes i and j are directly connected and a_{ij}= a_{ji}= 0 otherwise. The degree of node i of**A**is defined as ${k}_{i}={\sum}_{j=1}^{n}{a}_{ij}$; - Weighted Adjacency Matrix
**W**: it is possible to express the weighted Adjacency matrix**W**, in case to be available information about the connection strength between vertices of the graph G. Edge weights are expressed in terms of proximity and/or similarity between vertices. Thus, all of the weights are non-negative. That is, w_{ij}= w_{ji}≥ 0 if i and j are connected, w_{ij}= w_{ji}= 0 otherwise. The degree of a node i of**W**is defined as ${k}_{i}={\sum}_{j=1}^{n}{w}_{ij}$; - Un-normalized Laplacian Matrix
**L**: one of the main utilities of spectral graph theory is the Laplacian matrix [32] and both its un-normalized and normalized version [8]. Let**D**_{k}= diag(k_{i}) be the diagonal matrix of the vertex connectivity degrees, the Laplacian matrix is defined as the difference between**D**_{k}and the Adjacency matrix**A**(or the weighted Adjacency matrix**W**if it is considered a weighted graph). The un-normalized Laplacian matrix is defined by**L**=**D**_{k}−**A**(**L**=**D**_{k}−**W**); - Random Walk Normalized Laplacian Matrix
**L**_{rw}: it is closely related to a random walk representation. Its definition comes from the Laplacian matrix L being multiplied by the inverse of the diagonal matrix of the vertex connectivity degrees,**D**_{k}. Then, ${\mathit{L}}_{rw}={{\mathit{D}}_{k}}^{-1}\mathit{L}$ [33].

_{1}≤ … ≤ λ

_{n}. These properties are of main importance in the graph spectral theory.

#### 2.2. Network Eigenvalues

- The Largest eigenvalue (Spectral radius or Index) λ
_{1}: it refers to the Adjacency graph matrix**A**and it plays an important role in modelling a moving substance propagation in a network. It takes into account not only immediate neighbours of vertices, but also the neighbours of the neighbours [34]. Spectral radius concept is often introduced by using the example of how a virus spread in a network. The smaller the Spectral radius the larger the robustness of a network against the spread of any virus in it. In this regard, the epidemic threshold is proportional to the Inverse of Spectral radius 1/λ_{1}[35]. This fact can be explained as the number of walks in a connected graph is proportional to λ_{1}. The greater the number of walks of a network, the more intensive is the spread of the moving substance in it. The other way round, the higher the Spectral radius, the better is the communication into a network. - The Spectral gap ∆λ: it represents the difference between the first and second eigenvalue of an Adjacency matrix,
**A**. It is a measure of network connectivity strength. In particular, it quantifies the robustness of network connections and the presence of bottlenecks, articulation points, or bridges. This is of significant importance, as the removal of a bridge splits the network in two or more parts. The larger the Spectral gap the more robust is the network [36]. - The Multiplicity of zero eigenvalue m
_{0}: the multiplicity of the eigenvalue 0 of**L**is equal to the number of connected components**A**_{1}, …,**A**_{k}in the graph; thus, the matrix**L**has as many eigenvalues 0 as connected components [37]. - The Eigengap λ
_{k+}_{1}− λ_{k}: it is a spectral utility specifically designed for network clustering. A suitable number of clusters k may be chosen such that all eigenvalues λ_{1}, …, λ_{k}of Laplacian matrix**L**are very small, but λ_{k+}_{1}is relatively large [38]. The more significant the difference for a-priori proposing the number of clusters the better is the further clustering configuration. - The Second smallest eigenvalue (Algebraic connectivity) λ
_{2}: it refers to the Laplacian matrix. λ_{2}plays a special role in many graph theories related problems [39]. It quantifies the strength of network connections and its robustness to link failures. The larger the Algebraic connectivity is the more difficult to cut a graph into independent components. It is also related to the min-cut problem of a data set for spectral clustering [37].

_{1}, …, λ

_{5}of the Laplacian matrix for the four layout configurations of Example Network. It is noticeable that some eigenvalues are equal for all of the layouts. The first eigenvalue λ

_{1}is always equal to zero because the graph Laplacian matrix is positive semi-definite [37].

_{0}, is equal to 2. Consequently, also the second eigenvalue λ

_{2}(the Algebraic connectivity) is equal to zero (Table 1). This means that there are two separated subregions in the network, as the number of multiplicity of zero, m

_{0}, is equal to the number of the disconnected subregions. In all four layouts, the maximum eigengap occurs between the third eigenvalue λ

_{3}and the second eigenvalue λ

_{2}. This indicates that, from a topological point of view, the optimal number of clusters to split the network is two. These results match with those naturally expected by the Example Network construction and also by its visualization. It also important to highlight that the value of the eigengap decreases as the number of links between the two A) regions increases. This suggests that the eigengap criterion works better when the clusters in the network can be well defined (not overlapping).

#### 2.3. Network Eigenvectors

- Principal eigenvector: it corresponds to the largest A-eigenvalue, v
_{1}, of a connected graph. It gives the possibility to rank graph vertices by its coordinates with respect to the number of paths passing through them to connect two nodes in the network [44]. The number of paths can be seen as the “importance” (also called the centrality) of node i. In this regard, the eigenvector centrality attributes a score to each node equals to the corresponding coordinate of the principal eigenvector. Groups of highly interconnected nodes are more “important” for the communication in comparison to equally high connected nodes do not form groups, that is, whose neighbours are less connected than them (according to the social principle that “I am influential if I have influential friends”). An important Principal eigenvector application is on Web search engines as Google’s PageRank algorithm [45]; - The Fiedler eigenvector: it corresponds to the second smallest Laplacian (or normalized Laplacian) eigenvalue of a connected graph. Fiedler [39] first demonstrated that the eigenvector v
_{2}associated to the second smallest eigenvalue λ_{2}provides an approximate solution to the graph bi-partitioning problem. This is approached according to the signs of the components of v_{2}. A subgraph is encompassed by nodes with positive components in the Fiedler eigenvector. The other subgraph contains nodes that are related to negative Fiedler eigenvector components. The v_{2}values closer to 0 correspond to “better” splits. In this regard, if a number of clusters k ≥ 2 is needed, then it is useful to resort to the Recursive spectral bisection [46,47]. According to this, the Fiedler eigenvector is used to bi-divide the vertices of the graph by the sign of its coordinates and the process is iterated then for each defined sub-part until reach the targeted number k of clusters. - Other Eigenvector: an alternative to obtain a good graph partitioning for k ≥ 2 clusters is related to the first k smallest eigenvector of the Laplacian matrix (or normalized Laplacian). The approach is based on solving the relaxed versions of the RCut problem (NCut problem) to define the so-called spectral clustering (normalized spectral clustering). It has been demonstrated in literature [33] that the normalized spectral clustering, based on the Random Walk Normalized Laplacian Matrix
**L**_{rw}, shows a superior performance to other spectra alternatives to find a clustering configuration. The solution is simultaneously characterized by both a minimum number of cuts and a well-balanced clusters size. According to [33], the minimization of the NCut problem is equal to the minimization of the Rayleigh quotient.$$\mathrm{min}\left(NCut\left(x\right)\right)=\mathrm{min}\frac{{y}^{T}\left({\mathit{D}}_{k}-\mathit{A}\right)y}{{y}^{T}\mathit{D}y}$$

**D**−

**A**) matrix that is in correspondence to its smallest eigenvector. In this regard, the minimization of the NCut problem is related to the solution of the generalized eigenvalues system.

**L**= ${\mathit{D}}_{k}$ −

**A**, and pre-multiplying by ${\mathit{D}}_{k}^{-1}$, the problem is reduced to the classical eigenvalues system.

- definition of Adjacency matrix
**A**(or weighted Adjacency matrix**W**); - computation of the Laplacian
**L**; - computation of the first k eigenvectors of normalized Laplacian
**L**_{rw}matrix - definition of the matrix
**U**_{nxk}containing the first k eigenvectors as columns; and, - clustering the nodes of the network into clusters C
_{1}, …, C_{k}using the k-means algorithm applied to the rows of the**U**_{nxk}matrix.

_{k}. An important aspect according of the spectral algorithm is to change the representation of the nodes from Euclidean space to points in the

**U**

_{nxk}matrix. This new data space enhances important cluster-properties and the final configuration has an easier detection [37]. Successful applications for the water distribution networks can be found in [11,14].

_{1,i}is evaluated for layout D). Table 2 shows that the two most important nodes are the node 6 and the node 13 (marked in Table 2), as those nodes correspond the maximum value of the eigenvector. The connectivity degree for these nodes is k

_{i}= 4, and they are connected to other nodes with a connectivity degree k

_{i}= 4 (that is node 5 and node 13 are connected to node 6; node 14 and node 6 are connected to node 13). So, the two most important nodes, identified with the eigenvector centrality, are those nodes that have highly connected adjacent neighbour. These nodes 6 and 13 can consequently be considered “central” nodes for the communication of the network (from a topological point of view). Similar results are obtained also for the other Example Network layouts.

_{2}for the four layouts of Example Network are shown in Figure 5. The Fiedler eigenvector has a number of components (coordinates) equal to the number of nodes. It is clear that the coordinates have positive and negative values for the four layouts. In particular, it is possible to define two well separated groups. The first ranges from node 1 to node 9 (negative values), while the second is made by node 10 up to node 18 (positive values). By splitting the nodes of the network according to their coordinates for v

_{2}, it is possible to define a bisection of them.

**U**

_{nxk}for k = 2.

## 3. Case Study

_{0}= 1. This means that in both WDNs, there is only one connected component. It is interesting to note that also for complex network models (made by thousands of components) it is still easy to check if any anomaly observed in the water supply is caused by the decomposition of the original network in several subregions (as it is the case of unexpected pipe disruptions or valve malfunctions).

_{2}calculated on the Laplacian matrix. The values of the Spectral gap and the Algebraic connectivity aid and simplify the assessment of robustness of a WDN, as it was preliminary proposed by [12,13,14]. In the current case studies, it is clear that the corresponding values of the two spectral measures are small and near to zero, Δλ = 0.0685 and λ

_{2}= 0.0212 for Parete, while Δλ = 0.0303 and λ

_{2}= 0.0006 for C-Town. These small values are justified by the fact that WDNs are sparser than other networks as Internet or social networks. This is due to both geographical embedding and economic constraints [7,11].

_{k+}

_{1}− λ

_{k}could be interpreted as a measure of the surplus of the strength needed to split the network from k + 1 to k clusters. Once defining the maximum eigengap λ

_{k+}

_{1}− λ

_{k}, it is clear that, from a topological point of view, it is better to split the network at most up to k clusters, since a greater surplus of strength is needed to split the network in k + 1 and more clusters. For this reason, the maximum eigengap can be used to define the optimal number of clusters from a topological and connectivity point of view. Figure 6 shows the first ten eigenvalues of the Laplacian matrix for the graph of C-Town and Parete. It is clear that the first largest eigengap for C-Town, occurs between the sixth and the fifth eigenvalue (λ

_{6}− λ

_{5}= 0.002), while for Parete occurs between the fifth and the fourth eigenvalue (λ

_{5}− λ

_{4}= 0.042). This metric suggests that, an optimal number of clusters on which subdivided the water distribution networks of C-Town and Parete is, respectively, k = 5 and k = 4.

_{2}, for C-Town and Parete WDNs. It is clear, as it was shown on Example Network, that the coordinates of the second eigenvector, v

_{2}, easily define an optimal bipartition layout for the network. These divide the network nodes according to the signs (positive or negative) for the corresponding value of the Fiedler eigenvector. It is worth highlighting that this procedure ensures the continuity of each defined cluster, as each node of a cluster is linked at least to another node of the same cluster.

_{2}, can be used as input for a recursive bisection process. That is, for each cluster, the Fiedler eigenvector v

_{2}can be computed for the next clustering up to reach the targeted number of clusters. This network bisection can also represent a starting layout for other recursive algorithms that require an initial random choice of the clustering layout. Another GST based powerful tool for the optimal clustering layout of a water distribution network, is the Ncut spectral clustering [33], already explained in the Eigenvector techniques section, based on the use of other eigenvectors further than v

_{2}.

_{1}, of the Adjacency matrix. Ranking WDN nodes is useful for locating optimal nodes in which locate devices (i.e., chlorine stations, pressure regulation valves, quality sensors, flow meters, etc.). The identification of the most important nodes can also contribute as initial guess for further development of specific device location algorithms. The applications range, for instance, from detecting accidental or intentional contamination to control pipe flows and node pressures. These challenging tasks can be approached through GSTs, even when no other information is available rather than the network topology. As it is explained in the previous section, the eigenvector centrality can spot the most “influential” nodes, according to the number of neighbours of the adjacent nodes. The idea behind the network centrality concept is to identify which points are traversed by the greatest number of connections. Central nodes are thus considered as essential nodes for network connectivity and have influence over large network areas. Figure 8 points out also the most important nodes based on the eigenvector centrality criterion. The results show the highest centrality node per each DMA of the C-Town and Parete partitioned WDNs. After WDNs clustering, the process is focus on every single Adjacency matrix related to water distribution sub-networks. The eigenvector centrality provides most the important nodes per cluster or DMA, from a topological and connectivity point of view.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Four layouts of the Example Network with the same number of nodes and a different number of links.

**A**) two separated subregions;

**B**) a single edge links the two subregions;

**C**) two edges link the two subregions;

**D**) three edges link the two subregions.

**Figure 2.**Algebraic connectivity, Inverse Spectral radius and Spectral radius for the layout A, B, C, and D of Example Network.

**Figure 4.**Two most important nodes, computed by the eigenvector centrality, for the layout D of Example Network.

**Figure 6.**First 10 eigenvalues for the two case studies: (

**a**) C-Town network; and, (

**b**) Parete network.

**Figure 7.**Fiedler eigenvector v

_{2}coordinates for the two case studies: (

**a**) C-Town network; and, (

**b**) Parete network.

**Figure 8.**Optimal clustering layout for the two case studies with different colors for each clusters and highlighting the most important nodes of each cluster according to the eigenvector centrality of the partitioned networks: (

**a**) C-Town network (k = 5); and (

**b**) Parete network (k = 4).

Metric | Layout A | Layout B | Layout C | Layout D |
---|---|---|---|---|

Inverse of Spectral radius 1/λ_{1} | 0.354 | 0.332 | 0.320 | 0.311 |

Spectral gap Δλ | 0.000 | 0.275 | 0.422 | 0.555 |

Eigengap λ_{k}_{+1} − λ_{k} | 1.000 | 0.875 | 0.806 | 0.732 |

Multiplicity of zero m_{0} | 2 | 1 | 1 | 1 |

Algebraic connectivity λ_{2} | 0.000 | 0.125 | 0.194 | 0.268 |

n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

v_{1,i} | 0.12 | 0.21 | 0.26 | 0.16 | 0.30 | 0.37 | 0.12 | 0.21 | 0.26 | 0.26 | 0.21 | 0.12 | 0.37 | 0.30 | 0.16 | 0.26 | 0.21 | 0.12 |

**Table 3.**Main characteristics of water distribution network of C-Town and Parete. The symbol in brackets “-” indicates that the parameter is dimensionless.

Network | n (-) | m (-) | nr (-) | L_{TOT} (km) |
---|---|---|---|---|

C-Town | 396 | 444 | 1 | 56.7 |

Parete | 184 | 282 | 2 | 34.7 |

**Table 4.**Principal Eigenvalues of the Adjacency and Laplacian matrices of water distribution network of C-Town and Parete.

Network | m_{0} | Δλ | λ_{2} | 1/λ_{1} | λ_{k+}_{1} − λ_{k} |
---|---|---|---|---|---|

C-Town | 1 | 0.0303 | 0.0006 | 0.358 | 5 |

Parete | 1 | 0.0685 | 0.0212 | 0.303 | 4 |

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**MDPI and ACS Style**

Di Nardo, A.; Giudicianni, C.; Greco, R.; Herrera, M.; Santonastaso, G.F.
Applications of Graph Spectral Techniques to Water Distribution Network Management. *Water* **2018**, *10*, 45.
https://doi.org/10.3390/w10010045

**AMA Style**

Di Nardo A, Giudicianni C, Greco R, Herrera M, Santonastaso GF.
Applications of Graph Spectral Techniques to Water Distribution Network Management. *Water*. 2018; 10(1):45.
https://doi.org/10.3390/w10010045

**Chicago/Turabian Style**

Di Nardo, Armando, Carlo Giudicianni, Roberto Greco, Manuel Herrera, and Giovanni F. Santonastaso.
2018. "Applications of Graph Spectral Techniques to Water Distribution Network Management" *Water* 10, no. 1: 45.
https://doi.org/10.3390/w10010045