Application of graph-spectral methods in the vulnerability assessment of water supply networks∗ J.A. Guti´errez–P´erez ‡†, M. Herrera‡ , R. P´erez–Garc´ıa‡ and E. Ramos–Mart´ınez‡ (‡ ) Fluing–IMM, Universitat Polit`ecnica de Val`encia, C. de Vera s/n, 46022 Valencia, Spain.
November 8, 2011
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Introduction
There are various methods to approach a solution to graph clustering, among the most important are the spectral methods. The spectral methods are based on eigenvalues and eigenvectors of a block-diagonal matrix conveniently associated with the graph. Among spectral methods, graph-based ranking algorithms are essentially a way of deciding the importance of a vertex in a graph, by taking into account global information drawn from the graph structure. Addressing this issue, [5] and [7] investigated applications of graph theory and complex network principles in the analysis of vulnerability and ranking elements of WSNs. The aim of this paper is to define importance areas by a method of semisupervised clustering into WSNs, through ranking nodes. Our proposal uses ∗ This work has been supported by project IDAWAS, DPI2009-11591, of the Direcci´on General de Investigaci´ on of the Ministerio de Ciencia e Innovaci´on of Spain and the complementary support ACOMP/2011/188, of the Conseller´ıa de Educaci´o of the Generalitat Valenciana. † contact e-mail:
[email protected]
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spectral-based methods by the algebraical connectivity of the graph Laplacian matrix. In order to develop efficient models from these spectral methods, it will also be useful to adapt the PageRank [1] and HITS [6] algorithms. Both ranking algorithms provide relative importance measures of webpages.
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Adapting the algorithms to WSN
To adapt the algorithms to WSNs, we considered the physical structure of a WSN as a mathematical graph G = (V, E) in which the set of all graph vertices, V , respresents tanks, water sources and nodes or junctions (wich are the connection between pipes and points of water withdrawal). The set of the graph edges, E, represents pipes, valves and pumps. By understanding a WSN as a graph of special characteristics, it is possible to abstract the concept of web page looking at it a consumption node in a WSN. Links between pages are now understood as pipes connecting different nodes. Thus, both algorithms can be adapted. About it [3] demonstrated the possibilities of ranking algorithms through the adaptation of the PageRank algorithm to measure the importance of the nodes of a WSN.
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Experimental study
In order to show the methodology exposed we considered a real case, the WSN of the central area of Celaya, Mexico. This network is made out of 479 pipes and 339 nodes; 5 sources and 1 tank. Its total pipe length is 42.5 km, the node elevation average is 156 metres; and the total consumed flow rate amounts to 91 l/s. As stayed, our aim is to divide the WSN of our case study on importance clusters. To this purpose, we apply the procedure of semi-supervised clustering algorithm explained in [2]. After applying this process to the WSN data, the following results were obtained. In Figure 1 we can observe the division of the WSN into three importance clusters introducing the PageRank and HITS measures. In addition, this figure shows the distribution of the nodes with high values in each case. It is important to note that cluster 2 (Figure 1a) is the zone with high PageRank average. Therefore, we can say that this area is more critical than the others. It is because there are nodes with sensible connectivity and remove them could affect the performance of the
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WSN. In the case of HITS algorithm (Figure 1b), the results obtained show that the division of the clusters was not homogeneus, because one of the clusters has more of the 50 percent of the nodes. In Table 1 we can observe a summary of each cluster regarding PageRank and HITS.
Figure 1: Scheme of the division clusters by PageRank and HITS algorithms.
Table 1: Description of PageRank and HITS clusters. Cluster n Nodes PR avg. Cluster n Nodes HITS avg. C1 74 0.00298 C1 38 208.1E-7 C2 91 0.00304 C2 41 144.4E-6 C3 174 0.00289 C3 260 0.03243
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Conclusions
In the present work, we propose to introduce PageRank and HITS algorithms as importance measures to form clusters into a WSN by a semi-supervised clustering methodology. This let to know critical nodes into the WSN, and how would they affect the nodes are connected with them. It has demonstrated that PageRank and HITS algorithms can be an acceptable relative importance measures. These measures help us to understand the structure and performance of the network, identifying vulnerability points.
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References [1] Brin S. and Page L., 1998. The Anatomy of a Large-Scale Hypertextual Web Search Engine. http://infolab.stanford.edu/backrub/google.html. [2] Herrera M. Improving water network management by efficient division into supply clusters. PhD. thesis, Universitat Polit`ecnica de Val`encia, Spain, 2011. [3] Herrera M., Guti´errez-P´erez J. A., Izquierdo J., P´erez-Garc´ıa R., 2011. Ajustes en el modelo Page-Rank de Google para el estudio de la importancia relativa de los nodos de la red de abastecimiento. Proceedings, X Seminario Iberoamericano de Planificaci´on, Proyecto y Operaci´on de Sistemas de Abastecimiento de Agua (SEREA). Morelia, M´exico. [4] Herrera M., Canu S., Karatzoglou A., P´erez-Garc´ıa R., Izquierdo J., 2010. An approach to water supply clusters by semi-supervised learning. International Environmental Modelling and Software Society (iEMSs) 2010 International Congress on Environmental Modelling and Software. [5] Izquierdo J., Montalvo I., P´erez-Garc´ıa R., 2008. Sensitivity analysis to asses the relative importance of pipes in water distribution networks. Mathematical and Computing Modeling. 48:268-278. [6] Kleinberg S., 1999. Authoritative sources in a hyperlinked environment. Journal of the ACM, 48:604-632. [7] Michaud D. and Apostolakis G. E., 2006. Methodology for ranking the elements of water supply networks. Journal of Infrastructure Systems, 12(4):230-242.
