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Complex network theory (CNT) is an emerging topic based on the paradigm that quite all the natural and man-made physical systems work as networks, namely, their features derive from the internal connectivity among vertices exchanging information through edges. Water distribution infrastructures are networked systems connecting the vertices named nodes, by edges named pipes, and transferring water to customers. Therefore, water distribution networks (WDNs) fall into CNT and belongs to the class of spatial networks due to their urban constraints. CNT proposed several centrality metrics for quantifying the importance of vertex and, sometime, edges. Those metrics can be potentially very useful for analyzing the key features of the physical domain (i.e., the network) where WDN hydraulics occurs, but they need to be tailored to consider that (i) pipes/edges are the relevant physical components of the WDNs, (ii) some nodes/vertices (reservoirs and tanks) play a completely different hydraulic role from the majority of nodes (demand and connection nodes), and (iii) pipes/edges have different characteristics (length, diameter, hydraulic resistance, etc.). Accordingly, this work presents and discusses the need of tailoring the most suitable centrality metrics for spatial networks: betweenness, closeness, and degree. Then the capacity of the WDN-tailored edge betweenness is demonstrated and discussed using two real WDNs, showing that it can extract useful information from the domain, that is, the emerging hydraulic behavior due to the network connectivity structure. Therefore, the WDN-tailored edge betweenness can assist analysis, planning, and management actions before and after the hydraulic analysis.

Plain Language Summary The aim of the work is to develop a water distribution network (WDN)-tailored centrality metric from complex network theory for the analysis of the domain of the WDN hydraulics. The WDN-tailored edge betweenness is, then, developed. Tailoring is performed considering three specific WDN features: (1) The pipes are the relevant components of such infrastructure systems. This is in contrast with standard complex network theory centrality metrics that generally focus on nodes because they represent the objects exchanging information. (2) Pipes are material components characterized by asset features such as length, diameter, and hydraulic resistance. (3) Each node can represent a different system component. It can be a source of water, a connection among pipes, or a water outflow. Accordingly, the work defines source of water, connection, and demand nodes, respectively.