arXiv:2609.10800v2 Announce Type: replace Abstract: Water distribution networks (WDNs) are susceptible to various failures, including but not limited to human errors, cyber-attacks, and network modifications, necessitating the vulnerability analysis of WDNs. Graph-theoretic centrality measures---as a main class of centrality measures---aim to rank the network components solely based on their influence (i.e., criticality) on the WDN topology in the case of input changes, while overlooking the WDN dynamics. To overcome such a limitation, this paper uses a control-theoretic centrality measure to identify the network's most and least influential pipes in WDNs, by simultaneously incorporating the dynamics and topology of the WDN. First, given a WDN modeled by nonlinear differential-algebraic equations (NDAEs) consisting of transient flow dynamics as differential equation (DE) and conservation of water mass as algebraic equation (AE) and considering the pipe flow rates as states of the state-space (SS) representation, we extract a linearized system modeled by linear ordinary differential equations (LODEs) around the equilibrium (i.e., steady) flow rate vector. Second, treating pipe flow rates as SS nodes (states), we introduce a node centrality-based measure, namely \textit{vulnerability vector (VV)}, to rank the network pipes based on their influence on the dynamics and topology of the WDN in the case of input changes. In particular, the network's most and least influential pipes can be identified through such a centrality-based approach. This enables water engineers to understand the WDN's vulnerability better and effectively prioritize the maintenance and operational efforts on the most influential pipes within the WDN.