The reliability and robustness against failures of networked water distribution systems are central tenets of water supply system design and operation. The ability of such networks to continue to supply water when components are damaged or fail is dependent on the connectivity of the network and the role and location of the individual components. This paper employs a set of advanced network analysis techniques to study the connectivity of water distribution systems, its relationship with system robustness, and susceptibility to damage. Water distribution systems are modeled as weighted and directed networks by using the physical and hydraulic attributes of system components. A selection of descriptive measurements is utilized to quantify the structural properties of benchmark systems at both local (component) and global (network) scales. Moreover, a novel measure of component criticality, the demand-adjusted entropic degree, is proposed to support identification of critical nodes and their ranking according to failure impacts. The application and value of this metric is demonstrated through two case study networks in the USA and UK. Discussion focuses on the potential for gradual evolution of abstract graph-based tools and techniques to more practical network analysis methods, where a theoretical framework for the analysis of robustness and vulnerability of water distribution networks to better support planning and management decisions is presented.