In applied network analysis, one of the key questions involves identifying the most important nodes, typically characterized by various centrality measures. Nevertheless, any inaccuracies inherent in the measurements used for network construction or in the construction of the network itself will inevitably affect the centrality measures, potentially obscuring the key nodes. In this work, we rigorously study the influence of network noise on the recovery of Top-K nodes, focusing on degree centrality. We derive the conditions for consistent recovery and evaluate the feasibility of these conditions under a number of canonical network models. Additionally, we present findings on the infeasibility of detecting vital nodes under certain conditions and demonstrate the implications for network applications. For scenarios that fall between consistency and infeasibility under noise, we propose a confidence set that includes the vital nodes with high probability.