Networks created from real-world data contain some inaccuracies or noise, manifested as small changes in the network structure. An important question is whether these small changes can significantly affect the analysis results. In this paper, we study the effect of noise in changing ranks of the high centrality vertices. We compare, using the Jaccard Index (JI), how many of the top-k high centrality nodes from the original network are also part of the top-k ranked nodes from the noisy network. We deem a network as stable if the JI value is high. We observe two features that affect the stability. First, the stability is dependent on the number of top-ranked vertices considered. When the vertices are ordered according to their centrality values, they group into clusters. Perturbations to the network can change the relative ranking within the cluster, but vertices rarely move from one cluster to another. Second, the stability is dependent on the local connections of the high ranking vertices. The network is highly stable if the high ranking vertices are connected to each other. Our findings show that the stability of a network is affected by the local properties of high centrality vertices, rather than the global properties of the entire network. Based on these local properties we can identify the stability of a network, without explicitly applying a noise model.