Identifying key actors or nodes in a network is a relevant task regarding many applications. In general, the real-valued function that evaluates the nodes is called node centrality measure. Performing a relevance-based ranking on the list of nodes is also of high practical importance, since the most central nodes by a measure usually provide the highest contribution in explaining the behavior of the whole network. Stability of centrality measures against graph perturbation is an important concept, especially in the analysis of real world—often noise contaminated—datasets from different domains. In this paper, with the utilization of the formal definition of stability introduced by Segarra and Ribeiro (IEEE Trans Signal Process 64(3):543–555, 2015), we discuss three main perturbation categories and experimentally analyze the stability of several node centrality measures.