Centrality measures such as Degree, Eigenvector, and Katz play a pivotal role in understanding the structure and functioning of complex networks. However, these metrics may be sensitive to graph perturbations—changes like random node failures or targeted attacks. In this chapter, the authors examine how small perturbations, modelled via Uniform and Best Connected probabilistic failure models, affect different centrality metrics. They find that Eigenvector centrality is particularly susceptible under uniform perturbations, while in targeted (Best Connected) scenarios the degree of perturbation scales with the proportion of attacked nodes. This study sheds light on the robustness of centrality measures and offers guidance on their reliability in perturbed network environments.