Despite the huge importance that centrality metrics have in understanding the topology of a network, little is known about how small alterations in the network’s topology affect the norm of the centrality vector, which stores node centralities. This paper investigates that gap by formalizing centrality definitions and empirically examining three fundamental metrics (Degree, Eigenvector, and Katz centrality) under two probabilistic node-failure models: Uniform (each node removed independently with fixed probability) and Best Connected (removal probability proportional to node degree). The findings show that Degree centrality remains relatively stable under minor perturbations, while Eigenvector and Katz centralities can be extremely sensitive — even small changes may cause large distortions under specific conditions.