This paper introduces a formal definition of continuity and generalizes an existing notion of stability for node centrality measures in weighted graphs. It is shown that the frequently used measures of degree, closeness and eigenvector centrality are stable and continuous whereas betweenness centrality is neither. Numerical experiments in synthetic and real-world networks show that both stability and continuity are desirable in practice since they imply different levels of robustness in the presence of noisy data. In particular, a stable alternative of betweenness centrality is shown to exhibit resilience against noise while preserving its notion of centrality.
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