  We show that eigenvector centrality exhibits localization phenomena on
networks that can be easily partitioned by the removal of a vertex cut set, the
most extreme example being networks with a cut vertex. Three distinct types of
localization are identified in these structures. One is related to the
well-established hub node localization phenomenon and the other two are
introduced and characterized here. We gain insights into these problems by
deriving the relationship between eigenvector centrality and Katz centrality.
This leads to an interpretation of the principal eigenvector as an
approximation to more robust centrality measures which exist in the full span
of an eigenbasis of the adjacency matrix.
