  Eigenvector centrality is a common measure of the importance of nodes in a
network. Here we show that under common conditions the eigenvector centrality
displays a localization transition that causes most of the weight of the
centrality to concentrate on a small number of nodes in the network. In this
regime the measure is no longer useful for distinguishing among the remaining
nodes and its efficacy as a network metric is impaired. As a remedy, we propose
an alternative centrality measure based on the nonbacktracking matrix, which
gives results closely similar to the standard eigenvector centrality in dense
networks where the latter is well behaved, but avoids localization and gives
useful results in regimes where the standard centrality fails.
