title: Localization and centrality in networks.
abstract: 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.
authors.affiliation: Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, Michigan 48109, USA.
authors.forename: Travis
authors.initials: T
authors.lastname: Martin
authors.affiliation: Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA.
authors.forename: Xiao
authors.initials: X
authors.lastname: Zhang
authors.affiliation: Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48109, USA.
authors.forename: M E J
authors.initials: ME
authors.lastname: Newman