Title: Localization and centrality in networks

Abstract: Eigenvector centrality is a widely used 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 and renders the measure useless for most practical purposes. As a remedy, we propose an alternative centrality measure based on the non-backtracking 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.