  The spectrum of the non-backtracking matrix plays a crucial role in
determining various structural and dynamical properties of networked systems,
ranging from the threshold in bond percolation and non-recurrent epidemic
processes, to community structure, to node importance. Here we calculate the
largest eigenvalue of the non-backtracking matrix and the associated
non-backtracking centrality for uncorrelated random networks, finding
expressions in excellent agreement with numerical results. We show however that
the same formulas do not work well for many real-world networks. We identify
the mechanism responsible for this violation in the localization of the
non-backtracking centrality on network subgraphs whose formation is highly
unlikely in uncorrelated networks, but rather common in real-world structures.
Exploiting this knowledge we present an heuristic generalized formula for the
largest eigenvalue, which is remarkably accurate for all networks of a large
empirical dataset. We show that this newly uncovered localization phenomenon
allows to understand the failure of the message-passing prediction for the
percolation threshold in many real-world structures.
