Localized eigenvector of the non-backtracking matrix

Emergence of localized eigenvectors can cause a failure of standard spectral method for graph partitioning. The spectral method using the non-backtracking matrix was proposed as a way to overcome this problem. The numerical experiments on several examples of real networks show that, indeed, the non-backtracking matrix does not exhibit localization of eigenvectors. We show that, however, localized eigenvectors of the non-backtracking matrix out of the spectral band can exist, deteriorating the performance of graph partitioning.