Localized eigenvectors of the non-backtracking matrix

In the case of graph partitioning, the emergence of localized eigenvectors can cause the standard spectral method to fail. To overcome this problem, the spectral method using a non-backtracking matrix was proposed. Based on numerical experiments on several examples of real networks, it is clear that the non-backtracking matrix does not exhibit localization of eigenvectors. However, we show that localized eigenvectors of the non-backtracking matrix can exist outside the spectral band, which may lead to deterioration in the performance of graph partitioning.