Eigenvalues of the non-backtracking operator detached from the bulk

We describe the non-backtracking spectrum of a stochastic block model with connection probabilities $p_{$\mathrm{i}\mathrm{n}$}, p_{$\mathrm{o}\mathrm{u}\mathrm{t}$} = \omega(\log n)/n$. In this regime we answer a question posed in Dall’Amico and al. (2019) regarding the existence of a real eigenvalue `inside’ the bulk, close to the location $\frac{p_{$\mathrm{i}\mathrm{n}$}+ p_{$\mathrm{o}\mathrm{u}\mathrm{t}$}}{p_{$\mathrm{i}\mathrm{n}$}- p_{$\mathrm{o}\mathrm{u}\mathrm{t}$}}$. We also introduce a variant of the Bauer-Fike theorem well suited for perturbations of quadratic eigenvalue problems, and which could be of independent interest.