# Eigenvalues of the non-backtracking operator detached from the bulk

## TL;DR

This paper analyzes the non-backtracking spectrum of stochastic block models in a dense regime, identifying a key eigenvalue inside the bulk and introducing a new perturbation theorem for quadratic eigenvalue problems.

## Contribution

It provides a detailed spectral analysis of the non-backtracking operator in dense stochastic block models and introduces a novel Bauer-Fike variant for quadratic eigenvalue problems.

## Key findings

- Existence of a real eigenvalue inside the bulk near a specific location.
- Characterization of the non-backtracking spectrum in dense regimes.
- Introduction of a new Bauer-Fike theorem variant for quadratic eigenvalue problems.

## Abstract

We describe the non-backtracking spectrum of a stochastic block model with connection probabilities $p_{\mathrm{in}}, p_{\mathrm{out}} = \omega(\log n)/n$. 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 $\frac{p_{\mathrm{in}}+ p_{\mathrm{out}}}{p_{\mathrm{in}}- p_{\mathrm{out}}}$. 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.