Eigenvalues of the non-backtracking operator detached from the bulk

Simon Coste; Yizhe Zhu

doi:10.1142/s2010326321500283

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arXiv · 1907.05603

Eigenvalues of the non-backtracking operator detached from the bulk

Simon Coste·Yizhe Zhu

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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.

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Simon Coste, Yizhe Zhu. 2019-07-12. Eigenvalues of the non-backtracking operator detached from the bulk. https://doi.org/10.1142/s2010326321500283

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