Eigenvalues of the non-backtracking operator detached from the bulk

07/12/2019

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by   Simon Coste, et al.

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We describe the non-backtracking spectrum of a stochastic block model with connection probabilities p_in, p_out = ω( 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 p_in+ p_out/p_in- p_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.