{
  "id": "1907.05603",
  "version": "v1",
  "published": "2019-07-12T07:52:46.000Z",
  "updated": "2019-07-12T07:52:46.000Z",
  "title": "Eigenvalues of the non-backtracking operator detached from the bulk",
  "authors": [
    "Simon Coste",
    "Yizhe Zhu"
  ],
  "comment": "18 pages, 4 figures. Comments are welcome",
  "categories": [
    "math.PR",
    "cs.NA",
    "math.CO",
    "math.NA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-12T07:52:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-backtracking operator",
      "stochastic block model",
      "quadratic eigenvalue problems",
      "connection probabilities",
      "real eigenvalue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}