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Limiting empirical spectral measure of the normalized Laplacian in preferential attachment graphs

Malika Kharouf

Abstract

We study the empirical spectral distribution of the normalized Laplacian of linear preferential attachment graphs in the Barab{á}si-Albert regime with fixed out-degree. For the resulting sequence of random multigraphs, we prove that the empirical spectral distribution converges weakly in probability to a deterministic probability measure supported on the interval [0, 2]. The limit is characterized via the local weak limit of preferential attachment graphs (the P{ó}lya-point graph): the limiting Stieltjes transform is given by the expected diagonal Green function at the root of the normalized Laplacian operator on this infinite random graph. The proof combines a resolvent approach with a uniform Neumann-series expansion for the normalized Laplacian, a random-walk representation in terms of return probabilities on decorated neighborhoods, a truncation and Doob martingale-Azuma-Hoeffding concentration argument along the PA filtration, and an analytic continuation argument based on normal families.

Limiting empirical spectral measure of the normalized Laplacian in preferential attachment graphs

Abstract

We study the empirical spectral distribution of the normalized Laplacian of linear preferential attachment graphs in the Barab{á}si-Albert regime with fixed out-degree. For the resulting sequence of random multigraphs, we prove that the empirical spectral distribution converges weakly in probability to a deterministic probability measure supported on the interval [0, 2]. The limit is characterized via the local weak limit of preferential attachment graphs (the P{ó}lya-point graph): the limiting Stieltjes transform is given by the expected diagonal Green function at the root of the normalized Laplacian operator on this infinite random graph. The proof combines a resolvent approach with a uniform Neumann-series expansion for the normalized Laplacian, a random-walk representation in terms of return probabilities on decorated neighborhoods, a truncation and Doob martingale-Azuma-Hoeffding concentration argument along the PA filtration, and an analytic continuation argument based on normal families.
Paper Structure (39 sections, 18 theorems, 78 equations)

This paper contains 39 sections, 18 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Motivation: spectra of the normalized Laplacian in sparse, heterogeneous networks
  3. Preferential attachment: scale-free growth and correlations
  4. Local weak limits and the Pólya--point graph
  5. Spectral limits for sparse graphs
  6. Our contribution
  7. Organization
  8. Model, operators and main result
  9. The Barabási--Albert preferential attachment multigraph
  10. Initial condition.
  11. Growth step.
  12. Degrees and total degree.
  13. Adjacency, random-walk kernel, and normalized Laplacian
  14. Simple random walk and its kernel.
  15. Normalized adjacency and normalized Laplacian.
  16. ...and 24 more sections

Key Result

theorem 1

As $n\to\infty$, the rooted graphs $(G_n,U_n)$ converge in distribution for the local topology to a random infinite rooted graph $(G_\infty,o)$. The law of $(G_\infty,o)$ is the Pólya--point graph constructed in BergerBorgsChayesSaberi2014; see also BenjaminiSchramm2001AldousLyons2007 for the genera

Theorems & Definitions (41)

  • remark 1
  • remark 2: Uniform resolvent bound ReedSimon1980Bhatia1997
  • theorem 1: Local weak limit BergerBorgsChayesSaberi2014
  • theorem 2: Limiting empirical spectral measure
  • lemma 1: Uniform resolvent bound
  • proof
  • lemma 2: Herglotz property
  • proof
  • lemma 3: Neumann series and tail bound
  • proof
  • ...and 31 more