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A semicircle law for the normalized Laplacian of sparse random graphs

Yiming Chen, Zijun Chen, Yizhe Zhu

TL;DR

This work determines the limiting spectrum of the normalized Laplacian for sparse random graphs. By adopting a Moore-Penrose pseudoinverse convention for the degree matrix to handle isolated vertices, the authors prove that the empirical spectral distribution of a suitably scaled $\mathcal{L}_n$ converges to the semicircle law $\mu_{sc}$ as $np_n\to\infty$, with almost sure convergence under stronger growth. The key technique is a two-step comparison: replace $\mathcal{L}_n$ by a simpler matrix $\widetilde{\mathcal{L}}_n$ with a known semicircular limit, and show the two matrices have asymptotically identical spectral distributions using a combination of expectation control and concentration via Warnke's typical bounded differences. The results are extended to the Chung-Lu model, where the semicircle law is shown to hold for the normalized Laplacian itself under suitable minimum weight growth, not just for a proxy, thereby unifying the spectral behavior across homogeneous and inhomogeneous sparse graphs. These findings have implications for understanding random-walk dynamics and spectral clustering in sparse networks.

Abstract

We study the limiting spectral distribution of the normalized Laplacian $\mathcal L$ of an Erdős-Rényi graph $G(n,p)$. To account for the presence of isolated vertices in the sparse regime, we define $\mathcal L$ using the Moore-Penrose pseudoinverse of the degree matrix. Under this convention, we show that the empirical spectral distribution of a suitably normalized $\mathcal L$ converges weakly in probability to the semicircle law whenever $np\to\infty$, thereby providing a rigorous justification of a prediction made in (Akara-pipattana and Evnin, 2023). Moreover, if $np>\log n+ω(1)$, so that $G(n,p)$ has no isolated vertices with high probability, the same conclusion holds for the standard definition of $\mathcal L$. We further strengthen this result to almost sure convergence when $np=Ω(\log n)$. Finally, we extend our approach to the Chung-Lu random graph model, where we establish a semicircle law for $\mathcal L$ itself, improving upon (Chung, Lu, and Vu 2003), which obtained the semicircle law only for a proxy matrix.

A semicircle law for the normalized Laplacian of sparse random graphs

TL;DR

This work determines the limiting spectrum of the normalized Laplacian for sparse random graphs. By adopting a Moore-Penrose pseudoinverse convention for the degree matrix to handle isolated vertices, the authors prove that the empirical spectral distribution of a suitably scaled converges to the semicircle law as , with almost sure convergence under stronger growth. The key technique is a two-step comparison: replace by a simpler matrix with a known semicircular limit, and show the two matrices have asymptotically identical spectral distributions using a combination of expectation control and concentration via Warnke's typical bounded differences. The results are extended to the Chung-Lu model, where the semicircle law is shown to hold for the normalized Laplacian itself under suitable minimum weight growth, not just for a proxy, thereby unifying the spectral behavior across homogeneous and inhomogeneous sparse graphs. These findings have implications for understanding random-walk dynamics and spectral clustering in sparse networks.

Abstract

We study the limiting spectral distribution of the normalized Laplacian of an Erdős-Rényi graph . To account for the presence of isolated vertices in the sparse regime, we define using the Moore-Penrose pseudoinverse of the degree matrix. Under this convention, we show that the empirical spectral distribution of a suitably normalized converges weakly in probability to the semicircle law whenever , thereby providing a rigorous justification of a prediction made in (Akara-pipattana and Evnin, 2023). Moreover, if , so that has no isolated vertices with high probability, the same conclusion holds for the standard definition of . We further strengthen this result to almost sure convergence when . Finally, we extend our approach to the Chung-Lu random graph model, where we establish a semicircle law for itself, improving upon (Chung, Lu, and Vu 2003), which obtained the semicircle law only for a proxy matrix.
Paper Structure (11 sections, 10 theorems, 105 equations)

This paper contains 11 sections, 10 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. The Erdős-Rényi graph $G(n,p_n)$
  4. The Chung--Lu Model
  5. Preliminaries
  6. Proof of Theorems \ref{['thm2']} and \ref{['thm1']}
  7. Proof of Lemma \ref{['lem1']}
  8. Proof of Theorem \ref{['thm2']}
  9. Proof of Lemma \ref{['lem2']}
  10. Proof of Theorem \ref{['thm1']}
  11. Proof of Theorem \ref{['thm:CLV']}

Key Result

Theorem 1

Assume $\mathbf{D}_n^{-1}$ is defined using the pseudo-inverse convention. Suppose that there exists a fixed constant $\varepsilon_0 > 0$ such that $\sup_n p_n \le 1 - \varepsilon_0$ and Then the empirical spectral distribution of $\sqrt{\frac{np_n}{1-p_n}} (\mathbf{I}_n-\mathcal{L}_n)$ converges weakly to the semicircle law $\mu_{\mathrm{sc}}$ defined in sem, in probability as $n \to \infty$.

Theorems & Definitions (17)

  • Theorem 1: Semicircle law for $\mathcal{L}_n$
  • Remark 1: Optimality
  • Corollary 1
  • Theorem 2
  • Theorem 3: Semicircle law for the Chung-Lu model
  • Remark 2: Comparison with chung2003spectra
  • Definition 1: bounded Lipschitz metric
  • Lemma 1
  • proof
  • Lemma 2: Approximation of $\frac{1}{n}\mathbb{E}(\operatorname{tr}(\widetilde{\mathbf{\mathcal{L}}}_n - \mathbf{\mathcal{L}}_n)^2)$
  • ...and 7 more