A semicircle law for the normalized Laplacian of sparse random graphs (2512.20146v1)

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.

• The paper proves that the normalized Laplacian’s empirical spectral distribution converges to the semicircular law in sparse random graphs when np tends to infinity.
• It introduces the use of the Moore-Penrose pseudoinverse to handle isolated vertices, thereby relaxing technical conditions in both homogeneous and inhomogeneous models.
• Almost sure convergence is achieved under minimal concentration inequalities, reinforcing robust spectral algorithms and effective graph statistical inference.

Semicircle Law for the Normalized Laplacian of Sparse Random Graphs

Overview and Context

The paper "A semicircle law for the normalized Laplacian of sparse random graphs" (2512.20146) rigorously determines the limiting spectral distribution (LSD) for the normalized Laplacian $\mathcal L$ of random graphs—especially in the sparse regime. It addresses the Erdős–Rényi $G(n,p)$ model with $np\to\infty$ and the Chung-Lu model, establishing that under minimal conditions, the empirical spectral distribution (ESD) of a normalized version of $\mathcal L$ converges to the Wigner semicircle law. The analysis resolves previous conjectures, relaxes prior technical conditions, and advances the generality for both homogeneous and inhomogeneous random graphs.

Main Results

The central achievement is to prove that the semicircle law holds for the normalized Laplacian of $G(n,p)$ as soon as $np\to\infty$, sharply identifying the threshold as optimal. The authors resolve a conjecture (Akara-pipattana and Evnin, 2023) regarding the emergence of the semicircle law for the rescaled spectrum of the normalized adjacency operator, including graphs with isolated vertices by leveraging the Moore-Penrose pseudoinverse for $\mathbf{D}^{-1}$.

Semicircle Law for $G(n,p)$

• If the degree matrix $\mathbf{D}_n$ is defined via the Moore-Penrose pseudoinverse, and $np_n\to\infty$ with $p_n$ bounded away from 1, the ESD of

$$\sqrt{\frac{np_n}{1-p_n}}\left(\mathbf{I}_n - \mathcal L_n\right)$$

converges weakly (in probability) to the semicircle law $\mu_{sc}$.

• If $np_n > \log n + \omega(1)$, so that isolated vertices disappear w.h.p. and $\mathbf{D}_n$ is invertible, the same result holds for the standard definition of $\mathcal L$ (without pseudoinverse extension).
• When $np_n \gtrsim \log n$, almost sure convergence of the ESD is obtained, further strengthening standard results that typically give only convergence in probability.
• The optimality is established via comparison with previous works (e.g., Chi, 2016), which show that for $np_n=O(1)$, the LSD is no longer semicircular.

Chung-Lu Model

• Under the Chung-Lu (rank-one inhomogeneous) random graph model, and assuming the minimal expected degree $w_{n,\min}\gg\sqrt{\bar w_n}$, the ESD of

$$\sqrt{\bar w_n}\left(\mathbf{I}_n - \mathcal L_n\right)$$

again converges to $\mu_{sc}$ in probability.

• The result sharpens prior findings (Chung, Lu, Vu, 2003) where only proxy or averaged matrices were shown to be semicircular, now extending this to the normalized Laplacian operator itself.

Technical Contributions

The analysis leverages several innovative and robust probabilistic and spectral techniques:

• Pseudoinverse Extension: To handle isolated vertices in $G(n,p)$ for small $p$, the normalized Laplacian is defined using the pseudoinverse of the degree matrix, ensuring the operator is always well-defined.
• Approximation Arguments: Adapting the comparison method for ESDs, the authors show that the spectra of the normalized Laplacian and a simpler, deterministically-normalized matrix become indistinguishable as $n$ grows, under appropriate rescaling.
• Concentration Inequalities: A refined bounded differences approach (Warnke, 2016) achieves almost sure convergence of the ESDs under only $np\gtrsim \log n$—a significant improvement over earlier, more restrictive bounds.
• Perturbation Analysis: For the Chung-Lu model, the difference between degree-rescaled adjacency and expected degree-rescaled is shown to have vanishing spectral impact in the ESD sense, extending the semicircle law to the true normalized Laplacian.

Numerical and Optimality Statements

• The threshold $np\to\infty$ is shown to be both necessary and sufficient for the semicircle law to appear, which constitutes a sharp and optimal identification of the universality boundary for this result.
• In the Chung-Lu setting, the expansion $w_{n,\min}\gg\sqrt{\bar w_n}$, accompanied by careful tail bounds, demonstrates that the spectral bulk remains semicircular even as the graph’s degree sequence becomes highly heterogeneous, provided minimum degree diverges fast enough.
• The convergence of the ESD is shown to occur almost surely for $np\gtrsim\log n$, indicating strong self-averaging properties of the global spectrum even in very sparse graphs.

Implications and Future Directions

Theoretical Implications

• Universality and Spectral Bulk: These results establish that the bulk spectrum of normalized Laplacians for a wide family of sparse random graphs behaves as predicted by classical random matrix theory, up to the sharp sparsity threshold.
• Robustness to Degree Fluctuations: The use of the Moore-Penrose pseudoinverse and the demonstration of spectral convergence in the presence of extreme degree fluctuations and isolated vertices highlight the robustness of semicircular behavior under significant randomness in graph topology.
• Extension to Inhomogeneous Models: By handling the Chung-Lu setting, the approach signals that semicircular universality covers not only homogeneous but also a large region of inhomogeneous random graph ensembles.

Practical Implications

• Spectral Algorithms: The practical effectiveness and accuracy of spectral algorithms (such as spectral clustering and partitioning) rest, in part, on the regularity and predictability of the Laplacian spectrum. The sharp understanding of the spectral limit for the normalized Laplacian in sparse graphs provides theoretical guarantees for such methods, even near the connectivity threshold.
• Graph Statistical Inference: For model testing, anomaly detection, and inferential algorithms relying on spectral density estimations, the results guarantee that deviations from semicircularity reflect underlying structural deviations, not just sparse randomness.

Future Research

• Local Spectral Statistics: While this work resolves the global bulk behavior, local spectral statistics (e.g., edge universality or eigenvector delocalization) in the same sparse regime remain less understood.
• Beyond Rank-One Edge Probability: Extensions to more complex inhomogeneous variances (e.g., general $p_{ij}$ matrices) and models with heavy-tailed degree distributions or divergent second moments present directions where the same methodological framework could be further developed.
• Random Graphs with Dependency or Geometry: Analyzing spectra of random geometric graphs, dependent-edge models, or graphs with planted structures under normalized Laplacian transformations would further broaden the understanding of universality in the spectral domain.
• Non-Hermitian Generalizations: For directed graphs or those with signed/weighted normalizations, analogs of the semicircle law for (non-symmetric) normalized operators deserve further investigation.

Conclusion

This paper (2512.20146) establishes a precise characterization of the spectral bulk for normalized Laplacian matrices in sparse random graphs through rigorous weak and almost sure semicircle laws. The results significantly generalize and refine the spectral theory for random graphs, settling previously open conjectures, optimizing thresholds for convergence, and extending universality to both homogeneous and inhomogeneous sparse networks. The methods and conclusions will serve as a foundation for future research on the fine spectral properties of random graphs and for principled theoretical underpinnings of spectral graph algorithms in high-dimensional statistics and network science.