Hammerstein equations for sparse random matrices

TL;DR

This work establishes a systematic, numerically robust framework to recover spectra of sparse random matrices by solving Hammerstein integral equations. By deriving explicit 1D Hammerstein formulations for adjacency, ordinary Laplacian, and normalized Laplacian spectra of Erdős-Rényi graphs, and implementing Kumar-Sloan collocation with Laguerre and half-range Hermite bases, the authors achieve fast, deterministic spectral reconstructions that agree with large-scale simulations for moderate to high average degree $c$ and remain informative down to $c\approx 8$ (and $4$ for Laplacians). The approach offers a complementary alternative to cavity and population-dynamics methods, with potential applicability to a broader class of sparse random matrices and related problems in disordered systems and network theory. While low-$c$ regimes present numerical challenges due to sharp spectral features, the demonstrated stability and efficiency motivate further methodological refinements and extensions to non-Hermitian or more complex sparse ensembles.

Abstract

Finding eigenvalue distributions for a number of sparse random matrix ensembles can be reduced to solving nonlinear integral equations of the Hammerstein type. While a systematic mathematical theory of such equations exists, it has not been previously applied to sparse matrix problems. We close this gap in the literature by showing how one can employ numerical solutions of Hammerstein equations to accurately recover the spectra of adjacency matrices and Laplacians of random graphs. While our treatment focuses on random graphs for concreteness, the methodology has broad applications to more general sparse random matrices.

Figures (3)

• Figure 1: Solutions of the collocation problem (\ref{['collocadj']}) for Erdős-Rényi graph adjacency matrices at $J=10$, $\gamma=1$, converted to eigenvalue distribution estimates using (\ref{['padjbeta']}) and plotted as solid black lines for (left)$c=15$ and (right)$c=8$. As the distributions are reflection-symmetric, only the $\lambda>0$ part is plotted explicitly. The grey shaded areas represent empirical eigenvalue density histograms obtained from a sample of 1000 Erdős-Rényi graphs with 10000 vertices each at the corresponding values of $c$. Tiny undulations are seen near the center of the $c=8$ plot. These numerical artifacts become more problematic at smaller values of $c$.
• Figure 2: Results for the eigenvalue density of the ordinary graph Laplacian plotted as solid black lines: (left) at $c=8$ using the Laguerre collocation problem (\ref{['collocidL']}) with $J=10$, $\gamma=5$, from which the eigenvalue density is extracted via (\ref{['pL']}), (right) at $c=4$ using the general Kumar-Sloan collocation (\ref{['colloc']}) with the half-range Hermite expansion (\ref{['hrHermite']}) at $J=10$, followed by the integration in (\ref{['extractpL']}). Additionally, on the right panel, we add a dotted curve representing the result of solving (\ref{['collocidL']}) with $J=10$, $\gamma=5$ at $c=4$, showing more spurious undulations than the half-range Hermite method. The grey shaded areas represent empirical eigenvalue density histograms obtained from a sample of 1000 Erdős-Rényi graphs with 10000 vertices each at the corresponding values of $c$.
• Figure 3: Solutions of the collocation problem (\ref{['collocidLn']}) for the normalized Laplacians of Erdős-Rényi graphs at $J=9$, $\gamma=2$, converted to eigenvalue distribution estimates using (\ref{['pLn']}) and plotted as solid black lines for (left)$c=8$ and (right)$c=4$. As the distributions are reflection-symmetric, only the $\lambda>1$ part is plotted explicitly. We excise small regions near $\lambda=1$ where the convergence of our collocation scheme is compromised by the presence of sharp spikes in the actual eigenvalue distribution. (There are generically $\delta$-function spikes at $\lambda=1$ that become more and more prominent at small $c$. We comment on them further in the conclusions.) The grey shaded areas represent empirical eigenvalue density histograms obtained from a sample of 1000 Erdős-Rényi graphs with 10000 vertices each at the corresponding values of $c$.