Distributional Limits for Eigenvalues of Graphon Kernel Matrices
Behzad Aalipur, Mohammad S. M. Moakhar
TL;DR
This work establishes a distributional theory for individual eigenvalues of dense graphon-based kernel matrices under minimal $L^2\cap L^\infty$ regularity. It proves a sharp dichotomy: in the non-degenerate regime, normalized eigenvalues satisfy a central limit theorem with explicit variance $\lambda_r^2\sigma_r^2$, while in the degenerate regime the leading fluctuations converge to a weighted chi-square limit with coefficients $\frac{\lambda_r\lambda_k}{\lambda_r-\lambda_k}$ and an $L^2$-convergent series. The analysis relies on a second-order Rayleigh–Schrödinger expansion, Hoeffding decompositions of U-statistics, and operator-norm concentration, and shows that latent-position sampling drives $\sqrt{n}$-scale uncertainty with edge noise asymptotically negligible. The results hold for non-Lipschitz kernels, including stochastic block models with unequal blocks and alpha-Hölder kernels, broadening the scope of eigenvalue fluctuation theory beyond smooth graphons. These insights enable principled inference for eigenvalue-based spectral methods in network analysis, with implications for confidence intervals and model diagnostics in block-structured and inhomogeneous graphs.
Abstract
We study fluctuations of individual eigenvalues of kernel matrices arising from dense graphon-based random graphs. Under minimal integrability and boundedness assumptions on the graphon, we establish distributional limits for simple, well-separated eigenvalues of the associated integral operator. We show that a sharp dichotomy governs the asymptotic behavior. In the non-degenerate regime, the properly normalized empirical eigenvalue satisfies a central limit theorem with an explicit variance, whereas in the degenerate regime, the leading fluctuations vanish and the centered eigenvalue converges to an explicit weighted chi-square limit determined by the operator spectrum. Our analysis requires no smoothness or Lipschitz-type assumptions on the graphon. While earlier work under comparable integrability conditions established operator convergence and eigenspace consistency, the present results characterize the full fluctuation behavior of individual eigenvalues, thereby extending eigenvalue fluctuation theory beyond regimes accessible through operator convergence alone.