Distributional Limits for Eigenvalues of Graphon Kernel Matrices

TL;DR

This paper characterizes the distributional limits of individual eigenvalues of graphon-based kernel matrices, revealing a probabilistic dichotomy and extending fluctuation theory beyond classical operator convergence.

Contribution

It provides the first detailed distributional analysis of eigenvalue fluctuations for dense graphon kernels under minimal assumptions, including a sharp dichotomy in behavior.

Findings

Eigenvalues follow a CLT in the non-degenerate regime.

Centered eigenvalues converge to a weighted chi-square law in the degenerate regime.

The dominant randomness at the $\, ext{sqrt}(n)$ scale is from latent-position sampling.

Abstract

We study the fluctuation behavior 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. A sharp probabilistic dichotomy emerges: 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 stochastic term vanishes and the centered eigenvalue converges to a weighted chi-square law determined by the operator spectrum. The analysis requires no smoothness or Lipschitz conditions on the kernel. Prior work under comparable assumptions established only operator convergence and eigenspace consistency; the present results characterize the full distributional behavior of individual eigenvalues, extending fluctuation theory beyond the reach of classical operator-level arguments. The proofs combine second-order perturbation expansions, concentration bounds for kernel matrices, and Hoeffding decompositions for symmetric statistics, revealing that at the $\sqrt{n}$ scale the dominant randomness arises from latent-position sampling rather than Bernoulli edge noise.