Eigenvector fluctuations and limit results for random graphs with infinite rank kernels

TL;DR

This work develops a comprehensive spectral-embedding theory for random graphs generated from latent-position models with general, possibly infinite-rank and indefinite kernels. It provides sharp, uniform $2\to\infty$ perturbation bounds and rowwise asymptotic normality for leading embeddings $\widehat{U}|\widehat{\Lambda}|^{\alpha}$, valid in growing-dimension regimes, and extends the analysis from PSD to indefinite kernels. Building on these results, the authors derive entrywise bounds for estimating the edge-probability matrix $P$, and introduce a rank-adaptive, data-driven test for equality of latent positions whose null distribution is a weighted sum of independent $\chi^2$ variables (converging to $N(0,2)$ in the infinite-rank limit). The results unify and extend prior perturbation theory, enable robust inference for large graphs, and are supported by numerical experiments showing accurate finite-sample performance and practical rank adaptation. These contributions advance understanding of eigenvector fluctuations in high-dimensional graph embeddings and provide practical tools for graphon estimation and latent-position testing.

Abstract

This paper systematically studies the behavior of the leading eigenvectors for independent edge undirected random graphs generated from a general latent position model whose link function is possibly infinite rank and also possibly indefinite. We first derive uniform error bounds in the two-to-infinity norm as well as row-wise normal approximations for the leading sample eigenvectors. We then build on these results to tackle two graph inference problems, namely (i) entrywise bounds for graphon estimation and (ii) testing for the equality of latent positions, the latter of which is achieved by proposing a rank-adaptive test statistic that converges in distribution to a weighted sum of independent chi-square random variables under the null hypothesis. Our fine-grained theoretical guarantees and applications differ from the existing literature which primarily considers first order upper bounds and more restrictive low rank or positive semidefinite model assumptions. Further, our results collectively quantify the statistical properties of eigenvector-based spectral embeddings with growing dimensionality for large graphs.

Figures (3)

• Figure 1: Plots of the forty largest eigenvalues (left panel) and gap between consecutive eigenvalues (right panel) for $n \times n$ edge probability matrices $P$ when the link function is $\kappa(x,y) = \exp(-\|x - y\|)$, where $X_{1}, \dots, X_{n}$ are sampled i.i.d. from the uniform distribution on the unit sphere in $\mathbb{R}^{3}$, and $n = 8000, \rho_{n} = 1$. There are $10$ blue-colored lines (nearly indistinguishable) in each panel, one for each independent Monte Carlo replicate. The dashed line in the left panel corresponds to the best approximation of the form $\lambda_{r}/n \propto r^{-3/2}$.
• Figure 2: Empirical histograms, based on $500$ Monte Carlo replicates, for $T(\widehat{X}_i, \widehat{X}_j)$ under $\mathbb{H}_0 \colon X_i = X_j$ when the link function is $\kappa(x,y) = \exp(-\|x - y\|)$, the latent positions $\{X_i\}$ are sampled i.i.d. from the uniform distribution on the unit sphere in $\mathbb{R}^{3}$, and the sparsity parameter is $\rho_n = 0.4$. The red curve in each plot is the probability density function for a weighted sum of $\widehat{r}$ independent $\chi^2_1$ random variables as given in \ref{['eq:twosample_critical_estimate']}, where $\widehat{r}$ is chosen according to \ref{['eq:selection_r_test']} resulting in $\widehat{r} \equiv 4$ when $n = 4000$ and $\widehat{r} \equiv 9$ when $n = 8000$.
• Figure 3: Empirical histograms, based on $500$ Monte Carlo replicates, for $T(\widehat{X}_i, \widehat{X}_j)$ under $\mathbb{H}_0 \colon X_i = X_j$ when the link function is $\kappa(x,y) = \exp(-\tfrac{5}{2}\|x - y\|^2)$, the latent positions $\{X_i\}$ are sampled i.i.d. from the bivariate normal distribution with mean zero and identity covariance matrix, and the sparsity parameter is $\rho_n = 0.4$. The red curve in each plot is the probability density function for a weighted sum of $\widehat{r}$ independent $\chi^2_1$ random variables as given in \ref{['eq:twosample_critical_estimate']}, where $\widehat{r}$ is chosen via the criteria \ref{['eq:selection_r_test']} resulting in $\widehat{r} \equiv 3$ when $n = 2000$ and $\widehat{r} \equiv 6$ when $n = 4000$.

Theorems & Definitions (43)

• Definition 1: Latent position graph
• Remark 1: Latent position graph versus graphon
• Proposition 1: Properties of positive semidefinite kernels
• Remark 2: Population eigenvectors and coherence
• Theorem 1: Fine-grained eigenvector perturbation analysis for LPGs with positive semidefinite kernels
• Remark 3: Explicit constants and bounding the residual perturbation term
• Remark 4: On dimension selection and eigengaps in high rank models
• Corollary 1: Data-driven dimension selection in \ref{['thm:psd']}
• Remark 5: Eigenvalues with polynomial decay
• Remark 6: Eigenvalues with exponential decay