From Simple to Composite Perturbations: A Unified Decomposition Framework for Stochastic Block Models

TL;DR

This paper introduces a unified decomposition framework for analyzing perturbations in stochastic block models, improving understanding of error sources and refining asymptotic results for spectral test statistics.

Contribution

It develops a structured decomposition of composite perturbations, enabling precise error control and improved asymptotic theory for eigenvalue and spectral statistics.

Findings

Refined condition for largest eigenvalue statistic: $K=o(n^{1/6})$

Asymptotic normality of linear spectral statistics established

Distinction between simple and composite perturbations clarified

Abstract

Statistical inference for stochastic block models typically relies on the spectrum of the normalized adjacency matrix $\A^*$. In practice, the true probability matrix $\mathbf{B}$ is unknown and must be replaced by a plug-in estimator $\hat{\mathbf{B}}$. This substitution introduces two distinct types of estimation error: a simple perturbation $\boldsymbol{\Delta}$, arising when $\hat{\mathbf{B}}$ replaces $\mathbf{B}$ only in the numerator, and a composite perturbation $\tilde{\boldsymbol{\Delta}}$, arising when the replacement occurs in both the numerator and the denominator. Under both perturbation regimes, we decompose the total sum of squares into three components and conduct a detailed analysis of their asymptotic properties. This reveals a key, and perhaps surprising, distinction between simple and composite perturbations: the cross term $\tr({\A^*}\bDelta)$ is asymptotically negligible, whereas its composite counterpart $\tr({\A^*}\tilde{\bDelta})$ is not. Motivated by this, we develop a unified decomposition framework, expressing the composite perturbation matrix as $\tilde{\bDelta}=\check{\A}+\bDelta+\check{\bDelta}$, where $\check{\A}$ is a bias matrix of the normalized adjacency matrix, $\bDelta$ is the simple perturbation, and $\check{\bDelta}$ is a bias matrix of $\bDelta$. This structured decomposition allows us to precisely isolate and control each source of error, leading to a refined limiting theory for two key classes of test statistics. Concretely, for the largest eigenvalue statistic, we improve the existing condition from $K=O(n^{1/6-\tau})$ to the optimal rate $K=o(n^{1/6})$ under both simple and composite perturbations. For the linear spectral statistic, our unified decomposition framework provides the necessary structure to systematically control these errors term by term, leading to a complete and rigorous proof of asymptotic normality.