Limit Results for Estimation of Connectivity Matrix in Multi-layer Stochastic Block Models

TL;DR

This paper investigates the estimation and asymptotic distributional properties of connectivity matrices in multi-layer stochastic block models, providing new methods and theoretical insights for statistical inference in complex networks.

Contribution

It introduces a novel efficient estimation method for scaled connectivity matrices and establishes their asymptotic normality under multi-layer SBMs.

Findings

Proposed method outperforms existing techniques in simulations.

Asymptotic normality enables reliable interval estimation and hypothesis testing.

Application to real data yields interpretable results.

Abstract

Multi-layer networks arise naturally in various domains including biology, finance and sociology, among others. The multi-layer stochastic block model (multi-layer SBM) is commonly used for community detection in the multi-layer networks. Most of current literature focuses on statistical consistency of community detection methods under multi-layer SBMs. However, the asymptotic distributional properties are also indispensable which play an important role in statistical inference. In this work, we aim to study the estimation and asymptotic properties of the layer-wise scaled connectivity matrices in the multi-layer SBMs. We develop a novel and efficient method to estimate the scaled connectivity matrices. Under the multi-layer SBM and its variant multi-layer degree-corrected SBM, we establish the asymptotic normality of the estimated matrices under mild conditions, which can be used for interval estimation and hypothesis testing. Simulations show the superior performance of proposed method over existing methods in two considered statistical inference tasks. We also apply the method to a real dataset and obtain interpretable results.