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Community Detection with Heterogeneous Block Covariance Model

Xiang Li, Yunpeng Zhao, Qing Pan, Ning Hao

TL;DR

The paper addresses clustering of features by learning a covariance-based community structure with heterogeneity across features. It introduces the heterogeneous block covariance model (HBCM) and an efficient variational EM algorithm that adds a second latent layer to achieve closed-form updates and polynomial-time computation. Theoretical results establish identifiability and consistency of the estimated memberships under mild conditions, and extensive simulations show HBCM outperforms spectral clustering and related methods under diverse settings, including misspecified covariances. The approach is demonstrated on mouse embryo scRNA-seq data and stock prices, yielding improved clustering and clearer interpretation of block structures, with cross-validated selection of the number of communities. Overall, HBCM offers a principled, scalable framework for covariance-based feature clustering with strong theoretical guarantees and practical applicability to biological and financial datasets.

Abstract

Community detection is the task of clustering objects based on their pairwise relationships. Most of the model-based community detection methods, such as the stochastic block model and its variants, are designed for networks with binary (yes/no) edges. In many practical scenarios, edges often possess continuous weights, spanning positive and negative values, which reflect varying levels of connectivity. To address this challenge, we introduce the heterogeneous block covariance model (HBCM) that defines a community structure within the covariance matrix, where edges have signed and continuous weights. Furthermore, it takes into account the heterogeneity of objects when forming connections with other objects within a community. A novel variational expectation-maximization algorithm is proposed to estimate the group membership. The HBCM provides provable consistent estimates of memberships, and its promising performance is observed in numerical simulations with different setups. The model is applied to a single-cell RNA-seq dataset of a mouse embryo and a stock price dataset. Supplementary materials for this article are available online.

Community Detection with Heterogeneous Block Covariance Model

TL;DR

The paper addresses clustering of features by learning a covariance-based community structure with heterogeneity across features. It introduces the heterogeneous block covariance model (HBCM) and an efficient variational EM algorithm that adds a second latent layer to achieve closed-form updates and polynomial-time computation. Theoretical results establish identifiability and consistency of the estimated memberships under mild conditions, and extensive simulations show HBCM outperforms spectral clustering and related methods under diverse settings, including misspecified covariances. The approach is demonstrated on mouse embryo scRNA-seq data and stock prices, yielding improved clustering and clearer interpretation of block structures, with cross-validated selection of the number of communities. Overall, HBCM offers a principled, scalable framework for covariance-based feature clustering with strong theoretical guarantees and practical applicability to biological and financial datasets.

Abstract

Community detection is the task of clustering objects based on their pairwise relationships. Most of the model-based community detection methods, such as the stochastic block model and its variants, are designed for networks with binary (yes/no) edges. In many practical scenarios, edges often possess continuous weights, spanning positive and negative values, which reflect varying levels of connectivity. To address this challenge, we introduce the heterogeneous block covariance model (HBCM) that defines a community structure within the covariance matrix, where edges have signed and continuous weights. Furthermore, it takes into account the heterogeneity of objects when forming connections with other objects within a community. A novel variational expectation-maximization algorithm is proposed to estimate the group membership. The HBCM provides provable consistent estimates of memberships, and its promising performance is observed in numerical simulations with different setups. The model is applied to a single-cell RNA-seq dataset of a mouse embryo and a stock price dataset. Supplementary materials for this article are available online.

Paper Structure

This paper contains 23 sections, 12 theorems, 88 equations, 10 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Methodology
  3. Model description
  4. Variational expectation-maximization algorithm
  5. Theory
  6. Model Identifiability
  7. Asymptotic Properties
  8. Simulation Studies
  9. Numerical experiments with various sample sizes and dimensionality
  10. Numerical experiments with various parameters
  11. Cross-validation for selecting the number of communities
  12. Application on Mouse Embryo Single-Cell RNA-seq Data
  13. Discussion
  14. Additional simulation studies
  15. Comparison of HBCM with SBM and DC-SBM
  16. ...and 8 more sections

Key Result

Theorem 1

Two systems $\{\boldsymbol{\lambda},\boldsymbol{\Omega}, \mathbf{L}, \boldsymbol{\sigma}^2\}$ and $\{\tilde{\boldsymbol{\lambda}},\tilde{\boldsymbol{\Omega}},\tilde{\mathbf{L}},\tilde{\boldsymbol{\sigma}}^2\}$ under Condition 1 give the same covariance structure $\boldsymbol{\Sigma}$ if and only if

Figures (10)

  • Figure 1: Compare the clustering performance of HBCM and spectral clustering under various parameter setups. In the right plot, dashed line represents simulation data with variance correction and solid line indicates data without any variance correction.
  • Figure 2: Spectral clustering is misled by the individual signal $\lambda_j$ as the signal-to-noise ratio decreases.
  • Figure 3: Compare the clustering performance of HBCM and spectral clustering in the presence of noisy population covariance matrices.
  • Figure 4: Cross-validation method for selecting the number of communities.
  • Figure 5: Left panel: Cross-validation method for selecting the number of communities in the scRNA data. Right panel: between-cluster densities of the absolute correlation matrix and $\hat{\mathbf{\Omega}}$.
  • ...and 5 more figures

Theorems & Definitions (26)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Definition 1: Soft confusion matrix
  • Proposition 5
  • Proposition 6
  • Theorem 2
  • Proposition 7
  • ...and 16 more