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Estimating Mixed-Memberships Using the Symmetric Laplacian Inverse Matrix

Huan Qing, Jingli Wang

TL;DR

The paper introduces Mixed-SLIM, a spectral clustering approach on the symmetric Laplacian inverse matrix for mixed membership detection under the degree-corrected mixed membership (DCMM) model. It establishes finite-sample consistency results for a regularized variant and provides practical extensions for large networks through approximation. Empirically, Mixed-SLIM and its variants outperform several state-of-the-art methods on synthetic data and real networks, including SNAP ego-networks, highlighting robustness to degree heterogeneity and varying purity of mixed memberships. The work advances scalable, theoretically grounded mixed membership community detection with strong empirical performance and broad applicability to real-world networks.

Abstract

Mixed membership community detection is a challenging problem. In this paper, to detect mixed memberships, we propose a new method Mixed-SLIM which is a spectral clustering method on the symmetrized Laplacian inverse matrix under the degree-corrected mixed membership model. We provide theoretical bounds for the estimation error on the proposed algorithm and its regularized version under mild conditions. Meanwhile, we provide some extensions of the proposed method to deal with large networks in practice. These Mixed-SLIM methods outperform state-of-art methods in simulations and substantial empirical datasets for both community detection and mixed membership community detection problems.

Estimating Mixed-Memberships Using the Symmetric Laplacian Inverse Matrix

TL;DR

The paper introduces Mixed-SLIM, a spectral clustering approach on the symmetric Laplacian inverse matrix for mixed membership detection under the degree-corrected mixed membership (DCMM) model. It establishes finite-sample consistency results for a regularized variant and provides practical extensions for large networks through approximation. Empirically, Mixed-SLIM and its variants outperform several state-of-the-art methods on synthetic data and real networks, including SNAP ego-networks, highlighting robustness to degree heterogeneity and varying purity of mixed memberships. The work advances scalable, theoretically grounded mixed membership community detection with strong empirical performance and broad applicability to real-world networks.

Abstract

Mixed membership community detection is a challenging problem. In this paper, to detect mixed memberships, we propose a new method Mixed-SLIM which is a spectral clustering method on the symmetrized Laplacian inverse matrix under the degree-corrected mixed membership model. We provide theoretical bounds for the estimation error on the proposed algorithm and its regularized version under mild conditions. Meanwhile, we provide some extensions of the proposed method to deal with large networks in practice. These Mixed-SLIM methods outperform state-of-art methods in simulations and substantial empirical datasets for both community detection and mixed membership community detection problems.

Paper Structure

This paper contains 18 sections, 4 theorems, 20 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Degree-corrected mixed membership model
  3. Methodology
  4. Algorithm: mixed-SLIM
  5. Choice of tuning parameters
  6. Theoretical results
  7. Numerical Results
  8. Synthetic data experiments
  9. Fraction of pure nodes
  10. Connectivity across communities
  11. Purity of mixed nodes
  12. Degree heterogeneity
  13. Sparsity of mixing matrix
  14. Detectability of clusters
  15. Application to real-world datasets for community detection
  16. ...and 3 more sections

Key Result

Lemma 1

Under $DCMM(n, P, \Theta, \Pi)$, if assumption A1 holds, with probability at least $1-o(n^{-3})$,

Figures (2)

  • Figure 1: Estimation errors of Experiment 1 (y-axis: $\sum_{i=1}^{n}n^{-1}\|\hat{\pi}_{i}-\pi_{i}\|_{1}$).
  • Figure 2: Mean of mixed-Hamming error rates on SNAP ego-networks for Mixed-$\mathrm{SLIM}_{appro}$ based on different choice of $T$, where $T$ is in $\{1,2, \ldots, 20\}$. x-axis: $T$, y-axis: mean of mixed-Hamming error rates.

Theorems & Definitions (7)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 1