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Semi-Supervised Community Detection via Quasi-Stationary Distributions

Nicolas Fraiman, Michael Nisenzon

TL;DR

This work addresses semi-supervised community detection in a two-community PL-SBM by leveraging quasi-stationary distributions arising from absorbing revealed nodes. The authors develop a family of QSD-based estimators, including a pure QSD, simple voting, and a mixed QSD approach, and prove both minimax lower bounds and upper bounds on error rates, with the mixed method achieving the optimal connected-regime rate. The analysis hinges on a thorough entrywise eigenvector concentration framework, generalized Davis–Kahan perturbation results for submatrices, and careful large-deviation bounds for binomial differences, all tailored to the partially labeled setting. Empirically, QSD-based methods exhibit improvements over classical spectral approaches, especially in the bounded-degree regime, demonstrating the practical impact of integrating side information via absorbing-state random walks into spectral clustering. The work thus extends spectral methodology to quasi-stationary transitions, providing sharp performance guarantees and actionable algorithms for semi-supervised community detection.

Abstract

Spectral clustering is a widely used method for community detection in networks. We focus on a semi-supervised community detection scenario in the Partially Labeled Stochastic Block Model (PL-SBM) with two balanced communities, where a fixed portion of labels is known. Our approach leverages random walks in which the revealed nodes in each community act as absorbing states. By analyzing the quasi-stationary distributions associated with these random walks, we construct a classifier that distinguishes the two communities by examining differences in the associated eigenvectors. We establish upper and lower bounds on the error rate for a broad class of quasi-stationary algorithms, encompassing both spectral and voting-based approaches. In particular, we prove that this class of algorithms can achieve the optimal error rate in the connected regime. We further demonstrate empirically that our quasi-stationary approach improves performance on both real-world and simulated datasets.

Semi-Supervised Community Detection via Quasi-Stationary Distributions

TL;DR

This work addresses semi-supervised community detection in a two-community PL-SBM by leveraging quasi-stationary distributions arising from absorbing revealed nodes. The authors develop a family of QSD-based estimators, including a pure QSD, simple voting, and a mixed QSD approach, and prove both minimax lower bounds and upper bounds on error rates, with the mixed method achieving the optimal connected-regime rate. The analysis hinges on a thorough entrywise eigenvector concentration framework, generalized Davis–Kahan perturbation results for submatrices, and careful large-deviation bounds for binomial differences, all tailored to the partially labeled setting. Empirically, QSD-based methods exhibit improvements over classical spectral approaches, especially in the bounded-degree regime, demonstrating the practical impact of integrating side information via absorbing-state random walks into spectral clustering. The work thus extends spectral methodology to quasi-stationary transitions, providing sharp performance guarantees and actionable algorithms for semi-supervised community detection.

Abstract

Spectral clustering is a widely used method for community detection in networks. We focus on a semi-supervised community detection scenario in the Partially Labeled Stochastic Block Model (PL-SBM) with two balanced communities, where a fixed portion of labels is known. Our approach leverages random walks in which the revealed nodes in each community act as absorbing states. By analyzing the quasi-stationary distributions associated with these random walks, we construct a classifier that distinguishes the two communities by examining differences in the associated eigenvectors. We establish upper and lower bounds on the error rate for a broad class of quasi-stationary algorithms, encompassing both spectral and voting-based approaches. In particular, we prove that this class of algorithms can achieve the optimal error rate in the connected regime. We further demonstrate empirically that our quasi-stationary approach improves performance on both real-world and simulated datasets.

Paper Structure

This paper contains 16 sections, 22 theorems, 135 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related results
  3. Our contributions
  4. Main Results
  5. Setup and Notation
  6. Minimax bound for semi-supervised clustering
  7. Performance of QSD method
  8. Proofs
  9. Equivariance
  10. Normalized matrix bounds
  11. Eigenvalue bounds
  12. Bounding binomial differences
  13. Entrywise bounds
  14. Equivalence of left and right transition eigenvectors
  15. Entrywise error rate for right eigenvectors
  16. ...and 1 more sections

Key Result

Theorem 1

For equivariant estimators $\hat{\sigma}$ we have

Figures (3)

  • Figure 1: Revealed and unrevealed nodes up to permutation.
  • Figure 2: Submatrices of the transition matrix up to permutation.
  • Figure 3: QSD eigenvectors on connected SBM.

Theorems & Definitions (52)

  • Definition 1: Balanced partitions and partial labels
  • Definition 2: Partially Labeled Balanced SBM
  • Definition 3: Transition submatrices and eigenvectors
  • Definition 4: Mean--field eigenvector
  • Definition 5: Error rate
  • Definition 6: Equivariance
  • Theorem 1
  • proof
  • Definition 7
  • Definition 8
  • ...and 42 more