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Simplify to Amplify: Achieving Information-Theoretic Bounds with Fewer Steps in Spectral Community Detection

Sie Hendrata Dharmawan, Peter Chin

TL;DR

This work proposes a streamlined spectral algorithm for community detection in the two-community stochastic block model (SBM) under constant edge density assumptions that directly leverages the spectral properties of the adjacency matrix.

Abstract

We propose a streamlined spectral algorithm for community detection in the two-community stochastic block model (SBM) under constant edge density assumptions. By reducing algorithmic complexity through the elimination of non-essential preprocessing steps, our method directly leverages the spectral properties of the adjacency matrix. We demonstrate that our algorithm exploits specific characteristics of the second eigenvalue to achieve improved error bounds that approach information-theoretic limits, representing a significant improvement over existing methods. Theoretical analysis establishes that our error rates are tighter than previously reported bounds in the literature. Comprehensive experimental validation confirms our theoretical findings and demonstrates the practical effectiveness of the simplified approach. Our results suggest that algorithmic simplification, rather than increasing complexity, can lead to both computational efficiency and enhanced performance in spectral community detection.

Simplify to Amplify: Achieving Information-Theoretic Bounds with Fewer Steps in Spectral Community Detection

TL;DR

This work proposes a streamlined spectral algorithm for community detection in the two-community stochastic block model (SBM) under constant edge density assumptions that directly leverages the spectral properties of the adjacency matrix.

Abstract

We propose a streamlined spectral algorithm for community detection in the two-community stochastic block model (SBM) under constant edge density assumptions. By reducing algorithmic complexity through the elimination of non-essential preprocessing steps, our method directly leverages the spectral properties of the adjacency matrix. We demonstrate that our algorithm exploits specific characteristics of the second eigenvalue to achieve improved error bounds that approach information-theoretic limits, representing a significant improvement over existing methods. Theoretical analysis establishes that our error rates are tighter than previously reported bounds in the literature. Comprehensive experimental validation confirms our theoretical findings and demonstrates the practical effectiveness of the simplified approach. Our results suggest that algorithmic simplification, rather than increasing complexity, can lead to both computational efficiency and enhanced performance in spectral community detection.
Paper Structure (25 sections, 6 theorems, 55 equations, 5 figures)

This paper contains 25 sections, 6 theorems, 55 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Original Spectral Algorithm
  3. Our Modified Algorithm
  4. Improved Error Bounds for Spectral Partition
  5. Original Error Bounds
  6. Sharpness Of Theorem \ref{['theorem:gamma_bound_quad_Chin']}
  7. Statistical Properties of the Second Eigenvector
  8. Applying Chernoff Bounds to Relate $\gamma$ and $\sin \theta$
  9. Monte-Carlo Simulation and Normal Approximation to Relate $\gamma$ and $\sin \theta$
  10. Comparing Theoretical Predictions with Spectral Algorithm Results
  11. Scaling Behavior and Convergence Analysis
  12. Conclusion And Future Work
  13. Reproducibility Statement
  14. Appendix
  15. Proof of Theorem \ref{['theorem:M_bound_Chin']}
  16. ...and 10 more sections

Key Result

Theorem 1.2

For any constant $\gamma > 0$, there exist constants $C_1, C_2 >0$ such that if $a,b > C_1$ and $\frac{(a-b)^2}{a+b} > C_2 \log(a+b)$, one can find a $\gamma$-correct partition using a polynomial time algorithm.

Figures (5)

  • Figure 1: Spectral Partition
  • Figure 2: Correction
  • Figure 3: Partition
  • Figure 4: $\gamma$ as a function of $\sin \theta$ for various approaches
  • Figure 5: $\gamma$ as a function of $\sin \theta$ for various approaches including experimental results

Theorems & Definitions (8)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Theorem 3.2
  • proof