Community Detection for Contextual-LSBM: Theoretical Limitations of Misclassification Rate and Efficient Algorithms
Dian Jin, Yuqian Zhang, Qiaosheng Zhang
TL;DR
This work studies community detection in the Contextual-Labeled Stochastic Block Model (CLSBM), where network structure (LSBM) and Gaussian node attributes collectively inform community labels. It derives an information-theoretic lower bound on the misclassification rate, showing that any algorithm incurs at least $n \exp(-n D(\boldsymbol \alpha, \mathbf P, \boldsymbol{\mu}))$ misclassifications, where $D$ blends topological and attribute divergences. A practical, efficient spectral method is proposed via aggregation into a latent-factor model: $\mathbf S = \sum_l w_l \mathbf A_l + \frac{1}{n}\mathbf X^ op \mathbf X$, with a population mean $\mathbf M = \mathbf Z (\mathbf P_s + \frac{1}{n} \boldsymbol \mu^ op \boldsymbol \mu) \mathbf Z^ op - \text{diag}(\mathbf Z \mathbf P_s \mathbf Z^ op)$. Theoretical guarantees show a polynomial misclassification rate bound $s(\bar{\sigma}) \lesssim \frac{K}{n \cdot \mathrm{SNR}}$, where $\mathrm{SNR}$ captures the separation in the aggregated signal; this provides a valuable initialization for further refinement toward potentially achieving the optimal exponential rate. Overall, the paper establishes a benchmark for CLSBM performance, linking it to CH-divergence and Gaussian-mixture limits, and motivates refinement steps to bridge polynomial and exponential recovery regimes.
Abstract
The integration of network information and node attribute information has recently gained significant attention in the community detection literature. In this work, we consider community detection in the Contextual Labeled Stochastic Block Model (CLSBM), where the network follows an LSBM and node attributes follow a Gaussian Mixture Model (GMM). Our primary focus is the misclassification rate, which measures the expected number of nodes misclassified by community detection algorithms. We first establish a lower bound on the optimal misclassification rate that holds for any algorithm. When we specialize our setting to the LSBM (which preserves only network information) or the GMM (which preserves only node attribute information), our lower bound recovers prior results. Moreover, we present an efficient spectral-based algorithm tailored for the CLSBM and derive an upper bound on its misclassification rate. Although the algorithm does not attain the lower bound, it serves as a reliable starting point for designing more accurate community detection algorithms (as many algorithms use spectral method as an initial step, followed by refinement procedures to enhance accuracy).