Community Detection for Contextual-LSBM: Theoretical Limitations of Misclassification Rate and Efficient Algorithms

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).