Collapsed Structured Block Models for Community Detection in Complex Networks

2026-01-06

AI Summary

This work addresses the challenges of slow Markov chain mixing and difficult model comparison in traditional Bayesian stochastic block models when applied to binary, count, signed, directed, or multilayer networks. The authors propose a collapsed Bayesian stochastic block model framework that analytically integrates out redundant parameters using conjugate priors, retaining only the node partition and sufficient statistics. This enables efficient local updates and complexity control via marginal likelihood. For the first time, exact collapsed marginal likelihoods are derived for diverse edge types—including Beta-Bernoulli and Gamma-Poisson—and extended to networks with connectivity constraints, directionality, signed edges, and multiple layers, providing a unified approach for community detection in heterogeneous networks. Experiments on synthetic and real-world networks—such as email communication, hospital contact, and citation graphs—demonstrate high-accuracy community recovery, interpretable posterior summaries of inter-community interaction strengths, and favorable computational efficiency with modular design.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Bayesian Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Computational social science

Abstract

Community detection seeks to recover mesoscopic structure from network data that may be binary, count-valued, signed, directed, weighted, or multilayer. The stochastic block model (SBM) explains such structure by positing a latent partition of nodes and block-specific edge distributions. In Bayesian SBMs, standard MCMC alternates between updating the partition and sampling block parameters, which can hinder mixing and complicate principled comparison across different partitions and numbers of communities. We develop a collapsed Bayesian SBM framework in which block-specific nuisance parameters are analytically integrated out under conjugate priors, so the marginal likelihood p(Y|z) depends only on the partition z and blockwise sufficient statistics. This yields fast local Gibbs/Metropolis updates based on ratios of closed-form integrated likelihoods and provides evidence-based complexity control that discourages gratuitous over-partitioning. We derive exact collapsed marginals for the most common SBM edge types-Beta-Bernoulli (binary), Gamma-Poisson (counts), and Normal-Inverse-Gamma (Gaussian weights)-and we extend collapsing to gap-constrained SBMs via truncated conjugate priors that enforce explicit upper bounds on between-community connectivity. We further show that the same collapsed strategy supports directed SBMs that model reciprocity through dyad states, signed SBMs via categorical block models, and multiplex SBMs where multiple layers contribute additive evidence for a shared partition. Across synthetic benchmarks and real networks (including email communication, hospital contact counts, and citation graphs), collapsed inference produces accurate partitions and interpretable posterior block summaries of within- and between-community interaction strengths while remaining computationally simple and modular.

Problem

• Community Detection
• Stochastic Block Model
• Bayesian Inference
• Complex Networks
• Model Comparison

Innovation

• Collapsed Bayesian inference
• Stochastic Block Model
• Conjugate priors
• Community detection
• Marginal likelihood