Collapsed Structured Block Models for Community Detection in Complex Networks

TL;DR

The paper tackles the challenge of community detection in networks with rich within-block structure and tightly constrained between-block connectivity by introducing a collapsed Bayesian SBM framework. By analytically integrating out block-specific nuisance parameters, it yields a marginal likelihood $p(Y\mid z)$ that depends only on the partition and blockwise statistics, enabling fast local updates, automatic complexity control, and interpretable block summaries. The authors derive exact collapsed marginals for Beta--Bernoulli, Gamma--Poisson, and Normal--Inverse-Gamma blocks, extend collapsing to truncated priors for gap constraints, and support directed, signed, and multiplex extensions via a unified CSBM. Empirical results on synthetic and real networks demonstrate accurate partition recovery, robust performance in sparse and heavy-tailed settings, and compact, interpretable summaries of within- and between-community interactions, underscoring the approach’s practical utility and theoretical appeal.

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.

Figures (24)

• Figure 1: S1: collapsed Beta--Bernoulli SBM diagnostics. This multi-panel diagnostic is our standard reporting template: adjacency reordered by truth vs. MAP, posterior mean block probabilities under $z_{\mathrm{MAP}}$, posterior similarity matrix (PSM), and trace plots of collapsed log-posterior and ARI. The key empirical point is that the same local collapsed kernel yields both accurate recovery and a posterior uncertainty map (PSM) without sampling any $p_{rs}$ explicitly.
• Figure 2: S2: collapsed Gamma--Poisson SBM. Counts provide distributional separation (many zeros between blocks; heavier tails within blocks), so the collapsed model recovers the partition and learns interpretable interaction intensities.
• Figure 3: S3: gap-constrained SBM (binary). A hard between-block cap prevents "explaining away" cross edges via a large $p_{\rm out}$, thereby protecting community boundaries in extremely sparse regimes.
• Figure 4: S4: ZIP-SBM diagnostics. Top: adjacency heatmaps (true ordering vs. MAP ordering) using $\log(1+Y_{ij})$. Middle: posterior block means for $p_{rs}$ (activity), $\lambda_{rs}$ (intensity), and $\mu_{rs}=p_{rs}\lambda_{rs}$. Bottom: collapsed log-posterior trace (thinned).
• Figure 5: S5: multiplex collapsed SBM. Complementary layers contribute additive evidence for the same latent partition, concentrating the posterior near $z_{\mathrm{MAP}}$ even when any single layer is ambiguous.

Theorems & Definitions (8)

• Definition 3.1: CSBM collapsed likelihood
• Remark 3.2: Statistical role of type collapsing
• Theorem 3.5: Posterior concentration of the partition
• Theorem 3.6: Collapsed evidence induces an Occam penalty
• proof
• proof
• Lemma A.1: Laplace expansion for blockwise marginals
• proof : Proof of Lemma \ref{['lem:laplace_block_occam']}