Ordering Stochastic Block Models via prior transitivity

2026-06-20

AI Summary

Existing methods struggle to simultaneously capture community structure and ordered hierarchical relationships in directed networks, often neglecting inter-block transitivity or failing to identify structurally equivalent ordered groups. This work proposes the Transitive Stochastic Block Model (TSBM), which, within a Bayesian framework, disentangles total interaction volume from directional imbalance and imposes hierarchical constraints only on the latter. TSBM innovatively integrates weak and strong stochastic transitivity priors with a non-exchangeable age-ordered partition prior to jointly infer the number of blocks and node assignments. It provides the first probabilistic framework for assessing whether observed data support a transitive block structure. Experiments demonstrate that TSBM significantly improves link prediction accuracy and block recovery in sparse networks, outperforming baselines on four of six real-world networks, while also revealing scenarios—such as citation networks—that violate transitivity assumptions.

Abstract

In directed networks, nodes may form groups with similar interaction patterns, while these groups may themselves follow an ordered structure. Existing methods typically treat these features separately, either clustering nodes without enforcing a coherent block order, or ranking individual nodes without allowing for structurally equivalent groups. We introduce the Transitive Stochastic Block Model (TSBM), a Bayesian model for directed weighted networks that uses transitivity-inducing priors to infer ordered blocks. The model separates the total volume of interaction between two nodes from the direction of interaction conditional on interaction occurring, so that hierarchy is imposed on directional imbalance rather than interaction frequency. We consider two order-restricted specifications: a flexible weak-stochastic-transitivity version, which excludes cyclic dominance patterns while allowing heterogeneous block-pair strengths, and a Toeplitz strong-stochastic-transitivity version, in which directional advantage increases with rank separation. Posterior inference is performed through a Gibbs sampler using Pólya-Gamma data augmentation. Since ordered block labels are not exchangeable, we introduce an age-ordered partition prior to infer the number of blocks jointly with node allocation. Simulation studies show that order-constrained priors improve prediction and partition recovery, especially in sparse networks. Across six empirical directed networks, the TSBM improves predictive performance in four cases and yields partitions with clearer ordered structure. The results also identify cases, such as nearly deterministic dominance networks or non-transitive citation networks, where imposing ordered blocks can harm prediction. The TSBM therefore provides a probabilistic framework for estimating ordered groups and assessing when a transitive block structure is supported by the data.

Problem

• directed networks
• stochastic block models
• ordered blocks
• transitivity
• hierarchical structure

Innovation

• Transitive Stochastic Block Model
• order-restricted priors
• directed networks
• Bayesian inference
• stochastic transitivity