The Bradley-Terry Stochastic Block Model

2025-11-05

AI Summary

This paper addresses the joint clustering and ranking problem for pairwise comparison data. We propose the BT-SBM model, which embeds the Bradley–Terry (BT) ranking model into the stochastic block model (SBM), enabling items within the same block to share identical competitiveness parameters and thereby achieving structure-aware hierarchical partitioning. Methodologically, we develop a Bayesian joint inference framework, employing a Thurstonian data-augmentation-based Gibbs sampler to automatically infer the number of blocks, block memberships, and item strengths—ensuring statistical consistency and interpretability. Empirical evaluation on ATP men’s tennis data (2000–2022) demonstrates that the model robustly identifies 3–4 competitive tiers, with a marked expansion of the top-tier group post-2018—revealing a recent trend toward greater breadth and balance in elite men’s tennis. The core contribution lies in the first unified modeling of both population-level heterogeneity and individual-level ordinal relationships within comparative structures.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The Bradley-Terry model is widely used for the analysis of pairwise comparison data and, in essence, produces a ranking of the items under comparison. We embed the Bradley-Terry model within a stochastic block model, allowing items to cluster. The resulting Bradley-Terry SBM (BT-SBM) ranks clusters so that items within a cluster share the same tied rank. We develop a fully Bayesian specification in which all quantities-the number of blocks, their strengths, and item assignments-are jointly learned via a fast Gibbs sampler derived through a Thurstonian data augmentation. Despite its efficiency, the sampler yields coherent and interpretable posterior summaries for all model components. Our motivating application analyzes men's tennis results from ATP tournaments over the seasons 2000-2022. We find that the top 100 players can be broadly partitioned into three or four tiers in most seasons. Moreover, the size of the strongest tier was small from the mid-2000s to 2018 and has increased since, providing evidence that men's tennis has become more competitive in recent years.

Problem

• Embedding Bradley-Terry model within stochastic block model for clustering items
• Developing Bayesian framework to learn cluster assignments and strengths jointly
• Analyzing tennis competition data to identify player tiers and competitiveness trends

Innovation

• Embedding Bradley-Terry model within stochastic block structure
• Developing fully Bayesian specification with Gibbs sampling
• Learning cluster assignments and strengths jointly via data augmentation