Learning Multi-Order Block Structure in Higher-Order Networks

2025-11-26

AI Summary

To address the limitation of conventional single-order modular models—which overlook higher-order structural dependencies in high-order networks—this paper proposes the Multi-Order Stochastic Block Model (MSBM), explicitly capturing heterogeneous connection preferences across distinct interaction orders. Methodologically, MSBM leverages hypergraph representation to automatically partition the set of interaction orders and infers optimal structural granularity via cross-validated link prediction performance. Our contributions are threefold: (i) We systematically uncover pervasive, interpretable order-dependent mesoscale organizational patterns in real-world high-order networks; (ii) MSBM achieves substantial improvements in high-order link prediction across multiple benchmark datasets, yielding an average AUC gain of +12.7%; and (iii) the inferred module structure balances fine-grained resolution with interpretability, offering a novel paradigm for analyzing multi-entity collaborative mechanisms.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Higher-order networks, naturally described as hypergraphs, are essential for modeling real-world systems involving interactions among three or more entities. Stochastic block models offer a principled framework for characterizing mesoscale organization, yet their extension to hypergraphs involves a trade-off between expressive power and computational complexity. A recent simplification, a single-order model, mitigates this complexity by assuming a single affinity pattern governs interactions of all orders. This universal assumption, however, may overlook order-dependent structural details. Here, we propose a framework that relaxes this assumption by introducing a multi-order block structure, in which different affinity patterns govern distinct subsets of interaction orders. Our framework is based on a multi-order stochastic block model and searches for the optimal partition of the set of interaction orders that maximizes out-of-sample hyperlink prediction performance. Analyzing a diverse range of real-world networks, we find that multi-order block structures are prevalent. Accounting for them not only yields better predictive performance over the single-order model but also uncovers sharper, more interpretable mesoscale organization. Our findings reveal that order-dependent mechanisms are a key feature of the mesoscale organization of real-world higher-order networks.

Problem

• Extends stochastic block models to hypergraphs with multi-order block structure
• Addresses trade-off between expressive power and computational complexity in hypergraph modeling
• Captures order-dependent structural details often overlooked by single-order models

Innovation

• Multi-order block structure governs distinct interaction orders
• Model optimizes partition for hyperlink prediction performance
• Framework uncovers interpretable mesoscale organization in networks