Learning Multi-Order Block Structure in Higher-Order Networks

TL;DR

This paper introduces a multi-order stochastic block model for hypergraphs, capturing order-dependent interaction patterns, leading to improved prediction and more interpretable mesoscale structures in higher-order networks.

Contribution

It proposes a novel multi-order block structure framework that relaxes the single-order assumption, enhancing modeling flexibility and interpretability in higher-order network analysis.

Findings

Multi-order block structures are common in real-world networks.

Accounting for multiple orders improves hyperlink prediction performance.

The approach reveals more interpretable mesoscale organization.

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.