Learning Multi-Order Block Structure in Higher-Order Networks

TL;DR

The paper tackles the limitation of assuming a single universal rule for all interaction orders in higher-order networks by introducing HyperMOSBM, a multi-order stochastic block model for hypergraphs. By partitioning the set of hyperedge sizes $\mathcal{O}=\{2,\dots,D\}$ into subsets with shared affinity patterns and optimizing partitions via cross-validated hyperlink prediction (AUC), the method reveals order-dependent mesoscale structure. Across synthetic and 14 empirical hypergraphs, multi-order partitions are prevalent and yield superior predictive accuracy and more interpretable communities than both the single-order and full-order models, with a practical heuristic ($\Delta_{\text{AUC}} \ge 0.01$) guiding when to adopt the multi-order approach. The framework also enables interpretable case studies (e.g., co-citation networks) and supports extensions to dynamics, temporal hypergraphs, and integration of node attributes, marking a step toward descriptive and predictive modeling of real-world higher-order systems.

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.

Figures (10)

• Figure 1: Validation of the multi-order model on synthetic hypergraphs with tunable heterogeneity of affinity patterns across interaction orders. All results are averaged over 100 independent instances; shaded regions represent 95% confidence intervals estimated via bootstrapping. (a) Proportion of instances where each partition of interaction orders (2, 3, and 4) was selected as optimal by our AUC-driven framework. (b) Gain in hyperlink prediction performance ($\Delta_{\text{AUC}}$) achieved by the multi-order model with the selected partition compared to the single-order model. (c) Community recovery performance, measured by cosine similarity, for three models: the multi-order model with the selected partition, the single-order model, and the full-order model.
• Figure 2: Correspondence between ground-truth classes and inferred communities in the high-school contact network. The panels show the averaged membership matrix inferred by (a) the single-order model and (b) our multi-order model. Each row corresponds to a ground-truth class, and each column represents an inferred community. We generated community labels using a large language model based on representative members and their classes (see Methods).
• Figure 3: Inferred community structures in the co-citation network. The panels show the averaged membership matrix inferred by (a) the single-order model and (b) our multi-order model. Each row corresponds to a computer science subfield, and each column represents an inferred community. We assigned community labels based on their representative papers using a large language model (see Methods for details).
• Figure 4: Disentangling the NLP and IR community. The Venn diagram compares the sets of papers classified as representative of the NLP and IR community by the single-order (Hy-MMSBM) and our multi-order (HyperMOSBM) models.
• Figure 5: Comparison of community membership vectors for papers with the largest changes in their entropy. In each panel, the left and right columns correspond to the single-order (Hy-MMSBM) and our multi-order (HyperMOSBM) models, respectively. Top panel: Top three papers with the largest entropy decrease, selected from those that became representative of the NLP and IR community in the multi-order model. Bottom panel: Top three papers with the largest entropy increase, selected from those that are no longer classified as representative of the NLP and IR community in the multi-order model.