# Learning Multi-Order Block Structure in Higher-Order Networks

arXiv: https://arxiv.org/abs/2511.21350 · HTML: https://mlsanity.com/paper/2511.21350/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 55 of its 99 references that are in the map.
- First posted on arXiv 26 Nov 2025.

## Builds on (most cited first, 30 of 55)

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- [Heterogeneous Information Network Embedding for Recommendation](https://mlsanity.com/paper/1711.10730/) (2017)
- [Networks beyond pairwise interactions: structure and dynamics](https://mlsanity.com/paper/2006.01764/) (2020)
- [Community detection in networks: A user guide](https://mlsanity.com/paper/1608.00163/) (2016)
- [Multilayer Networks](https://mlsanity.com/paper/1309.7233/) (2013)
- [The physics of higher-order interactions in complex systems](https://mlsanity.com/paper/2110.06023/) (2021)
- [Contact patterns in a high school: a comparison between data collected using wearable sensors, contact diaries and friendship surveys](https://mlsanity.com/paper/1506.03645/) (2015)
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- [The structure and dynamics of multilayer networks](https://mlsanity.com/paper/1407.0742/) (2014)
- [Generative hypergraph clustering: from blockmodels to modularity](https://mlsanity.com/paper/2101.09611/) (2021)
- [A Review of Stochastic Block Models and Extensions for Graph Clustering](https://mlsanity.com/paper/1903.00114/) (2019)
- [The ground truth about metadata and community detection in networks](https://mlsanity.com/paper/1608.05878/) (2016)
- [Layer aggregation and reducibility of multilayer interconnected networks](https://mlsanity.com/paper/1405.0425/) (2014)
- [Clustering in graphs and hypergraphs with categorical edge labels](https://mlsanity.com/paper/1910.09943/) (2019)
- [Structure and inference in annotated networks](https://mlsanity.com/paper/1507.04001/) (2015)
- [Evolutionary dynamics of higher-order interactions in social networks](https://mlsanity.com/paper/2001.10313/) (2020)
- [Learning Latent Block Structure in Weighted Networks](https://mlsanity.com/paper/1404.0431/) (2014)
- [Consistency of Spectral Hypergraph Partitioning under Planted Partition Model](https://mlsanity.com/paper/1505.01582/) (2015)
- [Two's company, three (or more) is a simplex: Algebraic-topological tools for understanding higher-order structure in neural data](https://mlsanity.com/paper/1601.01704/) (2016)
- [How Do Hyperedges Overlap in Real-World Hypergraphs? -- Patterns, Measures, and Generators](https://mlsanity.com/paper/2101.07480/) (2021)
- [Data on face-to-face contacts in an office building suggests a low-cost vaccination strategy based on community linkers](https://mlsanity.com/paper/1409.7017/) (2014)
- [Community detection, link prediction, and layer interdependence in multilayer networks](https://mlsanity.com/paper/1701.01369/) (2017)
- [Evaluating Overfit and Underfit in Models of Network Community Structure](https://mlsanity.com/paper/1802.10582/) (2018)
- [A Survey on Hypergraph Mining: Patterns, Tools, and Generators](https://mlsanity.com/paper/2401.08878/) (2024)
- [Can co-location be used as a proxy for face-to-face contacts?](https://mlsanity.com/paper/1712.06346/) (2017)
- [Inference of hyperedges and overlapping communities in hypergraphs](https://mlsanity.com/paper/2204.05646/) (2022)
- [Higher-order interactions shape collective dynamics differently in hypergraphs and simplicial complexes](https://mlsanity.com/paper/2203.03060/) (2022)

## Abstract (arXiv)

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.