# Higher-order motif analysis in hypergraphs

arXiv: https://arxiv.org/abs/2108.03192 · HTML: https://mlsanity.com/paper/2108.03192/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Aug 2021.

## Built on by (most cited first, 6 of 6)

- [Inference of hyperedges and overlapping communities in hypergraphs](https://mlsanity.com/paper/2204.05646/) (2022)
- [Non-adaptive Learning of Random Hypergraphs with Queries](https://mlsanity.com/paper/2501.12771/) (2025)
- [Broad Spectrum Structure Discovery in Large-Scale Higher-Order Networks](https://mlsanity.com/paper/2505.21748/) (2025)
- [Degree Heterogeneity in Higher-Order Networks: Inference in the Hypergraph boldsymbolβ-Model](https://mlsanity.com/paper/2307.02818/) (2023)
- [MARIOH: Multiplicity-Aware Hypergraph Reconstruction](https://mlsanity.com/paper/2504.00522/) (2025)
- [Learning Multi-Order Block Structure in Higher-Order Networks](https://mlsanity.com/paper/2511.21350/) (2025)

## Abstract (arXiv)

A deluge of new data on social, technological and biological networked systems suggests that a large number of interactions among system units are not limited to pairs, but rather involve a higher number of nodes. To properly encode such higher-order interactions, richer mathematical frameworks such as hypergraphs are needed, where hyperlinks describe connections among an arbitrary number of nodes. Here we introduce the concept of higher-order motifs, small connected subgraphs where vertices may be linked by interactions of any order. We provide lower and upper bounds on the number of higher-order motifs as a function of the motif size, and propose an efficient algorithm to extract complete higher-order motif profiles from empirical data. We identify different families of hypergraphs, characterized by distinct higher-order connectivity patterns at the local scale. We also capture evidences of structural reinforcement, a mechanism that associates higher strengths of higher-order interactions for the nodes that interact more at the pairwise level. Our work highlights the informative power of higher-order motifs, providing a first way to extract higher-order fingerprints in hypergraphs at the network microscale.