# Combinatorial Complexes: Bridging the Gap Between Cell Complexes and Hypergraphs

arXiv: https://arxiv.org/abs/2312.09504 · HTML: https://mlsanity.com/paper/2312.09504/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 28 references that are in the map.
- First posted on arXiv 15 Dec 2023.

## Builds on (most cited first, 12 of 12)

- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Networks beyond pairwise interactions: structure and dynamics](https://mlsanity.com/paper/2006.01764/) (2020)
- [Simplicial Neural Networks](https://mlsanity.com/paper/2010.03633/) (2020)
- [Topological Deep Learning: Going Beyond Graph Data](https://mlsanity.com/paper/2206.00606/) (2022)
- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [Principled Simplicial Neural Networks for Trajectory Prediction](https://mlsanity.com/paper/2102.10058/) (2021)
- [Graph signal processing for machine learning: A review and new perspectives](https://mlsanity.com/paper/2007.16061/) (2020)
- [Cell Complex Neural Networks](https://mlsanity.com/paper/2010.00743/) (2020)
- [What are higher-order networks?](https://mlsanity.com/paper/2104.11329/) (2021)
- [The why, how, and when of representations for complex systems](https://mlsanity.com/paper/2006.02870/) (2020)
- [The topological Dirac equation of networks and simplicial complexes](https://mlsanity.com/paper/2106.02929/) (2021)
- [Higher-order signal processing with the Dirac operator](https://mlsanity.com/paper/2212.10196/) (2022)

## Built on by (most cited first, 15 of 15)

- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [TopoBench: A Framework for Benchmarking Topological Deep Learning](https://mlsanity.com/paper/2406.06642/) (2024)
- [Copresheaf Topological Neural Networks: A Generalized Deep Learning Framework](https://mlsanity.com/paper/2505.21251/) (2025)
- [Attending to Topological Spaces: The Cellular Transformer](https://mlsanity.com/paper/2405.14094/) (2024)
- [HOG-Diff: Higher-Order Guided Diffusion for Graph Generation](https://mlsanity.com/paper/2502.04308/) (2025)
- [Demystifying Higher-Order Graph Neural Networks](https://mlsanity.com/paper/2406.12841/) (2024)
- [Topological Neural Operators](https://mlsanity.com/paper/2606.09806/) (2026)
- [ICML Topological Deep Learning Challenge 2024: Beyond the Graph Domain](https://mlsanity.com/paper/2409.05211/) (2024)
- [HyperAgent: Leveraging Hypergraphs for Topology Optimization in Multi-Agent Communication](https://mlsanity.com/paper/2510.10611/) (2025)
- [CuMPerLay: Learning Cubical Multiparameter Persistence Vectorizations](https://mlsanity.com/paper/2510.12795/) (2025)
- [TopoOR: A Unified Topological Scene Representation for the Operating Room](https://mlsanity.com/paper/2603.09466/) (2026)
- [Weisfeiler Lehman Test on Combinatorial Complexes: Generalized Expressive Power of Topological Neural Networks](https://mlsanity.com/paper/2605.00725/) (2026)
- [No Triangulation Without Representation: Generalization in Topological Deep Learning](https://mlsanity.com/paper/2605.06467/) (2026)
- [Periodic Topological Deep Learning for Polymer Design and Discovery](https://mlsanity.com/paper/2605.26833/) (2026)
- [Collapsed Effective Operators for Higher-order Structures](https://mlsanity.com/paper/2606.23517/) (2026)

## Abstract (arXiv)

Graph-based signal processing techniques have become essential for handling data in non-Euclidean spaces. However, there is a growing awareness that these graph models might need to be expanded into `higher-order' domains to effectively represent the complex relations found in high-dimensional data. Such higher-order domains are typically modeled either as hypergraphs, or as simplicial, cubical or other cell complexes. In this context, cell complexes are often seen as a subclass of hypergraphs with additional algebraic structure that can be exploited, e.g., to develop a spectral theory. In this article, we promote an alternative perspective. We argue that hypergraphs and cell complexes emphasize different types of relations, which may have different utility depending on the application context. Whereas hypergraphs are effective in modeling set-type, multi-body relations between entities, cell complexes provide an effective means to model hierarchical, interior-to-boundary type relations. We discuss the relative advantages of these two choices and elaborate on the previously introduced concept of a combinatorial complex that enables co-existing set-type and hierarchical relations. Finally, we provide a brief numerical experiment to demonstrate that this modelling flexibility can be advantageous in learning tasks.