# Two's company, three (or more) is a simplex: Algebraic-topological tools for understanding higher-order structure in neural data

arXiv: https://arxiv.org/abs/1601.01704 · HTML: https://mlsanity.com/paper/1601.01704/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jan 2016.

## Built on by (most cited first, 22 of 22)

- [Cell Attention Networks](https://mlsanity.com/paper/2209.08179/) (2022)
- [E(n) Equivariant Topological Neural Networks](https://mlsanity.com/paper/2405.15429/) (2024)
- [SGAT: Simplicial Graph Attention Network](https://mlsanity.com/paper/2207.11761/) (2022)
- [CIN++: Enhancing Topological Message Passing](https://mlsanity.com/paper/2306.03561/) (2023)
- [Inference of hyperedges and overlapping communities in hypergraphs](https://mlsanity.com/paper/2204.05646/) (2022)
- [Signal processing on simplicial complexes](https://mlsanity.com/paper/2106.07471/) (2021)
- [MANTRA: The Manifold Triangulations Assemblage](https://mlsanity.com/paper/2410.02392/) (2024)
- [Dirac signal processing of higher-order topological signals](https://mlsanity.com/paper/2301.10137/) (2023)
- [Helmholtzian Eigenmap: Topological feature discovery & edge flow learning from point cloud data](https://mlsanity.com/paper/2103.07626/) (2021)
- [Structure and inference in hypergraphs with node attributes](https://mlsanity.com/paper/2311.03857/) (2023)
- [Minimal Cycle Representatives in Persistent Homology using Linear Programming: an Empirical Study with User's Guide](https://mlsanity.com/paper/2105.07025/) (2021)
- [A Topological Deep Learning Framework for Neural Spike Decoding](https://mlsanity.com/paper/2212.05037/) (2022)
- [Topological Dictionary Learning](https://mlsanity.com/paper/2503.11470/) (2025)
- [Stable and Transferable Hyper-Graph Neural Networks](https://mlsanity.com/paper/2211.06513/) (2022)
- [Persistent Homological State-Space Estimation of Functional Human Brain Networks at Rest](https://mlsanity.com/paper/2201.00087/) (2022)
- [Multiscale Hodge Scattering Networks for Data Analysis](https://mlsanity.com/paper/2311.10270/) (2023)
- [Dimensionality reduction for homological stability and global structure preservation](https://mlsanity.com/paper/2503.03156/) (2025)
- [Topological data analysis of truncated contagion maps](https://mlsanity.com/paper/2203.01720/) (2022)
- [Directed Semi-Simplicial Learning with Applications to Brain Activity Decoding](https://mlsanity.com/paper/2505.17939/) (2025)
- [The Human Brain as a Combinatorial Complex](https://mlsanity.com/paper/2511.20692/) (2025)
- [Learning Multi-Order Block Structure in Higher-Order Networks](https://mlsanity.com/paper/2511.21350/) (2025)
- [Colored Markov Random Fields for Probabilistic Topological Modeling](https://mlsanity.com/paper/2512.03727/) (2025)

## Abstract (arXiv)

The language of graph theory, or network science, has proven to be an exceptional tool for addressing myriad problems in neuroscience. Yet, the use of networks is predicated on a critical simplifying assumption: that the quintessential unit of interest in a brain is a dyad -- two nodes (neurons or brain regions) connected by an edge. While rarely mentioned, this fundamental assumption inherently limits the types of neural structure and function that graphs can be used to model. Here, we describe a generalization of graphs that overcomes these limitations, thereby offering a broad range of new possibilities in terms of modeling and measuring neural phenomena. Specifically, we explore the use of simplicial complexes, a theoretical notion developed in the field of mathematics known as algebraic topology, which is now becoming applicable to real data due to a rapidly growing computational toolset. We review the underlying mathematical formalism as well as the budding literature applying simplicial complexes to neural data, from electrophysiological recordings in animal models to hemodynamic fluctuations in humans. Based on the exceptional flexibility of the tools and recent ground-breaking insights into neural function, we posit that this framework has the potential to eclipse graph theory in unraveling the fundamental mysteries of cognition.