# The topological Dirac equation of networks and simplicial complexes

arXiv: https://arxiv.org/abs/2106.02929 · HTML: https://mlsanity.com/paper/2106.02929/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jun 2021.

## Built on by (most cited first, 11 of 11)

- [Generalized Simplicial Attention Neural Networks](https://mlsanity.com/paper/2309.02138/) (2023)
- [Gaussian Processes on Cellular Complexes](https://mlsanity.com/paper/2311.01198/) (2023)
- [Combinatorial Complexes: Bridging the Gap Between Cell Complexes and Hypergraphs](https://mlsanity.com/paper/2312.09504/) (2023)
- [Dirac signal processing of higher-order topological signals](https://mlsanity.com/paper/2301.10137/) (2023)
- [Topological Neural Operators](https://mlsanity.com/paper/2606.09806/) (2026)
- [Matched Topological Subspace Detector](https://mlsanity.com/paper/2504.05892/) (2025)
- [Dirac--Bianconi Graph Neural Networks -- Enabling Non-Diffusive Long-Range Graph Predictions](https://mlsanity.com/paper/2407.12419/) (2024)
- [Dirac-Equation Signal Processing: Physics Boosts Topological Machine Learning](https://mlsanity.com/paper/2412.05132/) (2024)
- [Stationarity and Spectral Characterization of Random Signals on Simplicial Complexes](https://mlsanity.com/paper/2602.03055/) (2026)
- [Topological Signal Processing: An Application-Oriented Tutorial](https://mlsanity.com/paper/2605.22853/) (2026)
- [Collapsed Effective Operators for Higher-order Structures](https://mlsanity.com/paper/2606.23517/) (2026)

## Abstract (arXiv)

We define the topological Dirac equation describing the evolution of a topological wave function on networks or on simplicial complexes. On networks, the topological wave function describes the dynamics of topological signals or cochains, i.e. dynamical signals defined both on nodes and on links. On simplicial complexes the wave function is also defined on higher-dimensional simplices. Therefore the topological wave function satisfies a relaxed condition of locality as it acquires the same value along simplices of dimension larger than zero. The topological Dirac equation defines eigenstates whose dispersion relation is determined by the spectral properties of the Dirac (or chiral) operator defined on networks and generalized network structures including simplicial complexes and multiplex networks. On simplicial complexes the Dirac equation leads to multiple energy bands. On multiplex networks the topological Dirac equation can be generalized to distinguish between different mutlilinks leading to a natural definition of rotations of the topological spinor. The topological Dirac equation is here initially formulated on a spatial network or simplicial complex for describing the evolution of the topological wave function in continuous time. This framework is also extended to treat the topological Dirac equation on 1+d spaces describing a discrete space-time with one temporal dimension and d spatial dimensions with d∈ 1,2,3. This work includes also the discussion of numerical results obtained by implementing the topological Dirac equation on simplicial complex models and on real simple and multiplex network data.