# Simplicial complexes: higher-order spectral dimension and dynamics

arXiv: https://arxiv.org/abs/2001.05934 · HTML: https://mlsanity.com/paper/2001.05934/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jan 2020.

## Built on by (most cited first, 9 of 9)

- [Signal processing on simplicial complexes](https://mlsanity.com/paper/2106.07471/) (2021)
- [Dirac signal processing of higher-order topological signals](https://mlsanity.com/paper/2301.10137/) (2023)
- [Homological Neural Networks: A Sparse Architecture for Multivariate Complexity](https://mlsanity.com/paper/2306.15337/) (2023)
- [Homological Convolutional Neural Networks](https://mlsanity.com/paper/2308.13816/) (2023)
- [SPHINX: Structural Prediction using Hypergraph Inference Network](https://mlsanity.com/paper/2410.03208/) (2024)
- [Dirac-Equation Signal Processing: Physics Boosts Topological Machine Learning](https://mlsanity.com/paper/2412.05132/) (2024)
- [Chordless cycle filtrations for dimensionality detection in complex networks via topological data analysis](https://mlsanity.com/paper/2509.08350/) (2025)
- [Deep Medial Voxels: Learned Medial Axis Approximations for Anatomical Shape Modeling](https://mlsanity.com/paper/2403.11790/) (2024)
- [A cohomology-based Gromov-Hausdorff metric approach for quantifying molecular similarity](https://mlsanity.com/paper/2411.13887/) (2024)

## Abstract (arXiv)

Simplicial complexes constitute the underlying topology of interacting complex systems including among the others brain and social interaction networks. They are generalized network structures that allow to go beyond the framework of pairwise interactions and to capture the many-body interactions between two or more nodes strongly affecting dynamical processes. In fact, the simplicial complexes topology allows to assign a dynamical variable not only to the nodes of the interacting complex systems but also to links, triangles, and so on. Here we show evidence that the dynamics defined on simplices of different dimensions can be significantly different even if we compare dynamics of simplices belonging to the same simplicial complex. By investigating the spectral properties of the simplicial complex model called "Network Geometry with Flavor" we provide evidence that the up and down higher-order Laplacians can have a finite spectral dimension whose value increases as the order of the Laplacian increases. Finally we discuss the implications of this result for higher-order diffusion defined on simplicial complexes.