# Simplicial Convolutional Neural Networks

arXiv: https://arxiv.org/abs/2110.02585 · HTML: https://mlsanity.com/paper/2110.02585/

## Key facts

- Cited by 40 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 18 references that are in the map.
- First posted on arXiv 6 Oct 2021.

## Builds on (most cited first, 13 of 13)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [Simplicial Neural Networks](https://mlsanity.com/paper/2010.03633/) (2020)
- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [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)
- [Graphs, Convolutions, and Neural Networks: From Graph Filters to Graph Neural Networks](https://mlsanity.com/paper/2003.03777/) (2020)
- [HodgeNet: Graph Neural Networks for Edge Data](https://mlsanity.com/paper/1912.02354/) (2019)
- [Flow Smoothing and Denoising: Graph Signal Processing in the Edge-Space](https://mlsanity.com/paper/1808.02111/) (2018)
- [EdgeNets:Edge Varying Graph Neural Networks](https://mlsanity.com/paper/2001.07620/) (2020)
- [Finite Impulse Response Filters for Simplicial Complexes](https://mlsanity.com/paper/2103.12587/) (2021)

## Built on by (most cited first, 30 of 40)

- [Architectures of Topological Deep Learning: A Survey of Message-Passing Topological Neural Networks](https://mlsanity.com/paper/2304.10031/) (2023)
- [Simplicial Attention Networks](https://mlsanity.com/paper/2204.09455/) (2022)
- [Convolutional Learning on Simplicial Complexes](https://mlsanity.com/paper/2301.11163/) (2023)
- [Generalized Simplicial Attention Neural Networks](https://mlsanity.com/paper/2309.02138/) (2023)
- [Simplicial Attention Neural Networks](https://mlsanity.com/paper/2203.07485/) (2022)
- [Simplicial Convolutional Filters](https://mlsanity.com/paper/2201.11720/) (2022)
- [Hodge-Compositional Edge Gaussian Processes](https://mlsanity.com/paper/2310.19450/) (2023)
- [Higher-order Graph Convolutional Network with Flower-Petals Laplacians on Simplicial Complexes](https://mlsanity.com/paper/2309.12971/) (2023)
- [Topo-MLP : A Simplicial Network Without Message Passing](https://mlsanity.com/paper/2312.11862/) (2023)
- [A Topological Deep Learning Framework for Neural Spike Decoding](https://mlsanity.com/paper/2212.05037/) (2022)
- [CAT-Walk: Inductive Hypergraph Learning via Set Walks](https://mlsanity.com/paper/2306.11147/) (2023)
- [Diss-l-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms](https://mlsanity.com/paper/2410.02622/) (2024)
- [Pooling Strategies for Simplicial Convolutional Networks](https://mlsanity.com/paper/2210.05490/) (2022)
- [Homological Neural Networks: A Sparse Architecture for Multivariate Complexity](https://mlsanity.com/paper/2306.15337/) (2023)
- [HOPSE: Scalable Higher-Order Positional and Structural Encoder for Combinatorial Representations](https://mlsanity.com/paper/2505.15405/) (2025)
- [Homological Convolutional Neural Networks](https://mlsanity.com/paper/2308.13816/) (2023)
- [Demystifying Higher-Order Graph Neural Networks](https://mlsanity.com/paper/2406.12841/) (2024)
- [Topological Schr"odinger Bridge Matching](https://mlsanity.com/paper/2504.04799/) (2025)
- [Binarized Simplicial Convolutional Neural Networks](https://mlsanity.com/paper/2405.04098/) (2024)
- [Imputation of Time-varying Edge Flows in Graphs by Multilinear Kernel Regression and Manifold Learning](https://mlsanity.com/paper/2409.05135/) (2024)
- [Matched Topological Subspace Detector](https://mlsanity.com/paper/2504.05892/) (2025)
- [Hypergraph Neural Sheaf Diffusion: A Symmetric Simplicial Set Framework for Higher-Order Learning](https://mlsanity.com/paper/2505.05702/) (2025)
- [Stable and Transferable Hyper-Graph Neural Networks](https://mlsanity.com/paper/2211.06513/) (2022)
- [Continuous Simplicial Neural Networks](https://mlsanity.com/paper/2503.12919/) (2025)
- [Graph Signal Adaptive Message Passing](https://mlsanity.com/paper/2410.17629/) (2024)
- [Stationarity and Spectral Characterization of Random Signals on Simplicial Complexes](https://mlsanity.com/paper/2602.03055/) (2026)
- [NervePool: A Simplicial Pooling Layer](https://mlsanity.com/paper/2305.06315/) (2023)
- [Spectral Convergence of Complexon Shift Operators](https://mlsanity.com/paper/2309.07169/) (2023)
- [Hodge-Aware Contrastive Learning](https://mlsanity.com/paper/2309.07364/) (2023)
- [Learning From Simplicial Data Based on Random Walks and 1D Convolutions](https://mlsanity.com/paper/2404.03434/) (2024)

## Abstract (arXiv)

Graphs can model networked data by representing them as nodes and their pairwise relationships as edges. Recently, signal processing and neural networks have been extended to process and learn from data on graphs, with achievements in tasks like graph signal reconstruction, graph or node classifications, and link prediction. However, these methods are only suitable for data defined on the nodes of a graph. In this paper, we propose a simplicial convolutional neural network (SCNN) architecture to learn from data defined on simplices, e.g., nodes, edges, triangles, etc. We study the SCNN permutation and orientation equivariance, complexity, and spectral analysis. Finally, we test the SCNN performance for imputing citations on a coauthorship complex.