# Higher-order Graph Convolutional Network with Flower-Petals Laplacians on Simplicial Complexes

arXiv: https://arxiv.org/abs/2309.12971 · HTML: https://mlsanity.com/paper/2309.12971/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 81 references that are in the map.
- First posted on arXiv 22 Sep 2023.

## Builds on (most cited first, 30 of 43)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Pitfalls of Graph Neural Network Evaluation](https://mlsanity.com/paper/1811.05868/) (2018)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Adaptive Universal Generalized PageRank Graph Neural Network](https://mlsanity.com/paper/2006.07988/) (2020)
- [Multi-scale Attributed Node Embedding](https://mlsanity.com/paper/1909.13021/) (2019)
- [Graph Neural Networks in Recommender Systems: A Survey](https://mlsanity.com/paper/2011.02260/) (2020)
- [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [Diffusion-Convolutional Neural Networks](https://mlsanity.com/paper/1511.02136/) (2015)
- [Modeling polypharmacy side effects with graph convolutional networks](https://mlsanity.com/paper/1802.00543/) (2018)
- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation](https://mlsanity.com/paper/2106.10994/) (2021)
- [Two Sides of the Same Coin: Heterophily and Oversmoothing in Graph Convolutional Neural Networks](https://mlsanity.com/paper/2102.06462/) (2021)
- [Finding Global Homophily in Graph Neural Networks When Meeting Heterophily](https://mlsanity.com/paper/2205.07308/) (2022)
- [How Powerful are Spectral Graph Neural Networks](https://mlsanity.com/paper/2205.11172/) (2022)
- [Networks beyond pairwise interactions: structure and dynamics](https://mlsanity.com/paper/2006.01764/) (2020)
- [Simplicial Neural Networks](https://mlsanity.com/paper/2010.03633/) (2020)
- [Simplicial Closure and higher-order link prediction](https://mlsanity.com/paper/1802.06916/) (2018)
- [Propagation Kernels](https://mlsanity.com/paper/1410.3314/) (2014)
- [Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian](https://mlsanity.com/paper/1807.05044/) (2018)

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

- [What Improves the Generalization of Graph Transformers? A Theoretical Dive into the Self-attention and Positional Encoding](https://mlsanity.com/paper/2406.01977/) (2024)
- [FSW-GNN: A Bi-Lipschitz WL-Equivalent Graph Neural Network](https://mlsanity.com/paper/2410.09118/) (2024)
- [HOG-Diff: Higher-Order Guided Diffusion for Graph Generation](https://mlsanity.com/paper/2502.04308/) (2025)
- [Democratizing Large Language Model-Based Graph Data Augmentation via Latent Knowledge Graphs](https://mlsanity.com/paper/2502.13555/) (2025)
- [Infinite-Horizon Graph Filters: Leveraging Power Series to Enhance Sparse Information Aggregation](https://mlsanity.com/paper/2401.09943/) (2024)
- [Graph Neural Networks Need Cluster-Normalize-Activate Modules](https://mlsanity.com/paper/2412.04064/) (2024)
- [Transfer Entropy in Graph Convolutional Neural Networks](https://mlsanity.com/paper/2406.06632/) (2024)
- [GrokFormer: Graph Fourier Kolmogorov-Arnold Transformers](https://mlsanity.com/paper/2411.17296/) (2024)
- [Cooperative Network Learning for Large-Scale and Decentralized Graphs](https://mlsanity.com/paper/2311.02117/) (2023)
- [HC-GAE: The Hierarchical Cluster-based Graph Auto-Encoder for Graph Representation Learning](https://mlsanity.com/paper/2405.14742/) (2024)
- [HoGA: Higher-Order Graph Attention via Diversity-Aware k-Hop Sampling](https://mlsanity.com/paper/2411.12052/) (2024)
- [Leveraging Personalized PageRank and Higher-Order Topological Structures for Heterophily Mitigation in Graph Neural Networks](https://mlsanity.com/paper/2507.16347/) (2025)
- [Weisfeiler Lehman Test on Combinatorial Complexes: Generalized Expressive Power of Topological Neural Networks](https://mlsanity.com/paper/2605.00725/) (2026)
- [Collapsed Effective Operators for Higher-order Structures](https://mlsanity.com/paper/2606.23517/) (2026)
- [Higher-Order Molecular Grammars for Generative and Foundation Models in Chemistry](https://mlsanity.com/paper/2610.02186/) (2026)

## Abstract (arXiv)

Despite the recent successes of vanilla Graph Neural Networks (GNNs) on various tasks, their foundation on pairwise networks inherently limits their capacity to discern latent higher-order interactions in complex systems. To bridge this capability gap, we propose a novel approach exploiting the rich mathematical theory of simplicial complexes (SCs) - a robust tool for modeling higher-order interactions. Current SC-based GNNs are burdened by high complexity and rigidity, and quantifying higher-order interaction strengths remains challenging. Innovatively, we present a higher-order Flower-Petals (FP) model, incorporating FP Laplacians into SCs. Further, we introduce a Higher-order Graph Convolutional Network (HiGCN) grounded in FP Laplacians, capable of discerning intrinsic features across varying topological scales. By employing learnable graph filters, a parameter group within each FP Laplacian domain, we can identify diverse patterns where the filters' weights serve as a quantifiable measure of higher-order interaction strengths. The theoretical underpinnings of HiGCN's advanced expressiveness are rigorously demonstrated. Additionally, our empirical investigations reveal that the proposed model accomplishes state-of-the-art performance on a range of graph tasks and provides a scalable and flexible solution to explore higher-order interactions in graphs. Codes and datasets are available at https://github.com/Yiminghh/HiGCN.