mlsanityData to 14 Oct 2026

Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond

arXiv 2101.05510 ↗First posted 14 Jan 2021cs.SI cs.LG physics.soc-phcited by 82

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Abstract (arXiv)

In this tutorial, we provide a didactic treatment of the emerging topic of signal processing on higher-order networks. Drawing analogies from discrete and graph signal processing, we introduce the building blocks for processing data on simplicial complexes and hypergraphs, two common higher-order network abstractions that can incorporate polyadic relationships. We provide brief introductions to simplicial complexes and hypergraphs, with a special emphasis on the concepts needed for the processing of signals supported on these structures. Specifically, we discuss Fourier analysis, signal denoising, signal interpolation, node embeddings, and nonlinear processing through neural networks, using these two higher-order network models. In the context of simplicial complexes, we specifically focus on signal processing using the Hodge Laplacian matrix, a multi-relational operator that leverages the special structure of simplicial complexes and generalizes desirable properties of the Laplacian matrix in graph signal processing. For hypergraphs, we present both matrix and tensor representations, and discuss the trade-offs in adopting one or the other. We also highlight limitations and potential research avenues, both to inform practitioners and to motivate the contribution of new researchers to the area.

Key facts. Cited by 82 papers in the map (data through 14 Oct 2026). Builds on 57 of its 146 references that are in the map. First posted on arXiv 14 Jan 2021.

Builds on

57 of 146 references.

References · 57

  1. 2016Semi-Supervised Classification with Graph Convolutional Networkscited by 11,361
  2. 2017Inductive Representation Learning on Large Graphscited by 6,320
  3. 2019A Comprehensive Survey on Graph Neural Networkscited by 2,919
  4. 2016Convolutional Neural Networks on Graphs with Fast Localized Spectral Filteringcited by 2,089
  5. 2016Variational Graph Auto-Encoderscited by 1,382
  6. 2016Geometric deep learning: going beyond Euclidean datacited by 1,048
  7. 2018Hypergraph Neural Networkscited by 488
  8. 2017Representation Learning on Graphs: Methods and Applicationscited by 457
  9. 2017Graph Signal Processing: Overview, Challenges and Applicationscited by 413
  10. 2019Hypergraph Convolution and Hypergraph Attentioncited by 228
  11. 2016Tensor Decomposition for Signal Processing and Machine Learningcited by 224
  12. 2019Stability Properties of Graph Neural Networkscited by 127
38 more
  1. 2018Connecting the Dots: Identifying Network Structure via Graph Signal Processingcited by 111
  2. 2020Simplicial Neural Networkscited by 110
  3. 2019Topological Signal Processing over Simplicial Complexescited by 96
  4. 2018Convolutional Neural Network Architectures for Signals Supported on Graphscited by 95
  5. 2018Simplicial Closure and higher-order link predictioncited by 93
  6. 2018Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplaciancited by 86
  7. 2019Hyper-SAGNN: a self-attention based graph neural network for hypergraphscited by 69
  8. 2016Network Topology Inference from Spectral Templatescited by 58
  9. 2020HyperSAGE: Generalizing Inductive Representation Learning on Hypergraphscited by 54
  10. 2020Heterogeneous Hypergraph Embedding for Graph Classificationcited by 52
  11. 2019Random Walks on Hypergraphs with Edge-Dependent Vertex Weightscited by 45
  12. 2019HodgeNet: Graph Neural Networks for Edge Datacited by 39
  13. 2018Flow Smoothing and Denoising: Graph Signal Processing in the Edge-Spacecited by 37
  14. 2013The Total Variation on Hypergraphs - Learning on Hypergraphs Revisitedcited by 35
  15. 2016Autoregressive Moving Average Graph Filteringcited by 35
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  17. 2017Structural Deep Embedding for Hyper-Networkscited by 33
  18. 2017Inhomogeneous Hypergraph Clustering with Applicationscited by 31
  19. 2020Signal Processing on Directed Graphscited by 28
  20. 2018Submodular Hypergraphs: p-Laplacians, Cheeger Inequalities and Spectral Clusteringcited by 25
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  22. 2019Hypernetwork Science via High-Order Hypergraph Walkscited by 20
  23. 2019Introducing Hypergraph Signal Processing: Theoretical Foundation and Practical Applicationscited by 20
  24. 2020Hypergraph Random Walks, Laplacians, and Clusteringcited by 20
  25. 2015Sampling of graph signals with successive local aggregationscited by 19
  26. 2015Tensor Spectral Clustering for Partitioning Higher-order Network Structurescited by 15
  27. 2016Uniform Hypergraph Partitioning: Provable Tensor Methods and Sampling Techniquescited by 13
  28. 2017Cheeger Inequalities for Submodular Transformationscited by 13
  29. 2013Reflection methods for user-friendly submodular optimizationcited by 11
  30. 2017Hypergraph p-Laplacian: A Differential Geometry Viewcited by 11
  31. 2018Three hypergraph eigenvector centralitiescited by 11
  32. 2020Line Hypergraph Convolution Network: Applying Graph Convolution for Hypergraphscited by 11
  33. 2014Hypergraph Markov Operators, Eigenvalues and Approximation Algorithmscited by 9
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  35. 2019Quadratic Decomposable Submodular Function Minimization: Theory and Practice (Computation and Analysis of PageRank over Hypergraphs)cited by 7
  36. 2019A Notion of Harmonic Clustering in Simplicial Complexescited by 7
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  38. 2017Decomposable Submodular Function Minimization: Discrete and Continuouscited by 5

50 most cited of 57 All 57 (JSON)

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 9; 2022: 10; 2023: 20; 2024: 14; 2025: 13; 2026: 16.

82 papers.

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  1. 2024Position: Topological Deep Learning is the New Frontier for Relational Learningcited by 66
  2. 2022Graph Filters for Signal Processing and Machine Learning on Graphscited by 53
  3. 2022Equivariant Hypergraph Diffusion Neural Operatorscited by 47
  4. 2023Architectures of Topological Deep Learning: A Survey of Message-Passing Topological Neural Networkscited by 43
  5. 2022Simplicial Attention Networkscited by 41
  6. 2021Simplicial Convolutional Neural Networkscited by 40
  7. 2021Signal Processing on Cell Complexescited by 31
  8. 2023Convolutional Learning on Simplicial Complexescited by 27
  9. 2023Generalized Simplicial Attention Neural Networkscited by 27
  10. 2024TopoBench: A Framework for Benchmarking Topological Deep Learningcited by 24
  11. 2024TopoX: A Suite of Python Packages for Machine Learning on Topological Domainscited by 22
  12. 2024E(n) Equivariant Topological Neural Networkscited by 22
38 more
  1. 2022Simplicial Attention Neural Networkscited by 21
  2. 2023From Latent Graph to Latent Topology Inference: Differentiable Cell Complex Modulecited by 19
  3. 2022Simplicial Convolutional Filterscited by 18
  4. 2022Geometric Scattering on Measure Spacescited by 17
  5. 2023Hodge-Compositional Edge Gaussian Processescited by 16
  6. 2021Finite Impulse Response Filters for Simplicial Complexescited by 15
  7. 2021Untrained Graph Neural Networks for Denoisingcited by 15
  8. 2023Combinatorial Complexes: Bridging the Gap Between Cell Complexes and Hypergraphscited by 15
  9. 2021Signal processing on simplicial complexescited by 12
  10. 2024Attending to Topological Spaces: The Cellular Transformercited by 11
  11. 2023Dirac signal processing of higher-order topological signalscited by 9
  12. 2021Data-Centric AI Requires Rethinking Data Notioncited by 8
  13. 2022Convolutional Filtering in Simplicial Complexescited by 8
  14. 2023Topo-MLP : A Simplicial Network Without Message Passingcited by 7
  15. 2023Topological Point Cloud Clusteringcited by 5
  16. 2023Representing Edge Flows on Graphs via Sparse Cell Complexescited by 5
  17. 2023Disentangling the Spectral Properties of the Hodge Laplacian: Not All Small Eigenvalues Are Equalcited by 5
  18. 2021Outlier Detection for Trajectories via Flow-embeddingscited by 4
  19. 2024Point-Level Topological Representation Learning on Point Cloudscited by 4
  20. 2024Demystifying Higher-Order Graph Neural Networkscited by 4
  21. 2025Topological Dictionary Learningcited by 4
  22. 2021Co-clustering Vertices and Hyperedges via Spectral Hypergraph Partitioningcited by 3
  23. 2023Learning graphs and simplicial complexes from datacited by 3
  24. 2024Imputation of Time-varying Edge Flows in Graphs by Multilinear Kernel Regression and Manifold Learningcited by 3
  25. 2024Graph Classification Gaussian Processes via Hodgelet Spectral Featurescited by 3
  26. 2025Matched Topological Subspace Detectorcited by 3
  27. 2021Free Energy Node Embedding via Generalized Skip-gram with Negative Samplingcited by 2
  28. 2022Multiscale methods for signal selection in single-cell datacited by 2
  29. 2022Stable and Transferable Hyper-Graph Neural Networkscited by 2
  30. 2023Zoo Guide to Network Embeddingcited by 2
  31. 2024Modeling Edge-Specific Node Features through Co-Representation Neural Hypergraph Diffusioncited by 2
  32. 2025Continuous Simplicial Neural Networkscited by 2
  33. 2025Topological Adaptive Least Mean Squares Algorithms over Simplicial Complexescited by 2
  34. 2025Simplicial Gaussian Models: Representation and Inferencecited by 2
  35. 2022Hypergraphs with Edge-Dependent Vertex Weights: p-Laplacians and Spectral Clusteringcited by 1
  36. 2023Hypergraphs with Edge-Dependent Vertex Weights: Spectral Clustering based on the 1-Laplaciancited by 1
  37. 2023A Unified View Between Tensor Hypergraph Neural Networks And Signal Denoisingcited by 1
  38. 2024DLGNet: Hyperedge Classification through Directed Line Graphs for Chemical Reactionscited by 1

50 most cited of 82 All 82 (JSON)