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Simplicial Neural Networks

Stefania Ebli, Michaël Defferrard, Gard Spreemann

We present simplicial neural networks (SNNs), a generalization of graph neural networks to data that live on a class of topological spaces called simplicial complexes. These are natural multi-dimensional extensions of graphs that encode not only pairwise relationships but also higher-order interactions between vertices—allowing us to consider richer data, including vector fields and n-fold collaboration networks. We define an appropriate notion of convolution that we leverage to construct the desired convolutional neural networks. We test the SNNs on the task of imputing missing data on coauthorship complexes.


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  1. Clone this repository.

    git clone
    cd simplicial_neural_networks
  2. Create the environment.

    CONDA_CHANNEL_PRIORITY=flexible conda env create -f environment.yml
    conda activate snn


Reproducing our results

Run the below to train a SNN to impute missing data (citations) on the simplicial complex (which encodes collaborations between authors).

python ./experiments/ ./data/s2_3_collaboration_complex ./experiments/output 150250 30


The data necessary to reproduce our experiment are found in the ./data/s2_3_collaboration_complex folder. The below three stages will recreate them.

  1. Download the full archive of the Open Research Corpus from Semantic Scholar, version 2018-05-03, which contains over 39 million published research papers in Computer Science, Neuroscience, and Biomedical.

    wget -i -P data/s2_1_raw/

    This step populates the ./data/s2_1_raw folder.

  2. Create a bipartite graph, whose vertices are papers (39,219,709 of them) in one part and authors (12,862,455 of them) in the other. A paper is connected to all its co-authors, and an author is connected to all the papers they wrote, leading to 139,268,795 edges. A citation count (the number of times the paper was cited) is available for each paper (from 0 to 37,230 citations per paper).

    # Create a bipartite graph from Semantic Scholar.
    # Clean and downsample that bipartite graph.
    # Project the bipartite graph to a graph between authors.

    Those steps populate the ./data/s2_2_bipartite_graph folder. Alternatively, that processed data is available at doi:10.5281/zenodo.4144319.

  3. Build the collaboration complex (where each collaboration of authors is represented by a simplex) and citation cochains (which are the number of citations attributed to the collaborations).

    # Downsample the bipartite graph to have a connected simplicial complex.
    # From a bipartite graph to a simplicial complex with k-cochains.
    # From a simplicial complex to k-degree Laplacians.
    # Artificially insert missing data on k-cochains.

    Those steps populate the ./data/s2_3_collaboration_complex folder.

License & citation

The content of this repository is released under the terms of the MIT license. Please cite our paper if you use it.

  title = {Simplicial Neural Networks},
  author = {Ebli, Stefania and Defferrard, Michaël and Spreemann, Gard},
  booktitle = {Topological Data Analysis and Beyond workshop at NeurIPS},
  year = {2020},
  archiveprefix = {arXiv},
  eprint = {2010.03633},
  url = {},


Simplicial neural networks (SNNs), a generalization of graph neural networks to data that live on a class of topological spaces called simplicial complexes.





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