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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.

Installation

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  1. Clone this repository.

    git clone https://github.com/stefaniaebli/simplicial_neural_networks.git
cd simplicial_neural_networks

  2. Create the environment.

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

Notebooks

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/impute_citations.py ./data/s2_3_collaboration_complex ./experiments/output 150250 30

Data

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 https://s3-us-west-2.amazonaws.com/ai2-s2-research-public/open-corpus/2018-05-03/manifest.txt -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.
python s2_1_corpus_to_bipartite.py
# Clean and downsample that bipartite graph.
python s2_2_downsample_bipartite.py
# Project the bipartite graph to a graph between authors.
python s2_3_bipartite_to_graphs.py

    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.
python s2_4_bipartite_to_downsampled.py
# From a bipartite graph to a simplicial complex with k-cochains.
python s2_5_bipartite_to_complex.py
# From a simplicial complex to k-degree Laplacians.
python s2_6_complex_to_laplacians.py
# Artificially insert missing data on k-cochains.
python s2_7_cochains_to_missingdata.py

    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.

@inproceedings{snn,
  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 = {https://arxiv.org/abs/2010.03633},
}