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Extending Undirected Graph Techniques to Directed Graphs via Category Theory

Author

Listed:
  • Sebastian Pardo-Guerra

    (Department of Bioengineering, University of California San Diego, La Jolla, CA 92037, USA
    Center for Engineered Natural Intelligence, University of California San Diego, La Jolla, CA 92037, USA
    These authors contributed equally to this work.)

  • Vivek Kurien George

    (Department of Bioengineering, University of California San Diego, La Jolla, CA 92037, USA
    Center for Engineered Natural Intelligence, University of California San Diego, La Jolla, CA 92037, USA
    These authors contributed equally to this work.)

  • Vikash Morar

    (Department of Bioengineering, University of California San Diego, La Jolla, CA 92037, USA
    Center for Engineered Natural Intelligence, University of California San Diego, La Jolla, CA 92037, USA
    These authors contributed equally to this work.)

  • Joshua Roldan

    (Department of Bioengineering, University of California San Diego, La Jolla, CA 92037, USA
    Center for Engineered Natural Intelligence, University of California San Diego, La Jolla, CA 92037, USA
    These authors contributed equally to this work.)

  • Gabriel Alex Silva

    (Department of Bioengineering, University of California San Diego, La Jolla, CA 92037, USA
    Center for Engineered Natural Intelligence, University of California San Diego, La Jolla, CA 92037, USA
    Department of Neurosciences, University of California San Diego, La Jolla, CA 92037, USA)

Abstract

We use Category Theory to construct a ‘bridge’ relating directed graphs with undirected graphs, such that the notion of direction is preserved. Specifically, we provide an isomorphism between the category of simple directed graphs and a category we call ‘prime graphs category’; this has as objects labeled undirected bipartite graphs (which we call prime graphs), and as morphisms undirected graph morphisms that preserve the labeling (which we call prime graph morphisms). This theoretical bridge allows us to extend undirected graph techniques to directed graphs by converting the directed graphs into prime graphs. To give a proof of concept, we show that our construction preserves topological features when applied to the problems of network alignment and spectral graph clustering.

Suggested Citation

  • Sebastian Pardo-Guerra & Vivek Kurien George & Vikash Morar & Joshua Roldan & Gabriel Alex Silva, 2024. "Extending Undirected Graph Techniques to Directed Graphs via Category Theory," Mathematics, MDPI, vol. 12(9), pages 1-20, April.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:9:p:1357-:d:1386067
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