# Path homologies of deep feedforward networks

arXiv: https://arxiv.org/abs/1910.07617 · HTML: https://mlsanity.com/paper/1910.07617/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Oct 2019.

## Built on by (most cited first, 6 of 6)

- [Predicting the generalization gap in neural networks using topological data analysis](https://mlsanity.com/paper/2203.12330/) (2022)
- [Characterizing and Measuring the Similarity of Neural Networks with Persistent Homology](https://mlsanity.com/paper/2101.07752/) (2021)
- [Persistent Homology Captures the Generalization of Neural Networks Without A Validation Set](https://mlsanity.com/paper/2106.00012/) (2021)
- [Betti numbers of attention graphs is all you really need](https://mlsanity.com/paper/2207.01903/) (2022)
- [Topology Structure Optimization of Reservoirs Using GLMY Homology](https://mlsanity.com/paper/2509.11612/) (2025)
- [Persistent reachability homology in machine learning applications](https://mlsanity.com/paper/2511.04825/) (2025)

## Abstract (arXiv)

We provide a characterization of two types of directed homology for fully-connected, feedforward neural network architectures. These exact characterizations of the directed homology structure of a neural network architecture are the first of their kind. We show that the directed flag homology of deep networks reduces to computing the simplicial homology of the underlying undirected graph, which is explicitly given by Euler characteristic computations. We also show that the path homology of these networks is non-trivial in higher dimensions and depends on the number and size of the layers within the network. These results provide a foundation for investigating homological differences between neural network architectures and their realized structure as implied by their parameters.