Recursive Computation of Path Homology for Stratified Digraphs

TL;DR

This paper introduces a recursive algorithm that efficiently computes high-dimensional path homologies of stratified digraphs, enabling analysis of complex neural network models with reduced computational effort.

Contribution

The paper presents a novel recursive method for calculating full-depth path homology in stratified digraphs, improving computational efficiency over existing algorithms.

Findings

Algorithm significantly reduces computation time for high-dimensional homologies.

Effective for full-depth persistent homologies and maximal path homology in acyclic digraphs.

Numerical experiments demonstrate superior performance with increasing graph depth.

Abstract

Stratified digraphs are popular models for feedforward neural networks. However, computation of their path homologies has been limited to low dimensions due to high computational complexity. A recursive algorithm is proposed to compute certain high-dimensional (reduced) path homologies of stratified digraphs. By recursion on matrix representations of homologies of subgraphs, the algorithm efficiently computes the full-depth path homology of a stratified digraph, i.e. homology with dimension equal to the depth of the graph. The algorithm can be used to compute full-depth persistent homologies and for acyclic digraphs, the maximal path homology, i.e., path homology with dimension equal to the maximum path length of a graph. Numerical experiments show that the algorithm has a significant advantage over the general algorithm in computation time as the depth of stratified digraph increases.