Recursive Computation of Path Homology for Stratified Digraphs

Abstract

Stratified digraphs are popular models for feedforward neural networks. However, computation of their path homologies has been limited to low dimensional ones 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 the maximal path homology of acyclic digraphs, i.e., path homology with dimension equal to the maximum path length of a graph. Numerical exper- iments show that the algorithm has a significant advantage over the general algorithm in computation time as the depth of stratified digraph increases.