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Recursive Computation of Path Homology for Stratified Digraphs

Zhengtong Zhu, Zhiyi Chi

TL;DR

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, which has a significant advantage over the general algorithm in computation time as the depth of stratified digraph increases.

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.

Recursive Computation of Path Homology for Stratified Digraphs

TL;DR

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, which has a significant advantage over the general algorithm in computation time as the depth of stratified digraph increases.

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.

Paper Structure

This paper contains 12 sections, 10 theorems, 15 equations, 1 figure, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Path homology
  4. A general algorithm to compute path homology
  5. Basic results
  6. Full-depth homology of stratified digraph
  7. Algorithms
  8. Recursive calculation of full-depth homology
  9. Auxiliary algorithms
  10. Experiments
  11. Experiments on computation time
  12. Full-depth persistent homology

Key Result

Lemma 2.1

Let $\eno\gamma n\in\Lambda_p$. If each one has an elementary $p$-path not in the others, then the $\gamma_i$'s are linearly independent.

Figures (1)

  • Figure 1: Modified persistence diagram for full-depth path homologies.

Theorems & Definitions (27)

  • Definition 1.1
  • Remark
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Definition 3.1: Cross sections of a path and support of an allowed path
  • Lemma 3.2
  • proof
  • Definition 3.3: Predecessor and successor
  • Proposition 3.4: Decomposition of allowed paths and cycles
  • ...and 17 more