Homology of weighted path complexes and directed hypergraphs

TL;DR

This paper develops a new homology theory for weighted directed hypergraphs, establishing conditions for homotopy invariance and providing examples to illustrate the theory's nontrivial aspects.

Contribution

It introduces weighted path homology for directed hypergraphs and explores its homotopy invariance, a novel extension in topological data analysis.

Findings

Weighted path homology groups are homotopy invariant under certain conditions.

Examples demonstrate the nontriviality of the introduced homology notions.

The framework extends topological methods to weighted directed hypergraphs.

Abstract

We introduce the weighted path homology on the category of weigh\-ted directed hypergraphs and describe conditions of homotopy invariance of weighted path homology groups. We give several examples that explain the nontriviality of the introduced notions.