Path homology and temporal networks

TL;DR

This paper introduces an algorithm for computing path homology in simple digraphs and demonstrates its application in analyzing the structure of temporal networks, revealing insights into their underlying motifs and behaviors.

Contribution

It presents a novel algorithm for path homology computation and applies it to analyze temporal networks, linking topological features to network dynamics.

Findings

Path homology helps identify key motifs in temporal networks.

Small digraphs contribute to homology in specific dimensions.

Path homology provides insights into network structure and behavior.

Abstract

We present an algorithm to compute path homology for simple digraphs, and use it to topologically analyze various small digraphs en route to an analysis of complex temporal networks which exhibit such digraphs as underlying motifs. The digraphs analyzed include all digraphs, directed acyclic graphs, and undirected graphs up to certain numbers of vertices, as well as some specially constructed cases. Using information from this analysis, we identify small digraphs contributing to path homology in dimension $2$ for three temporal networks, and relate these digraphs to network behavior. We conclude that path homology can provide insight into temporal network structure and vice versa.