# Persistent homology of unweighted complex networks via discrete Morse theory

## TL;DR

This paper introduces a discrete Morse theory-based method for analyzing the topological features of unweighted networks using persistent homology, improving computational efficiency and ability to distinguish network structures.

## Contribution

The paper presents a novel approach leveraging discrete Morse theory to efficiently compute persistent homology of unweighted networks, capturing higher-order topological features.

## Key findings

- Achieves near-minimal critical simplices in various networks
- Significantly improves computational efficiency of persistent homology
- Can distinguish different network models based on topology

## Abstract

Topological data analysis can reveal higher-order structure beyond pairwise connections between vertices in complex networks. We present a new method based on discrete Morse theory to study topological properties of unweighted and undirected networks using persistent homology. Leveraging on the features of discrete Morse theory, our method not only captures the topology of the clique complex of such graphs via the concept of critical simplices, but also achieves close to the theoretical minimum number of critical simplices in several analyzed model and real networks. This leads to a reduced filtration scheme based on the subsequence of the corresponding critical weights, thereby leading to a significant increase in computational efficiency. We have employed our filtration scheme to explore the persistent homology of several model and real-world networks. In particular, we show that our method can detect differences in the higher-order structure of networks, and the corresponding persistence diagrams can be used to distinguish between different model networks. In summary, our method based on discrete Morse theory further increases the applicability of persistent homology to investigate the global topology of complex networks.