Forman-Ricci curvature and Persistent homology of unweighted complex networks

TL;DR

This paper applies topological data analysis to unweighted networks by converting them into weighted networks using measures like Forman-Ricci curvature and edge betweenness, enabling persistent homology analysis to distinguish network types.

Contribution

It introduces a novel approach combining Forman-Ricci curvature and edge betweenness to analyze unweighted networks through persistent homology.

Findings

Persistent homology differentiates model and real networks.

Forman-Ricci curvature captures local topological features.

Edge betweenness reflects global network structure.

Abstract

We present the application of topological data analysis (TDA) to study unweighted complex networks via their persistent homology. By endowing appropriate weights that capture the inherent topological characteristics of such a network, we convert an unweighted network into a weighted one. Standard TDA tools are then used to compute their persistent homology. To this end, we use two main quantifiers: a local measure based on Forman's discretized version of Ricci curvature, and a global measure based on edge betweenness centrality. We have employed these methods to study various model and real-world networks. Our results show that persistent homology can be used to distinguish between model and real networks with different topological properties.