Forman curvature for complex networks

TL;DR

This paper adapts Forman's Ricci curvature to analyze undirected networks, revealing its potential to characterize network structure and vulnerability, especially through the distribution and correlation of curvature with network properties.

Contribution

The study introduces a novel application of Forman curvature to complex networks, providing insights into their organization and vulnerability that were not previously explored.

Findings

Most nodes and edges have negative curvature.

Distribution of curvature varies across network types.

Targeted removal of negatively curved nodes increases network vulnerability.

Abstract

We adapt Forman's discretization of Ricci curvature to the case of undirected networks, both weighted and unweighted, and investigate the measure in a variety of model and real-world networks. We find that most nodes and edges in model and real networks have a negative curvature. Furthermore, the distribution of Forman curvature of nodes and edges is narrow in random and small-world networks, while the distribution is broad in scale-free and real-world networks. In most networks, Forman curvature is found to display significant negative correlation with degree and centrality measures. However, Forman curvature is uncorrelated with clustering coefficient in most networks. Importantly, we find that both model and real networks are vulnerable to targeted deletion of nodes with highly negative Forman curvature. Our results suggest that Forman curvature can be employed to gain novel insights on the organization of complex networks.