Systematic evaluation of a new combinatorial curvature for complex networks

TL;DR

This paper evaluates a new combinatorial curvature measure, Forman curvature, for complex networks, comparing it with existing measures and analyzing its effectiveness in identifying important nodes and edges.

Contribution

It introduces and empirically assesses the effectiveness of Forman curvature as a network importance indicator, comparing two node curvature definitions.

Findings

Forman curvature outperforms embeddedness and dispersion in indicating edge importance.

Unnormalized Forman node curvature with combinatorial weights better indicates node importance.

Forman curvature provides a mathematically elegant and practical measure for complex network analysis.

Abstract

We have recently introduced Forman's discretization of Ricci curvature to the realm of complex networks. Forman curvature is an edge-based measure whose mathematical definition elegantly encapsulates the weights of nodes and edges in a complex network. In this contribution, we perform a comparative analysis of Forman curvature with other edge-based measures such as edge betweenness, embeddedness and dispersion in diverse model and real networks. We find that Forman curvature in comparison to embeddedness or dispersion is a better indicator of the importance of an edge for the large-scale connectivity of complex networks. Based on the definition of the Forman curvature of edges, there are two natural ways to define the Forman curvature of nodes in a network. In this contribution, we also examine these two possible definitions of Forman curvature of nodes in diverse model and real networks. Based on our empirical analysis, we find that in practice the unnormalized definition of the Forman curvature of nodes with the choice of combinatorial node weights is a better indicator of the importance of nodes in complex networks.