Systematic evaluation of a new combinatorial curvature for complex networks

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.

29 pages, 14 figures, 1 table. Accepted for publication in Chaos, Solitons and Fractals

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• Network-centric indicators for fragility in global financial indices
• Forman-Ricci curvature and Persistent homology of unweighted complex networks
• Community Detection on Networks with Ricci Flow
• Finding Influential Cores via Normalized Ricci Flows in Directed and Undirected Hypergraphs with Applications
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• Application of discrete Ricci curvature in pruning randomly wired neural networks: A case study with chest x-ray classification of COVID-19