Forman curvature for complex networks

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.

18 pages, 15 figures. Supplementary material available upon request from authors

References in corpus (15)

• Emergence of scaling in random networks
• Statistical mechanics of complex networks
• Community detection in graphs
• Lethality and centrality in protein networks
• Finding community structure in networks using the eigenvectors of matrices
• The large-scale organization of metabolic networks
• Self-similar community structure in organisations
• Growing Scale-Free Networks with Tunable Clustering
• Community Structure in Jazz
• Curvature of Co-Links Uncovers Hidden Thematic Layers in the World Wide Web
• Robust network community detection using balanced propagation
• The Large Scale Curvature of Networks
• Emergent Complex Network Geometry
• Ricci curvature on polyhedral surfaces via optimal transportation
• The Simplicial Ricci Tensor

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