Detecting hyperbolic geometry in networks: why triangles are not enough

Abstract

In the past decade, geometric network models have received vast attention in the literature. These models formalize the natural idea that similar vertices are likely to connect. Because of that, these models are able to adequately capture many common structural properties of real-world networks, such as self-invariance and high clustering. Indeed, many real-world networks can be accurately modeled by positioning vertices of a network graph in hyperbolic spaces. Nevertheless, if one observes only the network connections, the presence of geometry is not always evident. Currently, triangle counts and clustering coefficients are the standard statistics to signal the presence of geometry. In this paper we show that triangle counts or clustering coefficients are insufficient because they fail to detect geometry induced by hyperbolic spaces. We therefore introduce a novel triangle-based statistic, which weighs triangles based on their strength of evidence for geometry. We show analytically, as well as on synthetic and real-world data, that this is a powerful statistic to detect hyperbolic geometry in networks.

arXiv comment11 pages, 2 figures, 1 table

References in corpus (14)

• The structure of scientific collaboration networks
• Hyperbolic Geometry of Complex Networks
• The phase transition in inhomogeneous random graphs
• Random Geometric Graphs
• Self-similarity of complex networks and hidden metric spaces
• Network Geometry
• Clustering implies geometry in networks
• Characterizing the analogy between hyperbolic embedding and community structure of complex networks
• Explosion in weighted Hyperbolic Random Graphs and Geometric Inhomogeneous Random Graphs
• Clustering of random scale-free networks
• Local clustering in scale-free networks with hidden variables
• Variational principle for scale-free network motifs
• A geometry-induced topological phase transition in random graphs
• Cliques in geometric inhomogeneous random graph

Cited by in corpus (7)

• Model-independent methods for embedding directed networks into Euclidean and hyperbolic spaces
• Random graphs and real networks with weak geometric coupling
• Scaling of the clustering function in spatial inhomogeneous random graphs
• Real-World Networks are Low-Dimensional: Theoretical and Practical Assessment
• On the distances within cliques in a soft random geometric graph
• Inhomogeneous random graphs with infinite-mean fitness variables
• Localized geometry detection in scale-free random graphs