High Dimensional Hyperbolic Geometry of Complex Networks

Weihua Yang and David Rideout
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Weihua Yang: Faculty of Science, Beijing University of Technology, Beijing 100124, China
David Rideout: Department of Mathematics, University of California, San Diego, CA 92093, USA

Mathematics, 2020, vol. 8, issue 11, 1-39

Abstract: High dimensional embeddings of graph data into hyperbolic space have recently been shown to have great value in encoding hierarchical structures, especially in the area of natural language processing, named entity recognition, and machine generation of ontologies. Given the striking success of these approaches, we extend the famous hyperbolic geometric random graph models of Krioukov et al. to arbitrary dimension, providing a detailed analysis of the degree distribution behavior of the model in an expanded portion of the parameter space, considering several regimes which have yet to be considered. Our analysis includes a study of the asymptotic correlations of degree in the network, revealing a non-trivial dependence on the dimension and power law exponent. These results pave the way to using hyperbolic geometric random graph models in high dimensional contexts, which may provide a new window into the internal states of network nodes, manifested only by their external interconnectivity.

Keywords: hyperbolic geometry; complex network; degree distribution; asymptotic correlations of degree (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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