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High Dimensional Hyperbolic Geometry of Complex Networks

by 1,* and 2
1
Faculty of Science, Beijing University of Technology, Beijing 100124, China
2
Department of Mathematics, University of California, San Diego, CA 92093, USA
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(11), 1861; https://doi.org/10.3390/math8111861
Received: 24 August 2020 / Revised: 18 October 2020 / Accepted: 19 October 2020 / Published: 23 October 2020
(This article belongs to the Section Network Science)
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. View Full-Text
Keywords: hyperbolic geometry; complex network; degree distribution; asymptotic correlations of degree hyperbolic geometry; complex network; degree distribution; asymptotic correlations of degree
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MDPI and ACS Style

Yang, W.; Rideout, D. High Dimensional Hyperbolic Geometry of Complex Networks. Mathematics 2020, 8, 1861. https://doi.org/10.3390/math8111861

AMA Style

Yang W, Rideout D. High Dimensional Hyperbolic Geometry of Complex Networks. Mathematics. 2020; 8(11):1861. https://doi.org/10.3390/math8111861

Chicago/Turabian Style

Yang, Weihua; Rideout, David. 2020. "High Dimensional Hyperbolic Geometry of Complex Networks" Mathematics 8, no. 11: 1861. https://doi.org/10.3390/math8111861

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