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A simple differential geometry for complex networks

Published online by Cambridge University Press:  16 November 2020

Emil Saucan Open the ORCID record for Emil Saucan [Opens in a new window] ,
Areejit Samal  and
Jürgen Jost
Emil Saucan*
Affiliation:
Department of Applied Mathematics, ORT Braude College, Karmiel 2161002, Israel,
Areejit Samal
Affiliation:
The Institute of Mathematical Sciences (IMSc), Homi Bhabha National Institute (HBNI), Chennai 600113, India, Max Planck Institute for Mathematics in the Sciences, Leipzig 04103, Germany (email: asamal@imsc.res.in)
Jürgen Jost
Affiliation:
Max Planck Institute for Mathematics in the Sciences, Leipzig 04103, Germany (email: asamal@imsc.res.in) The Santa Fe Institute, Santa Fe, New Mexico 87501, USA (email: jost@mis.mpg.de)
*
*Corresponding author. Email: semil@braude.ac.il
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Abstract

We introduce new definitions of sectional, Ricci, and scalar curvatures for networks and their higher dimensional counterparts, derived from two classical notions of curvature for curves in general metric spaces, namely, the Menger curvature and the Haantjes curvature. These curvatures are applicable to unweighted or weighted and undirected or directed networks and are more intuitive and easier to compute than other network curvatures. In particular, the proposed curvatures based on the interpretation of Haantjes definition as geodesic curvature allow us to give a network analogue of the classical local Gauss–Bonnet theorem. Furthermore, we propose even simpler and more intuitive proxies for the Haantjes curvature that allow for even faster and easier computations in large-scale networks. In addition, we also investigate the embedding properties of the proposed Ricci curvatures. Lastly, we also investigate the behavior, both on model and real-world networks, of the curvatures introduced herein with more established notions of Ricci curvature and other widely used network measures.

Keywords

network curvatureMenger curvatureHaantjes curvatureRicci curvaturelocal Gauss–Bonnet theoremgeometric embedding
Type
Research Article
Information
Network Science , Volume 9 , Special Issue S1: Complex Networks 2019 , October 2021 , pp. S106 - S133
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

Action Editor: Hocine Cherifi

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