The modularity of random graphs on the hyperbolic plane

Jordan Chellig, Nikolaos Fountoulakis, Fiona Skerman

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Abstract

Modularity is a quantity which has been introduced in the context of complex networks in order to quantify how close a network is to an ideal modular network in which the nodes form small interconnected communities that are joined together with relatively few edges. In this paper, we consider this quantity on a probabilistic model of complex networks introduced by Krioukov et al. (Phys. Rev. E 2010).

This model views a complex network as an expression of hidden popularity hierarchies (i.e., nodes higher up in the hierarchies have more global reach), encapsulated by an underlying hyperbolic space. For certain parameters, this model was proved to have typical features that are observed in complex networks such as power law degree distribution, bounded average degree, clustering coefficient that is asymptotically bounded away from zero, and ultra-small typical distances. In the present work, we investigate its modularity and we show that, in this regime, it converges to one in probability.
Original languageEnglish
Article numbercnab051
JournalJournal of Complex Networks
Volume10
Issue number1
Early online date30 Dec 2021
DOIs
Publication statusPublished - Feb 2022

Keywords

  • modularity
  • random geometric graphs
  • hyperbolic plane
  • complex networks

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