Affiliation:
1. School of Mathematics, University of Birmingham, B15 2TT, UK
2. Department of Mathematics, Uppsala University, Uppsala 75106, Sweden
Abstract
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 article, we consider this quantity on a probabilistic model of complex networks introduced by Krioukov et al. (2010, Phys. Rev. E, 82, 036106). 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.
Funder
Engineering and Physical Sciences Research Council
Publisher
Oxford University Press (OUP)
Subject
Applied Mathematics,Computational Mathematics,Control and Optimization,Management Science and Operations Research,Computer Networks and Communications
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