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The diameter of KPKVB random graphs

Part of: Equilibrium statistical mechanics Geometric probability and stochastic geometry Graph theory

Published online by Cambridge University Press:  07 August 2019

Tobias Müller  and
Merlijn Staps
Tobias Müller*
Affiliation:
Utrecht University
Merlijn Staps*
Affiliation:
Utrecht University
*
*Current address: Bernoulli Institute, University of Groningen, Nijenborgh 9, 9747 AG Groningen, The Netherlands. Email address: tobias.muller@rug.nl
**Current address: Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08544, USA. Email address: merlijnstaps@gmail.com
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Abstract

We consider a random graph model that was recently proposed as a model for complex networks by Krioukov et al. (2010). In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most a certain hyperbolic distance from each other. It has previously been shown that this model has various properties associated with complex networks, including a power-law degree distribution and a strictly positive clustering coefficient. The model is specified using three parameters: the number of nodes N, which we think of as going to infinity, and $\alpha, \nu > 0$, which we think of as constant. Roughly speaking, $\alpha$ controls the power-law exponent of the degree sequence and $\nu$ the average degree. Earlier work of Kiwi and Mitsche (2015) has shown that, when $\alpha \lt 1$ (which corresponds to the exponent of the power law degree sequence being $\lt 3$), the diameter of the largest component is asymptotically almost surely (a.a.s.) at most polylogarithmic in N. Friedrich and Krohmer (2015) showed it was a.a.s. $\Omega(\log N)$ and improved the exponent of the polynomial in $\log N$ in the upper bound. Here we show the maximum diameter over all components is a.a.s. $O(\log N),$ thus giving a bound that is tight up to a multiplicative constant.

Keywords

Random graphcomplex networkhyperbolic planegraph diameter

MSC classification

Primary: 05C80: Random graphs
Secondary: 05C82: Small world graphs, complex networks 05C12: Distance in graphs 60D05: Geometric probability and stochastic geometry 82B43: Percolation
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 2 , June 2019 , pp. 358 - 377
Copyright
© Applied Probability Trust 2019 

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Footnotes

Supported in part by the Netherlands Organisation for Scientific Research (NWO) under project numbers 612.001.409 and 639.032.529.

This paper is the result of this author’s MSc thesis [16], carried out at Utrecht University under the supervision of the first author.

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