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Queueing Systems

, Volume 93, Issue 3–4, pp 309–331 | Cite as

The age-dependent random connection model

  • Peter Gracar
  • Arne Grauer
  • Lukas Lüchtrath
  • Peter MörtersEmail author
Article

Abstract

We investigate a class of growing graphs embedded into the d-dimensional torus where new vertices arrive according to a Poisson process in time, are randomly placed in space and connect to existing vertices with a probability depending on time, their spatial distance and their relative birth times. This simple model for a scale-free network is called the age-based spatial preferential attachment network and is based on the idea of preferential attachment with spatially induced clustering. We show that the graphs converge weakly locally to a variant of the random connection model, which we call the age-dependent random connection model. This is a natural infinite graph on a Poisson point process where points are marked by a uniformly distributed age and connected with a probability depending on their spatial distance and both ages. We use the limiting structure to investigate asymptotic degree distribution, clustering coefficients and typical edge lengths in the age-based spatial preferential attachment network.

Keywords

Scale-free networks Benjamini–Schramm limit Random connection model Preferential attachment Geometric random graphs Spatially embedded graphs Clustering coefficient Power-law degree distribution Edge lengths 

Mathematics Subject Classification

Primary 05C80 Secondary 60K35 

Notes

Acknowledgements

We would like to thank Sergey Foss for the invitation to the Stochastic Networks 2018 workshop at ICMS, Edinburgh, where this paper was first presented. We would also like to thank two anonymous referees for valuable comments which led to significant improvements in the paper.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department Mathematik/InformatikUniversität zu KölnCologneGermany

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