## 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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