## 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.

This is a preview of subscription content, access via your institution.

## References

Aiello, W., Bonato, A., Cooper, C., Janssen, J., Prałat, P.: A spatial web graph model with local influence regions. Int. Math.

**5**(1–2), 175–196 (2008)Barabási, A.-L., Albert, R.: Emergence of scaling in random networks. Science

**286**(5439), 509–512 (1999)Benjamini, I., Schramm, O.: Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6:13 pp. (2001)

Bollobás, B., Riordan, O., Spencer, J., Tusnády, G.: The degree sequence of a scale-free random graph process. Random Struct. Algorithms

**18**(3), 279–290 (2001)Cooper, C., Frieze, A., Prałat, P.: Some typical properties of the spatial preferred attachment model. In: Algorithms and Models for the Web Graph, Volume 7323 of Lecture Notes in Computer Science, pp. 29–40. Springer, Heidelberg (2012)

Deijfen, M., van der Hofstad, R., Hooghiemstra, G.: Scale-free percolation. Ann. Inst. Henri Poincaré Probab. Stat.

**49**(3), 817–838 (2013)Deprez, P., Wüthrich, M.V.: Scale-free percolation in continuum space. Commun. Math. Stat. (2018). https://doi.org/10.1007/s40304-018-0142-0

Dereich, S., Mönch, C., Mörters, P.: Typical distances in ultrasmall random networks. Adv. Appl. Probab.

**44**(2), 583–601 (2012)Dereich, S., Mörters, P.: Random networks with sublinear preferential attachment: the giant component. Ann. Probab.

**41**(1), 329–384 (2013)Dommers, S., van der Hofstad, R., Hooghiemstra, G.: Diameters in preferential attachment models. J. Stat. Phys.

**139**(1), 72–107 (2010)Flaxman, A.D., Frieze, A.M., Vera, J.: A geometric preferential attachment model of networks. Int. Math.

**3**(2), 187–205 (2006)Flaxman, A.D., Frieze, A.M., Vera, J.: A geometric preferential attachment model of networks. II. In: Algorithms and Models for the Web-Graph, Volume 4863 of Lecture Notes in Computer Science, pp. 41–55. Springer, Berlin (2007)

Hirsch, C., Mönch, C.: Distances and large deviations in the spatial preferential attachment model. ArXiv e-prints (2018)

Jacob, E., Mörters, P.: Spatial preferential attachment networks: power laws and clustering coefficients. Ann. Appl. Probab.

**25**(2), 632–662 (2015)Jacob, E., Mörters, P.: Robustness of scale-free spatial networks. Ann. Probab.

**45**(3), 1680–1722 (2017)Janssen, J., Prałat, P., Wilson, R.: Geometric graph properties of the spatial preferred attachment model. Adv. Appl. Math.

**50**(2), 243–267 (2013)Jordan, J.: Degree sequences of geometric preferential attachment graphs. Adv. Appl. Probab.

**42**(2), 319–330 (2010)Jordan, J.: Geometric preferential attachment in non-uniform metric spaces. Electron. J. Probab.

**18**(8), 15 (2013)Jordan, J., Wade, A.R.: Phase transitions for random geometric preferential attachment graphs. Adv. Appl. Probab.

**47**(2), 565–588 (2015)Last, G., Nestmann, F., Schulte, M.: The random connection model and functions of edge-marked Poisson processes: second order properties and normal approximation. ArXiv e-prints (2018)

Last, G., Penrose, M.: Lectures on the Poisson Process. Cambridge University Press, Cambridge (2017)

Manna, S.S., Sen, P.: Modulated scale-free network in Euclidean space. Phys. Rev. E

**66**, 066114 (2002)Meester, R., Roy, R.: Continuum Percolation. Cambridge University Press, Cambridge (1996)

Mourrat, J.-C., Valesin, D.: Spatial Gibbs random graphs. Ann. Appl. Probab.

**28**(2), 751–789 (2018)Newman, M.E.J., Strogatz, S.H., Watts, D.J.: Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E

**64**, 026118 (2001)Penrose, M.D., Yukich, J.E.: Weak laws of large numbers in geometric probability. Ann. Appl. Probab.

**13**(1), 277–303 (2003)Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature

**393**, 440–442 (1998)

## 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.

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Appendix A: Simulation of the model

### Appendix A: Simulation of the model

In this section, we give an overview of the code used to generate the pictures shown throughout the paper. It is also used for estimating the limiting average clustering coefficient in Sect. 5. The code can be freely accessed at: http://www.mi.uni-koeln.de/~moerters/LoadPapers/adrc-model.R.

The main objective of the code is to sample neighbours of a given vertex (*x*, *u*) in the age-dependent random connection model in dimension 1 for given parameters \(\beta \) and \(\gamma \) and the profile function \(\varphi \). Due to Proposition 4.1, which gives an explicit description of the neighbourhood of a given vertex, we can use rejection sampling to achieve this. The distribution in (5.1), defined on \({\mathbb {R}} \times (0,1]\), that we use to sample the neighbours of (*x*, *u*) may be unbounded and heavy tailed in the first parameter. To deal with this, we restrict the sampling to a region with mass \(q = 0.99\) with respect to this distribution. This sampling works for arbitrary but reasonable choices of the profile function \(\varphi \) and parameters \(\beta \), \(\gamma \); we provide and use an optimised sampling algorithm for with \(a \ge \frac{1}{2}\). The advantage of studying this class of \(\varphi \) is that expressions can be analytically simplified, which allows us to improve the algorithm by dividing the region from which the points are sampled into sub-regions with equal mass with respect to \(\varphi \), thus increasing the acceptance rate for points sampled far away from (*x*, *u*). That is, the code first selects one of these equally likely sub-regions uniformly at random and then points are sampled therein until one is accepted. The numerical optimisation method nlminb is used to calculate the boundaries of the ranges, i.e., quantiles of the distribution from (5.1).

A first application of the sampling is the estimation of the expected local clustering coefficient of a vertex (0, *u*) in the age-dependent random connection model (see Fig. 3) and by Theorem 5.1 also the average clustering coefficient for the age-based preferential attachment network (see Fig. 4). To this end, the code samples pairs of neighbours of (0, *u*) and averages the probability that the pair is connected. A second application of the sampling is generating heatmaps of the neighbourhoods of a given vertex (see Fig. 2). The heatmaps are generated using the R library MASS and function kde2d by estimating the heat kernel for the sampled neighbouring vertices. Further properties thereof can be studied with additional heatmap generating functions that we provide.

## Rights and permissions

## About this article

### Cite this article

Gracar, P., Grauer, A., Lüchtrath, L. *et al.* The age-dependent random connection model.
*Queueing Syst* **93**, 309–331 (2019). https://doi.org/10.1007/s11134-019-09625-y

Received:

Revised:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s11134-019-09625-y

### 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