The age-dependent random connection model
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.
KeywordsScale-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 ClassificationPrimary 05C80 Secondary 60K35
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.
- 1.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)Google Scholar
- 3.Benjamini, I., Schramm, O.: Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6:13 pp. (2001)Google Scholar
- 5.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)Google Scholar
- 11.Flaxman, A.D., Frieze, A.M., Vera, J.: A geometric preferential attachment model of networks. Int. Math. 3(2), 187–205 (2006)Google Scholar
- 12.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)Google Scholar
- 13.Hirsch, C., Mönch, C.: Distances and large deviations in the spatial preferential attachment model. ArXiv e-prints (2018)Google Scholar
- 18.Jordan, J.: Geometric preferential attachment in non-uniform metric spaces. Electron. J. Probab. 18(8), 15 (2013)Google Scholar
- 20.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)Google Scholar