• Bernoulli
  • Volume 26, Number 2 (2020), 927-947.

Distances and large deviations in the spatial preferential attachment model

Christian Hirsch and Christian Mönch

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This paper considers two asymptotic properties of a spatial preferential-attachment model introduced by E. Jacob and P. Mörters (In Algorithms and Models for the Web Graph (2013) 14–25 Springer). First, in a regime of strong linear reinforcement, we show that typical distances are at most of doubly-logarithmic order. Second, we derive a large deviation principle for the empirical neighbourhood structure and express the rate function as solution to an entropy minimisation problem in the space of stationary marked point processes.

Article information

Bernoulli, Volume 26, Number 2 (2020), 927-947.

Received: September 2018
Revised: January 2019
First available in Project Euclid: 31 January 2020

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

distances large deviation principle Poisson point process preferential attachment


Hirsch, Christian; Mönch, Christian. Distances and large deviations in the spatial preferential attachment model. Bernoulli 26 (2020), no. 2, 927--947. doi:10.3150/19-BEJ1121.

Export citation


  • [1] Bordenave, C. and Caputo, P. (2015). Large deviations of empirical neighborhood distribution in sparse random graphs. Probab. Theory Related Fields 163 149–222.
  • [2] Choi, J. and Sethuraman, S. (2013). Large deviations for the degree structure in preferential attachment schemes. Ann. Appl. Probab. 23 722–763.
  • [3] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Applications of Mathematics (New York) 38. New York: Springer.
  • [4] Dereich, S. and Mörters, P. (2009). Random networks with sublinear preferential attachment: Degree evolutions. Electron. J. Probab. 14 1222–1267.
  • [5] Deuschel, J.-D. and Pisztora, A. (1996). Surface order large deviations for high-density percolation. Probab. Theory Related Fields 104 467–482.
  • [6] Eichelsbacher, P. and Schmock, U. (1998). Exponential approximations in completely regular topological spaces and extensions of Sanov’s theorem. Stochastic Process. Appl. 77 233–251.
  • [7] Feng, J. and Kurtz, T.G. (2006). Large Deviations for Stochastic Processes. Mathematical Surveys and Monographs 131. Providence, RI: Amer. Math. Soc.
  • [8] Franceschetti, M. and Meester, R. (2007). Random Networks for Communication: From Statistical Physics to Information Systems. Cambridge Series in Statistical and Probabilistic Mathematics 24. Cambridge: Cambridge Univ. Press.
  • [9] Georgii, H.-O. and Zessin, H. (1993). Large deviations and the maximum entropy principle for marked point random fields. Probab. Theory Related Fields 96 177–204.
  • [10] Gracar, P., Grauer, A., Lüchtrath, L. and Mörters, P. (2018). The age-dependent random connection model. Preprint. Available at arXiv:1810.03429.
  • [11] Jacob, E. and Mörters, P. (2013). A spatial preferential attachment model with local clustering. In Algorithms and Models for the Web Graph. Lecture Notes in Computer Science 8305 14–25. Cham: Springer.
  • [12] Jacob, E. and Mörters, P. (2015). Spatial preferential attachment networks: Power laws and clustering coefficients. Ann. Appl. Probab. 25 632–662.
  • [13] Jacob, E. and Mörters, P. (2017). Robustness of scale-free spatial networks. Ann. Probab. 45 1680–1722.
  • [14] Last, G. and Penrose, M. (2017). Lectures on the Poisson Process. Institute of Mathematical Statistics Textbooks 7. Cambridge: Cambridge Univ. Press.