Advances in Applied Probability

Infection spread in random geometric graphs

Ghurumuruhan Ganesan

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


In this paper we study the speed of infection spread and the survival of the contact process in the random geometric graph G = G(n, rn, f) of n nodes independently distributed in S = [-½, ½]2 according to a certain density f(·). In the first part of the paper we assume that infection spreads from one node to another at unit rate and that infected nodes stay in the same state forever. We provide an explicit lower bound on the speed of infection spread and prove that infection spreads in G with speed at least D1nrn2. In the second part of the paper we consider the contact process ξt on G where infection spreads at rate λ > 0 from one node to another and each node independently recovers at unit rate. We prove that, for every λ > 0, with high probability, the contact process on G survives for an exponentially long time; there exist positive constants c1 and c2 such that, with probability at least 1 - c1 / n4, the contact process ξ1t starting with all nodes infected survives up to time tn = exp(c2n/logn) for all n.

Article information

Adv. in Appl. Probab., Volume 47, Number 1 (2015), 164-181.

First available in Project Euclid: 31 March 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60C05: Combinatorial probability 62E10: Characterization and structure theory

Random geometric graph speed of infection spread survival time of contact process


Ganesan, Ghurumuruhan. Infection spread in random geometric graphs. Adv. in Appl. Probab. 47 (2015), no. 1, 164--181. doi:10.1239/aap/1427814586.

Export citation


  • Chatterjee, S. and Durrett, R. (2009). Contact processes on random graphs with power-law degree distributions have critical value $0$. Ann. Prob. 37, 2332–2356.
  • Durrett, R. (1984). Oriented percolation in two dimensions. Ann. Prob. 12, 999–1040.
  • Franceschetti, M., Dousse, O., Tse, D. N. C. and Thiran, P. (2007). Closing the gap in the capacity of wireless networks via percolation theory. IEEE Trans. Inf. Theory 53, 1009–1018.
  • Ganesan, G. (2013). Size of the giant component in a random geometric graph. Ann. Inst. H. Poincaré Prob. Statist. 49, 1130–1140.
  • Gupta, P. and Kumar, P. R. (1999). Critical power for asymptotic connectivity in wireless networks. In Stochastic Analysis, Control, Optimization and Applications, Birkhäuser, Boston, MA, pp. 547–566.
  • Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin.
  • Meester, R. and Roy, R. (1996). Continuum Percolation. (Camb. Tracts Math. 119). Cambridge University Press.
  • Mountford, T., Mourrat, J.-C., Valesin, D. and Yao, Q. (2012). Exponential extinction time of the contact process on finite graphs. Preprint. Available at
  • Penrose, M. (2003). Random Geometric Graphs. Oxford University Press.
  • Sarkar, A. (1996). Some problems in continuum percolation. Doctoral Thesis, Indian Statistical Institute, Delhi.
  • Ganesan, G. (2014). Phase transitions for Erdős–Rényi graphs. Preprint. Available at
  • Ganesan, G. (2014). First passage percolation with nonidentical passage times. Preprint. Available at
  • Smythe, R. T. and Wierman, J. C. (1978). First-Passage Percolation on the Square Lattice. Springer, Berlin.