Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-19T11:59:08.386Z Has data issue: false hasContentIssue false
  • English
  • Français

Super-Exponential Extinction Time of the Contact Process on Random Geometric Graphs

Published online by Cambridge University Press:  01 August 2017

VAN HAO CAN
VAN HAO CAN*
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307 Ha Noi, Vietnam (e-mail: cvhao89@gmail.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we prove lower and upper bounds for the extinction time of the contact process on random geometric graphs with connection radius tending to infinity. We obtain that for any infection rate λ > 0, the contact process on these graphs survives a time super-exponential in the number of vertices.

Keywords

Primary 82C22Secondary 05C80
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 2 , March 2018 , pp. 162 - 185
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Berger, N., Borgs, C., Chayes, J. T. and Saberi, A. (2005) On the spread of viruses on the internet. In SODA '05: Proc. 16th Annual ACM–SIAM Symposium on Discrete Algorithms, SIAM, pp. 301–310.Google Scholar
[2] Can, V. H. (2015) Metastability for the contact process on the preferential attachment graph. Internet Math., to appear. doi:10.24166/im.08.2017 Google Scholar
[3] Can, V. H. (2016) Processus de contact sur des graphes aléatoires. PhD thesis, University of Aix–Marseille.Google Scholar
[4] Can, V. H. and Schapira, B. (2015) Metastability for the contact process on the configuration model with infinite mean degree. Electron. J. Probab. 20 #26.Google Scholar
[5] Chatterjee, S. and Durrett, R. (2009) Contact processes on random graphs with degree power law distribution have critical value zero. Ann. Probab. 37 2332–2356.Google Scholar
[6] Cranston, M., Mountford, T., Mourrat, J.-C. and Valesin, D. (2014) The contact process on finite homogeneous trees revisited. ALEA: Lat. Am. J. Probab. Math. Statist. 11 385–408.Google Scholar
[7] Durrett, R. (1984) Oriented percolation in two dimensions. Ann. Probab. 12 999–1040.Google Scholar
[8] Durrett, R. and Schonmann, R. H. (1988) Large deviation for the contact process and two dimensional percolation. Probab. Theory Rel. Fields 77 583–603.Google Scholar
[9] Friedrich, T., Sauerwald, T. and Stauffer, A. (2013) Diameter and broadcast time of random geometric graphs in arbitrary dimensions. Algorithmica 67 65–88.Google Scholar
[10] Ganesan, G. (2015) Infection spread in random geometric graphs. Adv. Appl. Probab. 47 164–181.Google Scholar
[11] Grimmett, G. (1999) Percolation, Vol. 321 of Grundlehren der Mathematischen Wissenschaften, Springer.Google Scholar
[12] Grimmett, G. (1985) Long paths and cycles in a random lattice. Ann. Discrete Math. 33 69–76.Google Scholar
[13] Gupta, P. and Kumar, P. R. (1999) Critical power for asymptotic connectivity in wireless networks. In Stochastic Analysis, Control, Optimization and Applications (McEneaney, W. M. et al., eds), Springer, pp. 547–566.Google Scholar
[14] Liggett, T. (1999) Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Vol. 324 of Grundlehren der Mathematischen Wissenschaften, Springer.Google Scholar
[15] Liggett, T., Schonmann, R. and Stacey, A. (1997) Domination by product measure. Ann. Probab. 25 71–95.Google Scholar
[16] Mountford, T. (1993) A metastable result for the finite multidimensional contact process. Canad. Math. Bull. 36 216–226.Google Scholar
[17] Mountford, T., Mourrat, J.-C., Valesin, D. and Yao, Q. (2016) Exponential extinction time of the contact process on finite graphs. Stochastic Process. Appl. 126 1974–2013.Google Scholar
[18] Mountford, T., Valesin, D. and Yao, Q. (2013) Metastable densities for contact processes on random graphs. Electron. J. Probab. 18 #103.Google Scholar
[19] Mourrat, J.-C. and Valesin, D. (2016) Phase transition of the contact process on random regular graphs. Electron. J. Probab. 21 #31.CrossRefGoogle Scholar
[20] Ménard, L. and Singh, A. (2016) Percolation by cumulative merging and phase transition for the contact process on random graphs. Ann. Sci. École Norm. Supér. 49 1189–1238.Google Scholar
[21] Penrose, M. (2003) Random Geometric Graphs, Oxford University Press.Google Scholar
[22] Preciado, V. M. and Jadbabaie, A. (2009) Spectral analysis of virus spreading in random geometric networks. In Proc. IEEE Conference on Decision and Control, IEEE, pp. 4802–4807.Google Scholar
[23] Scheinerman, E. R. (1990) An evolution of interval graphs. Discrete Math. 82 287–302.Google Scholar