The Annals of Applied Probability

Connectivity of soft random geometric graphs

Mathew D. Penrose

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Abstract

Consider a graph on $n$ uniform random points in the unit square, each pair being connected by an edge with probability $p$ if the inter-point distance is at most $r$. We show that as $n\to\infty$ the probability of full connectivity is governed by that of having no isolated vertices, itself governed by a Poisson approximation for the number of isolated vertices, uniformly over all choices of $p,r$. We determine the asymptotic probability of connectivity for all $(p_{n},r_{n})$ subject to $r_{n}=O(n^{-\varepsilon })$, some $\varepsilon >0$. We generalize the first result to higher dimensions and to a larger class of connection probability functions.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 2 (2016), 986-1028.

Dates
Received: April 2014
Revised: January 2015
First available in Project Euclid: 22 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1458651826

Digital Object Identifier
doi:10.1214/15-AAP1110

Mathematical Reviews number (MathSciNet)
MR3476631

Zentralblatt MATH identifier
1339.05369

Subjects
Primary: 05C80: Random graphs [See also 60B20] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 05C40: Connectivity 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Random graph stochastic geometry random connection model connectivity isolated points continuum percolation

Citation

Penrose, Mathew D. Connectivity of soft random geometric graphs. Ann. Appl. Probab. 26 (2016), no. 2, 986--1028. doi:10.1214/15-AAP1110. https://projecteuclid.org/euclid.aoap/1458651826


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References

  • [1] Balogh, J., Bollobás, B., Krivelevich, M., Müller, T. and Walters, M. (2011). Hamilton cycles in random geometric graphs. Ann. Appl. Probab. 21 1053–1072.
    Mathematical Reviews (MathSciNet): MR2830612
    Zentralblatt MATH: 1222.05235
    Digital Object Identifier: doi:10.1214/10-AAP718
    Project Euclid: euclid.aoap/1307020391
  • [2] Bollobás, B. (2001). Random Graphs, 2nd ed. Cambridge Studies in Advanced Mathematics 73. Cambridge Univ. Press, Cambridge.
    Mathematical Reviews (MathSciNet): MR1864966
  • [3] Broutin, N., Devroye, L., Fraiman, N. and Lugosi, G. (2014). Connectivity threshold of bluetooth graphs. Random Structures Algorithms 44 45–66.
    Mathematical Reviews (MathSciNet): MR3143590
  • [4] Coon, J., Dettmann, C. P. and Georgiou, O. (2012). Full connectivity: Corners, edges and faces. J. Stat. Phys. 147 758–778.
    Mathematical Reviews (MathSciNet): MR2930579
    Zentralblatt MATH: 1246.82046
    Digital Object Identifier: doi:10.1007/s10955-012-0493-y
  • [5] Diaz, J., Petit, J. and Serna, M. (2000). Faulty random geometric networks. Parallel Process. Lett. 10 343–357.
    Mathematical Reviews (MathSciNet): MR1841124
    Digital Object Identifier: doi:10.1142/S0129626400000329
  • [6] Erdős, P. and Rényi, A. (1959). On random graphs. I. Publ. Math. Debrecen 6 290–297.
    Mathematical Reviews (MathSciNet): MR120167
    Zentralblatt MATH: 0092.15705
  • [7] Gupta, P. and Kumar, P. R. (1999). Critical power for asymptotic connectivity in wireless networks. In Stochastic Analysis, Control, Optimization and Applications 547–566. Birkhäuser, Boston, MA.
    Mathematical Reviews (MathSciNet): MR1702981
    Zentralblatt MATH: 0916.90101
    Digital Object Identifier: doi:10.1007/978-1-4612-1784-8_33
  • [8] Gupta, P. and Kumar, P. R. (2000). The capacity of wireless networks. IEEE Trans. Inform. Theory 46 388–404.
    Mathematical Reviews (MathSciNet): MR1748976
    Digital Object Identifier: doi:10.1109/18.825799
  • [9] Krishnan, B. S., Ganesh, A. and Manjunath, D. (2013). On connectivity thresholds in superposition of random key graphs on random geometric graphs. In Information Theory Proceedings (ISIT), 2013 IEEE International Symposium on 7–12 July 2013 2389–2393. IEEE, New York.
  • [10] Mao, G. and Anderson, B. D. O. (2012). Towards a better understanding of large-scale network models. IEEE/ACM Transactions on Networking 20 408–421.
  • [11] Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge Tracts in Mathematics 119. Cambridge Univ. Press, Cambridge.
    Mathematical Reviews (MathSciNet): MR1409145
  • [12] Penrose, M. (2003). Random Geometric Graphs. Oxford Studies in Probability 5. Oxford Univ. Press, Oxford.
    Mathematical Reviews (MathSciNet): MR1986198
  • [13] Penrose, M. D. (1991). On a continuum percolation model. Adv. in Appl. Probab. 23 536–556.
    Mathematical Reviews (MathSciNet): MR1122874
    Zentralblatt MATH: 0729.60106
    Digital Object Identifier: doi:10.2307/1427621
  • [14] Penrose, M. D. (1997). The longest edge of the random minimal spanning tree. Ann. Appl. Probab. 7 340–361.
    Mathematical Reviews (MathSciNet): MR1442317
    Zentralblatt MATH: 0884.60042
    Digital Object Identifier: doi:10.1214/aoap/1034625335
    Project Euclid: euclid.aoap/1034625335
  • [15] Penrose, M. D. (1999). On $k$-connectivity for a geometric random graph. Random Structures Algorithms 15 145–164.
    Mathematical Reviews (MathSciNet): MR1704341
  • [16] Tse, D. and Viswanath, P. (2005). Fundamentals of Wireless Communication. Cambridge Univ. Press, Cambridge.
  • [17] Yağan, O. (2012). Performance of the Eschenauer–Gligor key distribution scheme under an ON/OFF channel. IEEE Trans. Inform. Theory 58 3821–3835.
    Mathematical Reviews (MathSciNet): MR2924403
    Digital Object Identifier: doi:10.1109/TIT.2012.2189353
  • [18] Yi, C.-W., Wan, P.-J., Lin, K.-W. and Huang, C.-H. (2006) Asymptotic distribution of the number of isolated nodes in wireless ad hoc networks with unreliable nodes and links. In Global Telecommunications Conference 2006, GLOBECOM’06. IEEE, New York.

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The Annals of Applied Probability

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