## The Annals of Applied Probability

### Connectivity of soft random geometric graphs

Mathew D. Penrose

#### 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
Revised: January 2015
First available in Project Euclid: 22 March 2016

https://projecteuclid.org/euclid.aoap/1458651826

Digital Object Identifier
doi:10.1214/15-AAP1110

Mathematical Reviews number (MathSciNet)
MR3476631

Zentralblatt MATH identifier
1339.05369

#### 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

#### 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.
• [2] Bollobás, B. (2001). Random Graphs, 2nd ed. Cambridge Studies in Advanced Mathematics 73. Cambridge Univ. Press, Cambridge.
• [3] Broutin, N., Devroye, L., Fraiman, N. and Lugosi, G. (2014). Connectivity threshold of bluetooth graphs. Random Structures Algorithms 44 45–66.
• [4] Coon, J., Dettmann, C. P. and Georgiou, O. (2012). Full connectivity: Corners, edges and faces. J. Stat. Phys. 147 758–778.
• [5] Diaz, J., Petit, J. and Serna, M. (2000). Faulty random geometric networks. Parallel Process. Lett. 10 343–357.
• [6] Erdős, P. and Rényi, A. (1959). On random graphs. I. Publ. Math. Debrecen 6 290–297.
• [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.
• [8] Gupta, P. and Kumar, P. R. (2000). The capacity of wireless networks. IEEE Trans. Inform. Theory 46 388–404.
• [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 712 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.
• [12] Penrose, M. (2003). Random Geometric Graphs. Oxford Studies in Probability 5. Oxford Univ. Press, Oxford.
• [13] Penrose, M. D. (1991). On a continuum percolation model. Adv. in Appl. Probab. 23 536–556.
• [14] Penrose, M. D. (1997). The longest edge of the random minimal spanning tree. Ann. Appl. Probab. 7 340–361.
• [15] Penrose, M. D. (1999). On $k$-connectivity for a geometric random graph. Random Structures Algorithms 15 145–164.
• [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.
• [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.