Annals of Applied Probability

Almost optimal sparsification of random geometric graphs

Nicolas Broutin, Luc Devroye, and Gábor Lugosi

Full-text: Open access

Abstract

A random geometric irrigation graph $\Gamma_{n}(r_{n},\xi)$ has $n$ vertices identified by $n$ independent uniformly distributed points $X_{1},\ldots,X_{n}$ in the unit square $[0,1]^{2}$. Each point $X_{i}$ selects $\xi_{i}$ neighbors at random, without replacement, among those points $X_{j}$ ($j\neq i$) for which $\Vert X_{i}-X_{j}\Vert <r_{n}$, and the selected vertices are connected to $X_{i}$ by an edge. The number $\xi_{i}$ of the neighbors is an integer-valued random variable, chosen independently with identical distribution for each $X_{i}$ such that $\xi_{i}$ satisfies $\xi_{i}\ge1$. We prove that when $r_{n}=\gamma_{n}\sqrt{\log n/n}$ for $\gamma_{n}\to\infty$ with $\gamma_{n}=o(n^{1/6}/\log^{5/6}n)$, the random geometric irrigation graph experiences explosive percolation in the sense that if ${\mathbf{E} \xi_{i}=1}$, then the largest connected component has $o(n)$ vertices but if $\mathbf{E} \xi_{i}>1$, then the number of vertices of the largest connected component is, with high probability, $n-o(n)$. This offers a natural noncentralized sparsification of a random geometric graph that is mostly connected.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 5 (2016), 3078-3109.

Dates
Received: November 2014
Revised: September 2015
First available in Project Euclid: 19 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1476884312

Digital Object Identifier
doi:10.1214/15-AAP1170

Mathematical Reviews number (MathSciNet)
MR3563202

Zentralblatt MATH identifier
1375.60031

Subjects
Primary: 60C05: Combinatorial probability 05C80: Random graphs [See also 60B20]

Keywords
Random geometric graph connectivity irrigation graph

Citation

Broutin, Nicolas; Devroye, Luc; Lugosi, Gábor. Almost optimal sparsification of random geometric graphs. Ann. Appl. Probab. 26 (2016), no. 5, 3078--3109. doi:10.1214/15-AAP1170. https://projecteuclid.org/euclid.aoap/1476884312


Export citation

References

  • [1] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Die Grundlehren der mathematischen Wissenschaften 196. Springer, New York.
  • [2] Bender, E. A. (1974). Asymptotic methods in enumeration. SIAM Rev. 16 485–515.
  • [3] Bender, E. A. (1975). An asymptotic expansion for the coefficients of some formal power series. J. Lond. Math. Soc. (2) 9 451–458.
  • [4] Bollobás, B. (2001). Random Graphs, 2nd ed. Cambridge Studies in Advanced Mathematics 73. Cambridge Univ. Press, Cambridge.
  • [5] Boucheron, S., Lugosi, G. and Massart, P. (2013). Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford Univ. Press, Oxford.
  • [6] Broutin, N., Devroye, L., Fraiman, N. and Lugosi, G. (2014). Connectivity threshold of Bluetooth graphs. Random Structures Algorithms 44 45–66.
  • [7] Broutin, N., Devroye, L. and Lugosi, G. (2015). Connectivity of sparse Bluetooth networks. Electron. Commun. Probab. 20 Art. ID 48.
  • [8] Chernoff, H. (1952). A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Stat. 23 493–507.
  • [9] Crescenzi, P., Nocentini, C., Pietracaprina, A. and Pucci, G. (2009). On the connectivity of Bluetooth-based ad hoc networks. Concurrency and Computation: Practice and Experience 21 875–887.
  • [10] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Applications of Mathematics (New York) 38. Springer, New York.
  • [11] Deuschel, J.-D. and Pisztora, A. (1996). Surface order large deviations for high-density percolation. Probab. Theory Related Fields 104 467–482.
  • [12] Dubhashi, D., Häggström, O., Mambrini, G., Panconesi, A. and Petrioli, C. (2007). Blue pleiades, a new solution for device discovery and scatternet formation in multi-hop Bluetooth networks. Wireless Networks 13 107–125.
  • [13] Dubhashi, D., Johansson, C., Häggström, O., Panconesi, A. and Sozio, M. (2005). Irrigating ad hoc networks in constant time. In Proceedings of the Seventeenth Annual ACM Symposium on Parallelism in Algorithms and Architectures 106–115. ACM, New York.
  • [14] Einmahl, U. (1989). Extensions of results of Komlós, Major, and Tusnády to the multivariate case. J. Multivariate Anal. 28 20–68.
  • [15] Fenner, T. I. and Frieze, A. M. (1982). On the connectivity of random $m$-orientable graphs and digraphs. Combinatorica 2 347–359.
  • [16] Ferraguto, F., Mambrini, G., Panconesi, A. and Petrioli, C. (2004). A new approach to device discovery and scatternet formation in Bluetooth networks. In Proceedings of the 18th International Parallel and Distributed Processing Symposium.
  • [17] Flajolet, P. and Odlyzko, A. M. (1990). Random mapping statistics. In Advances in Cryptology—EUROCRYPT ’89 (Houthalen, 1989). Lecture Notes in Computer Science 434 329–354. Springer, Berlin.
  • [18] Gilbert, E. N. (1961). Random plane networks. J. Soc. Indust. Appl. Math. 9 533–543.
  • [19] Häggström, O. and Meester, R. (1996). Nearest neighbor and hard sphere models in continuum percolation. Random Structures Algorithms 9 295–315.
  • [20] Hammersley, J. M. (1980). A generalization of McDiarmid’s theorem for mixed Bernoulli percolation. Math. Proc. Cambridge Philos. Soc. 88 167–170.
  • [21] Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs. Wiley-Interscience, New York.
  • [22] Kolchin, V. F. (1986). Random Mappings. Optimization Software, Inc., Publications Division, New York.
  • [23] Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent $\mathrm{RV}$’s and the sample $\mathrm{DF}$. I. Z. Wahrsch. Verw. Gebiete 32 111–131.
  • [24] Panagiotou, K., Spöhel, R., Steger, A. and Thomas, H. (2011). Explosive percolation in Erdős–Rényi-like random graph processes. Electron. Notes Discrete Math. 38 699–704.
  • [25] Penrose, M. (2003). Random Geometric Graphs. Oxford Studies in Probability 5. Oxford Univ. Press, Oxford.
  • [26] Penrose, M. D. (1996). Continuum percolation and Euclidean minimal spanning trees in high dimensions. Ann. Appl. Probab. 6 528–544.
  • [27] Penrose, M. D. (2016). Connectivity of soft random geometric graphs. Ann. Appl. Probab. 26 986–1028.
  • [28] Pettarin, A., Pietracaprina, A. and Pucci, G. (2009). On the expansion and diameter of Bluetooth-like topologies. In Algorithms—ESA 2009. Lecture Notes in Computer Science 5757 528–539. Springer, Berlin.
  • [29] Zaitsev, A. Y. (1998). Multidimensional version of the results of Komlós, Major and Tusnády for vectors with finite exponential moments. ESAIM Probab. Stat. 2 41–108 (electronic).