Electronic Journal of Probability

High-Dimensional Random Geometric Graphs and their Clique Number

Luc Devroye, András György, Gábor Lugosi, and Frederic Udina

Full-text: Open access

Abstract

We study the behavior of random geometric graphs in high dimensions. We show that as the dimension grows, the graph becomes similar to an Erdös-Rényi random graph. We pay particular attention to the clique number of such graphs and show that it is very close to that of the corresponding Erdös-Rényi graph when the dimension is larger than $\log^3(n)$ where $n$ is the number of vertices. The problem is motivated by a statistical problem of testing dependencies.

Article information

Source
Electron. J. Probab. Volume 16 (2011), paper no. 90, 2481-2508.

Dates
Accepted: 30 November 2011
First available in Project Euclid: 1 June 2016

Permanent link to this document
http://projecteuclid.org/euclid.ejp/1464820259

Digital Object Identifier
doi:10.1214/EJP.v16-967

Mathematical Reviews number (MathSciNet)
MR2861682

Zentralblatt MATH identifier
1244.05200

Subjects
Primary: 05C80: Random graphs [See also 60B20]
Secondary: 62H15: Hypothesis testing

Keywords
Clique number dependency testing geometric graphs random graphs

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Devroye, Luc; György, András; Lugosi, Gábor; Udina, Frederic. High-Dimensional Random Geometric Graphs and their Clique Number. Electron. J. Probab. 16 (2011), paper no. 90, 2481--2508. doi:10.1214/EJP.v16-967. http://projecteuclid.org/euclid.ejp/1464820259.


Export citation

References

  • Alon, Noga; Krivelevich, Michael; Sudakov, Benny. Finding a large hidden clique in a random graph. Proceedings of the Eighth International Conference "Random Structures and Algorithms” (Poznan, 1997). Random Structures Algorithms 13 (1998), no. 3-4, 457–466.
  • Alon, Noga; Spencer, Joel H. The probabilistic method. With an appendix by Paul Erdós. Wiley-Interscience Series in Discrete Mathematics and Optimization. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1992. xvi+254 pp. ISBN: 0-471-53588-5.
  • Ané, Cécile; Blachère, Sébastien; Chafaï, Djalil; Fougères, Pierre; Gentil, Ivan; Malrieu, Florent; Roberto, Cyril; Scheffer, Grégory. Sur les inégalités de Sobolev logarithmiques. (French) [Logarithmic Sobolev inequalities] With a preface by Dominique Bakry and Michel Ledoux. Panoramas et Synthèses [Panoramas and Syntheses], 10. Société Mathématique de France, Paris, 2000. xvi+217 pp. ISBN: 2-85629-105-8.
  • Bentkus, V. On the dependence of the Berry-Esseen bound on dimension. J. Statist. Plann. Inference 113 (2003), no. 2, 385–402.
  • Bollobás, Béla. Random graphs. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985. xvi+447 pp. ISBN: 0-12-111755-3; 0-12-111756-1.
  • Brieden, Andreas; Gritzmann, Peter; Kannan, Ravindran; Klee, Victor; Lovász, László; Simonovits, Miklós. Deterministic and randomized polynomial-time approximation of radii. Mathematika 48 (2001), no. 1-2, 63–105 (2003).
  • Chernoff, Herman. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statistics 23, (1952). 493–507.
  • Danzer, Ludwig; Grünbaum, Branko; Klee, Victor. Helly's theorem and its relatives. 1963 Proc. Sympos. Pure Math., Vol. VII pp. 101–180 Amer. Math. Soc., Providence, R.I.
  • Dudley, R. M. Central limit theorems for empirical measures. Ann. Probab. 6 no. 6, 899–929 (1979).
  • Erdós, P.; Rényi, A. On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutat' Int. Közl. 5 1960 17–61.
  • Janson, Svante; Łuczak, Tomasz; Rucinski, Andrzej. Random graphs. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York, 2000. xii+333 pp. ISBN: 0-471-17541-2.
  • Jung, H. Über die kleinste Kugel, die eine räumliche Figur einschliesst. J. Reine Angew. Math. 123 1901 241–257.
  • Massart, Pascal. Concentration inequalities and model selection. Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, July 6–23, 2003. With a foreword by Jean Picard. Lecture Notes in Mathematics, 1896. Springer, Berlin, 2007. xiv+337 pp. ISBN: 978-3-540-48497-4; 3-540-48497-3
  • Nadakudit, R.R.; Silverstein, J.W. Fundamental limit of sample generalized eigenvalue based detection of signals in noise using relatively few signal-bearing and noise-only samples Technical Report. 2009.
  • Palmer, Edgar M. Graphical evolution. An introduction to the theory of random graphs. Wiley-Interscience Series in Discrete Mathematics. A Wiley-Interscience Publication. John Wiley & Sons, Ltd., Chichester, 1985. xvii+177 pp. ISBN: 0-471-81577-2.
  • Penrose, Mathew. Random geometric graphs. Oxford Studies in Probability, 5. Oxford University Press, Oxford, 2003. xiv+330 pp. ISBN: 0-19-850626-0
  • Raič, M.. Normalna aproksimacija po Steinovi metodi. PhD thesis, Univerza v Ljubljani, 2009.
  • Vapnik, V. N.; Chervonenkis, A. Ya. \cyr Teoriya raspoznavaniya obrazov. Statisticheskie problemy obucheniya. (Russian) [Theory of pattern recognition. Statistical problems of learning] Izdat. "Nauka" WHERE article_id=Moscow, 1974. 415 pp.