Advertisement

Cliques in High-Dimensional Random Geometric Graphs

  • Konstantin Avrachenkov
  • Andrei BobuEmail author
Conference paper
Part of the Studies in Computational Intelligence book series (SCI, volume 881)

Abstract

Random geometric graphs are good examples of random graphs with a tendency to demonstrate community structure. Vertices of such a graph are represented by points in Euclid space \(\mathbb R^d\), and edge appearance depends on the distance between the points. Random geometric graphs were extensively explored and many of their basic properties are revealed. However, in the case of growing dimension \(d\rightarrow \infty \) practically nothing is known; this regime corresponds to the case of data with many features, a case commonly appearing in practice. In this paper, we focus on the cliques of these graphs in the situation when average vertex degree grows significantly slower than the number of vertices n with \(n\rightarrow \infty \) and \(d\rightarrow \infty \). We show that under these conditions random geometric graphs do not contain cliques of size 4 a.s. As for the size 3, we will present new bounds on the expected number of triangles in the case \(\log ^2 n \ll d \ll \log ^3 n\) that improve previously known results.

Keywords

Random geometric graphs High dimension Clique number Triangles 

References

  1. 1.
    Erdős, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. Ser. A 5(1), 17–60 (1960)Google Scholar
  2. 2.
    Bollobás, B.: Random Graphs. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  3. 3.
    Alon, N., Spencer, J.H.: The Probabilistic Method. Wiley, New York (2004)zbMATHGoogle Scholar
  4. 4.
    Preciado, V.M., Jadbabaie, A.: Spectral analysis of virus spreading in random geometric networks. In: Proceedings of the 48th IEEE Conference on Decision and Control (CDC) held Jointly with 2009 28th Chinese Control Conference, pp. 4802–4807 (2009).  https://doi.org/10.1109/CDC.2009.5400615
  5. 5.
    Haenggi, M., Andrews, J.G., Baccelli, F., Dousse, O., Franceschetti, M.: Stochastic geometry and random graphs for the analysis and design of wireless networks. IEEE J. Sel. Areas Commun. 27(7), 1029–1046 (2009).  https://doi.org/10.1109/JSAC.2009.090902CrossRefGoogle Scholar
  6. 6.
    Pottie, G.J., Kaiser, W.J.: Wireless integrated network sensors. Commun. ACM 43(5), 51–58 (2000).  https://doi.org/10.1145/332833.332838CrossRefGoogle Scholar
  7. 7.
    Nekovee, M.: Worm epidemics in wireless ad hoc networks. New J. Phys. 9(6), 189 (2007).  https://doi.org/10.1088/1367-2630/9/6/189CrossRefGoogle Scholar
  8. 8.
    Xiao, H., Yeh, E.M.: Cascading link failure in the power grid: a percolation-based analysis. In: 2011 IEEE International Conference on Communications Workshops (ICC), pp. 1–6 (2011).  https://doi.org/10.1109/iccw.2011.5963573
  9. 9.
    Higham, D.J., Ras̆ajski, M., Prz̆ulj, N.: Fitting a geometric graph to a protein-protein interaction network. Bioinformatics 24(8), 1093–1099 (2008).  https://doi.org/10.1093/bioinformatics/btn079CrossRefGoogle Scholar
  10. 10.
    Arias-Castro, E., Bubeck, S., Lugosi, G.: Detecting positive correlations in a multivariate sample. Bernoulli 21(1), 209–241 (2015).  https://doi.org/10.3150/13-BEJ565MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Penrose, M.: Random Geometric Graphs, vol. 5. Oxford University Press, Oxford (2003)CrossRefGoogle Scholar
  12. 12.
    Penrose, M.D.: On \(k\)-connectivity for a geometric random graph. Random Struct. Algorithms 15(2), 145–164 (1999). https://doi.org/10.1002/(SICI)1098-2418(199909)15:2%3C145::AID-RSA2%3E3.0.CO;2-GMathSciNetCrossRefGoogle Scholar
  13. 13.
    Appel, M.J., Russo, R.P.: The connectivity of a graph on uniform points on \([0, 1]^d\). Stat. Probab. Lett. 60(4), 351–357 (2002).  https://doi.org/10.1016/S0167-7152(02)00233-XMathSciNetCrossRefGoogle Scholar
  14. 14.
    McDiarmid, C.: Random channel assignment in the plane. Random Struct. Algorithms 22(2), 187–212 (2003).  https://doi.org/10.1002/rsa.10077MathSciNetCrossRefGoogle Scholar
  15. 15.
    Müller, T.: Two-point concentration in random geometric graphs. Combinatorica 28(5), 529 (2008).  https://doi.org/10.1007/s00493-008-2283-3MathSciNetCrossRefGoogle Scholar
  16. 16.
    McDiarmid, C., Müller, T.: On the chromatic number of random geometric graphs. Combinatorica 31(4), 423–488 (2011).  https://doi.org/10.1007/s00493-011-2403-3MathSciNetCrossRefGoogle Scholar
  17. 17.
    Devroye, L., György, A., Lugosi, G., Udina, F.: High-dimensional random geometric graphs and their clique number. Electron. J. Probab. 16, 2481–2508 (2011).  https://doi.org/10.1214/EJP.v16-967MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bubeck, S., Ding, J., Eldan, R., Rácz, M.Z.: Testing for high-dimensional geometry in random graphs. Random Struct. Algorithms 49(3), 503–532 (2016).  https://doi.org/10.1002/rsa.20633MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lee, Y., Kim, W.C.: Concise formulas for the surface area of the intersection of two hyperspherical caps. KAIST Technical Report (2014)Google Scholar
  20. 20.
    Brieden, A., Gritzmann, P., Kannan, R., Klee, V., Lovász, L., Simonovits, M.: Deterministic and randomized polynomial-time approximation of radii. Mathematika 48(1–2), 63–105 (2001).  https://doi.org/10.1112/S0025579300014364MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.InriaValbonneFrance

Personalised recommendations