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Percolation, Connectivity, Coverage and Colouring of Random Geometric Graphs

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Handbook of Large-Scale Random Networks

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 18))

Abstract

In this review paper, we shall discuss some recent results concerning several models of random geometric graphs, including the Gilbert disc model G r , the k-nearest neighbour model G nn k and the Voronoi model G P . Many of the results concern finite versions of these models. In passing, we shall mention some of the applications to engineering and biology.

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References

  1. M. Aizenman, Scaling limit for the incipient spanning clusters, in Mathematics of multiscale materials (Minneapolis, MN, 1995–1996), IMA Vol. Math. Appl., 99 (1998), 1–24.

    MathSciNet  Google Scholar 

  2. R. Arratia, L. Goldstein and L. Gordon, Two moments suffice for Poisson approximations: The Chen-Stein method, Ann. Probab., 17 (1989), 9–25.

    Article  MATH  MathSciNet  Google Scholar 

  3. A. Bagchi and S. Bansal, On the metric distortion of nearest-neighbour graphs on random point sets, available at http://arxiv.org/abs/0804.3784.

    Google Scholar 

  4. P. Balister, B. Bollobás, S. Kumar and A. Sarkar, Reliable density estimates for deployment of sensors in thin strips (detailed proofs), Technical Report, University of Memphis, 2007. Available at http://umdrive.memphis.edu/pbalistr/public/ThinStripComplete.pdf

    Google Scholar 

  5. P. Balister, B. Bollobás and A. Quas, Percolation in Voronoi tilings, Random Structures and Algorithms, 26 (2005), 310–318.

    Article  MATH  MathSciNet  Google Scholar 

  6. P. Balister, B. Bollobás, A. Sarkar and S. Kumar, Reliable density estimates for coverage and connectivity in thin strips of finite length, ACM MobiCom, Montréal, Canada (2007), 75–86.

    Google Scholar 

  7. P. Balister, B. Bollobás, A. Sarkar and M. Walters, Connectivity of random knearest neighbour graphs, Advances in Applied Probability, 37 (2005), 1–24.

    Article  MATH  MathSciNet  Google Scholar 

  8. P. Balister, B. Bollobás, A. Sarkar and M. Walters, Connectivity of a gaussian network, International Journal of Ad-Hoc and Ubiquitous Computing, 3 (2008), 204–213.

    Article  Google Scholar 

  9. P. Balister, B. Bollobás, A. Sarkar and M. Walters, Highly connected random geometric graphs, Discrete Applied Mathematics, 157 (2009), 309–320.

    Article  MATH  MathSciNet  Google Scholar 

  10. P. Balister, B. Bollobás, A. Sarkar and M. Walters, A critical constant for the k-nearest neighbour model, Advances in Applied Probability, 41 (2009), 1–12.

    Article  MATH  MathSciNet  Google Scholar 

  11. P. Balister, B. Bollobás, A. Sarkar and M. Walters, Sentry selection in wireless networks, submitted.

    Google Scholar 

  12. P. Balister, B. Bollobás and M. Walters, Continuum percolation with steps in an annulus, Annals of Applied Probability, 14 (2004), 1869–1879.

    Article  MATH  MathSciNet  Google Scholar 

  13. P. Balister, B. Bollobás and M. Walters, Continuum percolation with steps in the square or the disc, Random Structures and Algorithms, 26 (2005), 392–403.

    Article  MATH  MathSciNet  Google Scholar 

  14. P. Balister, B. Bollobás and M. Walters, Random transceiver networks, to appear in Advances in Applied Probability.

    Google Scholar 

  15. P. Balister, B. Bollobás and M. Walters, Percolation in the k-nearest neighbour model, submitted.

    Google Scholar 

  16. I. Benjamini and O. Schramm, Conformal invariance of Voronoi percólation, Commun. Math. Phys., 197 (1998), 75–107.

    Article  MATH  MathSciNet  Google Scholar 

  17. B. Bollobás, Random Graphs, second edition, Cambridge University Press, 2001.

    Google Scholar 

  18. B. Bollobás and O. M. Riordan, Percolation, Cambridge University Press, 2006, x + 323pp.

    Google Scholar 

  19. B. Bollobás and O. M. Riordan, The critical probability for random Voronoi percolation in the plane is 1/2, Probability Theory and Related Fields, 136 (2006), 417–468.

    Article  MATH  MathSciNet  Google Scholar 

  20. B. Bollobás and O. M. Riordan, A short proof of the Harris-Kesten Theorem, Bull. London Math. Soc., 38 (2006), 470–484.

    Article  MATH  MathSciNet  Google Scholar 

  21. B. Bollobás and O. M. Riordan, Percolation on random Johnson-Mehl tessellations and related models, Probability Theory and Related Fields, 140 (2008), 319–343.

    Article  MATH  MathSciNet  Google Scholar 

  22. A. Delesse, Procédé méchanique pour déterminer la composition des roches, Ann. des Mines (4th Ser.), 13 (1848), 379–388.

    Google Scholar 

  23. G. L. Dirichlet, Über die Reduktion der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen, Journal für die Reine und Angewandte Mathematik, 40 (1850), 209–227.

    Article  MATH  Google Scholar 

  24. D. W. Etherington, C. K. Hoge and A. J. Parkes, Global surrogates, manuscript, 2003.

    Google Scholar 

  25. J. W. Evans, Random and cooperative adsorption, Rev. Mod. Phys., 65 (1993), 1281–1329.

    Article  Google Scholar 

  26. M. Fanfoni and M. Tomellini, The Johnson-Mehl-Avrami-Kolmogorov model — a brief review, Nuovo Cimento della Societa Italiana di Fisica. D, 20 (7–8) (1998), 1171–1182.

    Article  Google Scholar 

  27. M. Fanfoni and M. Tomellini, Film growth viewed as stochastic dot processes, J. Phys.: Condens. Matter, 17 (2005), R571–R605.

    Article  Google Scholar 

  28. M. Franceschetti, L. Booth, M. Cook, R. Meester and J. Bruck, Continuum percolation with unreliable and spread-out connections, Journal of Statistical Physics, 118 (2005), 721–734.

    Article  MATH  MathSciNet  Google Scholar 

  29. M. H. Freedman, Percolation on the projective plane, Math. Res. Lett., 4 (1997), 889–894.

    MATH  MathSciNet  Google Scholar 

  30. H. L. Frisch and J. M. Hammersley, Percolation processes and related topics, J. Soc. Indust. Appl. Math., 11 (1963), 894–918.

    Article  MathSciNet  Google Scholar 

  31. E.N. Gilbert, Random plane networks, J. Soc. Indust. Appl. Math., 9 (1961), 533–543.

    Article  MATH  MathSciNet  Google Scholar 

  32. E. N. Gilbert, The probability of covering a sphere with N circular caps, Biometrika, 56 (1965), 323–330.

    Google Scholar 

  33. J. M. Gonzáles-Barrios and A. J. Quiroz, A clustering procedure based on the comparison between the k nearest neighbors graph and the minimal spanning tree, Statistics and Probability Letters, 62 (2003), 23–34.

    Article  MathSciNet  Google Scholar 

  34. O. Häggström and R. Meester, Nearest neighbor and hard sphere models in continuum percolation, Random Structures and Algorithms, 9 (1996), 295–315.

    Article  MATH  MathSciNet  Google Scholar 

  35. P. Hall, On the coverage of k-dimensional space by k-dimensional spheres, Annals of Probability, 13 (1985), 991–1002.

    Article  MATH  MathSciNet  Google Scholar 

  36. P. Hall, On continuum percolation, Annals of Probability, 13 (1985), 1250–1266.

    Article  MATH  MathSciNet  Google Scholar 

  37. T. E. Harris, A lower bound for the critical probability in a certain percolation process, Proc. Cam. Philos. Soc., 56 (1960), 13–20.

    Article  MATH  Google Scholar 

  38. S. Janson, Random coverings in several dimensions, Acta Mathematica, 156 (1986), 83–118.

    Article  MATH  MathSciNet  Google Scholar 

  39. G. R. Jerauld, J. C. Hatfield, L. E. Scriven and H. T. Davis, Percolation and conduction on Voronoi and triangular networks: a case study in topological disorder, J. Physics C: Solid State Physics, 17 (1984), 1519–1529.

    Article  Google Scholar 

  40. G. R. Jerauld, L. E. Scriven and H. T. Davis, Percolation and conduction on the 3D Voronoi and regular networks: a second case study in topological disorder, J. Physics C: Solid State Physics, 17 (1984), 3429–3439.

    Article  Google Scholar 

  41. H. Kesten, The critical probability of bond percolation on the square lattice equals 1/2, Comm. Math. Phys., 74 (1980), 41–59.

    Article  MATH  MathSciNet  Google Scholar 

  42. R. Langlands, P. Pouliot and Y. Saint-Aubin, Conformal invariance in two-dimensional percolation, Bull. Amer. Math. Soc., 30 (1994), 1–61.

    Article  MATH  MathSciNet  Google Scholar 

  43. R. Meester and R. Roy, Continuum Percolation, Cambridge University Press, 1996.

    Google Scholar 

  44. G. L. Miller, S. H. Teng and S. A. Vavasis, An unified geometric approach to graph separators, in IEEE 32nd Annual Symposium on Foundations of Computer Science, 1991, 538–547.

    Google Scholar 

  45. P. A. P. Moran and S. Fazekas de St Groth, Random circles on a sphere, Biometrika, 49 (1962), 389–396.

    MATH  MathSciNet  Google Scholar 

  46. B. Pacchiarotti, M. Fanfoni and M. Tomellini, Roughness in the Kolmogorov-Johnson-Mehl-Avrami framework: extension to (2+1)D of the Trofimov-Park model, Physica A, 358 (2005), 379–392.

    Article  Google Scholar 

  47. M. D. Penrose, Continuum percolation and Euclidean minimal spanning trees in high dimensions, Annals of Applied Probability, 6 (1996), 528–544.

    Article  MATH  MathSciNet  Google Scholar 

  48. M. D. Penrose, The longest edge of the random minimal spanning tree, Annals of Applied Probability, 7 (1997), 340–361.

    Article  MATH  MathSciNet  Google Scholar 

  49. M. D. Penrose, Random Geometric Graphs, Oxford University Press, 2003.

    Google Scholar 

  50. J. Quintanilla, S. Torquato and R. M. Ziff, Efficient measurement of the percolation threshold for fully penetrable discs, J. Phys. A, 33 (42): L399–L407 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  51. R. A. Ramos, P. A. Rikvold and M. A. Novotny, Test of the Kolmogorov-Johnson-Mehl-Avrami picture of metastable decay in a model with microscopic dynamics, Phys. Rev. B, 59 (1999), 9053–9069.

    Article  Google Scholar 

  52. S. Smirnov, Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits, Comptes Rendus de l’Académie des Sciences. Série I. Mathématique, 333 (2001), 239–244.

    Article  MATH  Google Scholar 

  53. S. Teng and F. Yao, k-nearest-neighbor clustering and percolation theory, Algorithmica, 49 (2007), 192–211.

    Article  MATH  MathSciNet  Google Scholar 

  54. M. Tomellini, M. Fanfoni and M. Volpe, Spatially correlated nuclei: How the Johnson-Mehl-Avrami-Kolmogorov formula is modified in the case of simultaneous nucleation, Phys. Rev. B, 62 (2000), 11300–11303.

    Article  Google Scholar 

  55. M. Tomellini, M. Fanfoni and M. Volpe, Phase transition kinetics in the case of nonrandom nucleation, Phys. Rev. B, 65 (2002), 140301-1–140301-4.

    Article  Google Scholar 

  56. M. Q. Vahidi-Asl and J. C. Wierman, First-passage percolation on the Voronoi tessellation and Delaunay triangulation, in Random graphs’ 87 (Poznań, 1987), Wiley, Chichester (1990), pp. 341–359.

    Google Scholar 

  57. M. Q. Vahidi-Asl and J. C. Wierman, A shape result for first-passage percolation on the Voronoi tessellation and Delaunay triangulation, in: Random graphs, Vol. 2 (Poznań, 1989), Wiley-Intersci. Publ., Wiley, New York (1992), pp. 247–262.

    Google Scholar 

  58. M. Q. Vahidi-Asl and J. C. Wierman, Upper and lower bounds for the route length of first-passage percolation in Voronoi tessellations, Bull. Iranian Math. Soc., 19 (1993), 15–28.

    MATH  MathSciNet  Google Scholar 

  59. G. Voronoi, Nouvelles applications des paramètres continus à la théorie des formes quadratiques, Journal für die Reine und Angewandte Mathematik, 133 (1908), 97–178.

    Article  MATH  Google Scholar 

  60. P. Wan and C.W. Yi, Asymtotic critical transmission radius and critical neighbor for k-connectivity in wireless ad hoc networks, ACM MobiHoc, Roppongi, Japan (2004).

    Google Scholar 

  61. P. H. Winterfeld, L. E. Scriven and H. T. Davis, Percolation and conductivity of random two-dimensional composites, J. Physics C, 14 (1981), 2361–2376.

    Article  Google Scholar 

  62. F. Xue and P. R. Kumar, The number of neighbors needed for connectivity of wireless networks, Wireless Networks, 10 (2004), 169–181.

    Article  Google Scholar 

  63. F. Xue and P. R. Kumar, On the theta-coverage and connectivity of large random networks, IEEE Transactions on Information Theory, 52 (2006), 2289–2399.

    Article  MathSciNet  Google Scholar 

  64. A. Zvavitch, The critical probability for Voronoi percolation, MSc. thesis, Weizmann Institute of Science (1996).

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Memphis, Memphis, TN, 38152, USA

    Paul Balister & Béla Bollobás

  2. Department of Mathematics, Western Washington University, Bellingham, WA, 98225, USA

    Amites Sarkar

  3. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB3 0WB, UK

    Béla Bollobás

Authors
  1. Paul Balister
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  2. Amites Sarkar
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  3. Béla Bollobás
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Editor information

Editors and Affiliations

  1. Department of Pure Mathematics & Mathematical Statistics, University of Cambridge, Centre for Mathematical Sciences, Cambridge, CB3 0WB, UK

    Béla Bollobás

  2. Dept. Mathematical Sciences, University of Memphis, Memphis, TN, 38152, USA

    Béla Bollobás  & Robert Kozma  & 

  3. Alfréd Rényi Inst. of Mathematics, Reáltanoda u. 13-15, Budapest, 1053, Hungary

    Dezső Miklós

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© 2008 János Bolyai Mathematical Society and Springer-Verlag

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Balister, P., Sarkar, A., Bollobás, B. (2008). Percolation, Connectivity, Coverage and Colouring of Random Geometric Graphs. In: Bollobás, B., Kozma, R., Miklós, D. (eds) Handbook of Large-Scale Random Networks. Bolyai Society Mathematical Studies, vol 18. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69395-6_2

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