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SCIENTIA SINICA Mathematica, Volume 42, Issue 8: 821-826(2012) https://doi.org/10.1360/012011-371

A central limit theorem of random biased connected graphs

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  • AcceptedMay 15, 2012
  • PublishedAug 6, 2012

Abstract

In this paper, we consider a class of random connected graphs with random vertices and random edges, the randomness of vertices decided by a Poisson point process with intensity n and the randomness of the latter by a connected function. The central limit theorem on the total lengths of all random edges was obtained.


References

[1] Grimmett G. Percolation, 2nd ed. Berlin: Springer, 1999. Google Scholar

[2] Bollobás B, Riordan O. Percolation. Cambridge: Cambridge University Press, 2006. Google Scholar

[3] Zhang Y. A martingale approach in the study of percolation clusters on the Zd lattice. J Theor Probab, 2001, 14:165-187. CrossRef Google Scholar

[4] Kuulasmaa K. The spatial general epidemic and locally dependent random graphs. J Appl Probab, 1982, 19: 745-758. CrossRef Google Scholar

[5] Berg J, van den Grimmett G R, Schinazi R. Dependent random graphs and spatial epidemics. Ann Appl Probab,1998, 8: 317-336. CrossRef Google Scholar

[6] Xu Z H, Han D. Limit theorems for size of the clusters of dependent percolation process on Z2. Chinese J Appl Probab Statistic, 2010, 26: 649-661. Google Scholar

[7] Benjamini I, Berger N. The diameter of long-range percolation clusters on finite cycles. Random Struct Alg, 2001, 19:102-111. Google Scholar

[8] Coppersmith D, Gamarnik D, Sviridenko M. The diameter of a long range percolation graph. Random Struct Alg,2002, 21: 1-13. Google Scholar

[9] Penrose M D. Random Geometric Graphs. In: Oxford Studies in Probability, vol. 5. Oxford: Oxford University Press,2003. Google Scholar

[10] Penrose M D. Central limit theorems for k-nearest neighbour distances. Stoch Proc Appl, 2000, 85: 295-320. Google Scholar

[11] Xue F, Kumar P R. The number of neighbors needed for connectivity of wireless networks. Wireless Networks, 2004,10: 169-181. CrossRef Google Scholar

[12] Meester R, Roy R. Continuum Percolation. Cambridge: Cambridge University Press, 1996. Google Scholar

[13] Bickel P J, Breiman L. Sums of functions of nearest neighbor distances, moment bounds, and a goodness of t test. Ann Probab, 1983, 11: 185-214. CrossRef Google Scholar

[14] Evans D, Jones A J. A proof of Gamma test. Proc R Soc London A, 2000, 458: 2759-2799. Google Scholar

[15] Henze N. On the fraction of random points with speci ed nearest-neighbour interrelations and degree of attraction. Adv Appl Probab, 1987, 19: 873-895. CrossRef Google Scholar

[16] Xu Z H, Higuchi Y, Hu C H. A strong law of large numbers for random biased connected graphs. Theoret Math Phys, in press. Google Scholar

[17] Baldi P, Rinott Y. On normal approximations of distributions in terms of dependency graphs. Ann Probab, 1989, 17:1646-1650. CrossRef Google Scholar

[18] Miller G L, Teng S H, Vavasis S. A uni ed geometric approach to graph separators. In: Proceedings of the 32nd Annual Symposium on Foundation of Computer Science. Boston: IEEE, 1991, 538-547. Google Scholar

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