Limit theory for the random on-line nearest-neighbor graph

Random Structures and Algorithms

Volumn 32, Issue 2, 2008, Pages 125-156

  Penrose, Mathew D ^a   Wade, Andrew R ^b  

^a University of Bath   (United Kingdom)

^b UNIVERSITY OF BRISTOL   (United Kingdom)

Author keywords

Divide and conquer; Fixed point equation; Nearest neighbor graph; Spatial network evolution; Weak convergence

Indexed keywords

EID: 52349123319     PISSN: 10429832     EISSN: 10982418     Source Type: Journal    
DOI: 10.1002/rsa.20185     Document Type: Article

References (28)

1. M. Abramowitz and I. A. Stegun, Eds. Handbook of mathematical functions, National Bureau of Standards (Applied mathematics series, Vol. 55), U.S. Government Printing Office, Washington, DC, 1965.
2. D. J. Aldous and A. Bandyopadhyay, A survey of max-type recursive distributional equations, Ann Appl Probab 15 (2005), 1047-1110.
3. N. Berger, B. Bollobás, C. Borgs, J. Chayes, and O. Riordan, "Degree distribution of the FKP model," In Automata, languages and programming (Lecture notes in computer science, Vol. 2719), J. C. M. Baeten, J. K. Lenstra, J. Parrow, G. J. Woeginger (Editors), Springer, Heidelberg, 2003, pp. 725-738.
4. J. Bertoin and A. Gnedin, Asymptotic laws for nonconservative selfsimilar fragmentations, Electr J Probab 9 (2004), 575-593.
5. P. Billingsley, Convergence of probability measures, 2nd edition, Wiley, New York, 1999.
6. B. Bollobás and O. M. Riordan, "Mathematical results on scale-free random graphs," In Handbook of graphs and networks, S. Bornholdt, H. G. Schuster (Editors), Wiley-VCH, Weinheim, 2003, pp. 1-34.
7. D. A. Darling, On a class of problems related to the random division of an interval, Ann Math Stats 24 (1953), 239-253.
8. S. N. Dorogovstev and J. F. F. Medes, Evolution of networks, Adv Phys 51 (2002), 1079-1187.
9. A. Fabrikant, E. Koutsoupias, and C. H. Papadimitriou, "Heuristically optimized trade-offs: A new paradigm for power laws in the internet," In Automata, languages and programming (Lecture notes in computer science, Vol. 2380), P. Widmayer, F. Triguero, R. Morales, M. Hennessy, S. Eidenbenz, R. Conejo (Editors), Springer, Berlin, 2002, pp. 110-122.
10. K. Huang, Statistical mechanics, 2nd edition, Wiley, New York, 1987.
11. H. Kesten and S. Lee, The central limit theorem for weighted minimal spanning trees on random points, Ann Appl Probab 6 (1996), 495-527.
12. R. Neininger and L. Rüschendorf, A general limit theorem for recursive algorithms and combinatorial structures, Ann Appl Probab 14 (2004), 378-418.
13. M. E. J. Newman, The structure and function of complex networks, SIAM Rev 45 (2003), 167-256.
14. M. Penrose, Random geometric graphs (Oxford studies in probability, Vol. 6), Clarendon Press, Oxford, 2003.
15. M. D. Penrose, Multivariate spatial central limit theorems with applications to percolation and spatial graphs, Ann Probab 33 (2005), 1945-1991.
16. M. D. Penrose and A. R. Wade, On the total length of the random minimal directed spanning tree, Adv Appl Probab 38 (2006), 336-372.
17. M. D. Penrose and J. E. Yukich, Central limit theorems for some graphs in computational geometry, Ann Appl Probab 11 (2001), 1005-1041.
18. M. D. Penrose and J. E. Yukich, Weak laws of large numbers in geometric probability, Ann Appl Probab 13 (2003), 277-303.
19. M. D. Penrose and J. E. Yukich, "Normal approximation in geometric probability," In Stein's method and applications (Lecture notes series, Institute for Mathematical Sciences, National University of Singapore, Vol. 5), A. D. Barbour, L. H. Y. Chen (Editors), Singapore University Press, Singapore, 2005, pp. 37-58.
20. R. Pyke, Spacings, J Royal Stat Soc Ser B 27 (1965), 395-449.
21. S. T. Rachev, Probability metrics and the stability of stochastic models, Wiley, Chichester, 1991.
22. U. Rösler, A fixed point theorem for distributions, Stoch Process Appl 42 (1992), 195-214.
23. U. R̈osler and L. Rüschendorf, The contraction method for recursive algorithms, Algorithmica 29 (2001), 3-33.
24. J. M. Steele, Probability theory and combinatorial optimization, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997.
25. A. R. Wade, Limiting behaviour of random spatial graphs and asymptotically homogeneous RWRE, Ph.D. Thesis, University of Durham, 2005.
26. A. R. Wade, Explicit laws of large numbers for random nearest-neighbour type graphs, Adv Appl Probab 39 (2007), 326-342.
27. A. R. Wade, Central limit theorem for the on-line nearest-neighbour graph, Preprint available from http://arxiv.org/abs/math.PR/0702414.
28. J. E. Yukich, Probability theory of classical Euclidean optimization problems (Lecture notes in mathematics, Vol. 1675), Springer, Berlin, 1998.