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Sharp thresholds for monotone properties in random geometric graphs. (English) Zbl 1192.05145

Proceedings of the 36th annual ACM symposium on theory of computing (STOC 2004), Chicago, IL, USA, June 13 - 15, 2004. New York, NY: ACM Press (ISBN 1-58113-852-0). 580-586, electronic only (2004).

MSC:

05C80 Random graphs (graph-theoretic aspects)

References:

[1] Martin J. B. Appel and Ralph P. Russo, The connectivity of a graph on uniform points on {0,1}d, Statistics & Probability Letters, 60 (2002), 351-7. · Zbl 1039.05054
[2] Lorna Booth, Jehoshua Bruck, Massimo Franceschetti, and Ronald Meester, Covering algorithms, continuum percolation and the geometry of wireless networks, The Annals of Applied Probability, 13 (2003), no. 2, 722-41. · Zbl 1029.60077
[3] Jean Bourgain and Gil Kalai, Influences of variables and threshold intervals under group symmetries, Geometric Functional Analysis (1997). · Zbl 0982.20004
[4] Béla Bollobás, Random graphs, Cambridge studies in advanced mathematics, vol. 73, Cambridge University Press, New York, 2001. · Zbl 0979.05003
[5] Josep Díaz, Mathew D. Penrose, Jordi Petit, and María Sema, Approximating layout problems on geometric random grapphs, Tech. Report TR-417-98, Algorithms and Complexity in Information Technology, 1998.
[6] Richard Durrett, Probability: Theory and Examples, second ed., Duxbury Press, 1996.
[7] Pál Erdos and Alfréd Renyi, On random graphs I, Publicationes Mathimaticae Debrecen (1959). · Zbl 0092.15705
[8] Pál Erdos and Alfréd Renyi, On the evolution of random graphs, Magyar Tud. Akad. Mat. Kut. Int. Kozl., 5 (1960), 17-61. · Zbl 0103.16301
[9] Ehud Friedgut and Gil Kalai, Every monotone graph property has a sharp threshold, Proceedings of the American Mathematical Society 124 (1996), 2993-3002. · Zbl 0864.05078
[10] Erhard Godehardt and Jerzy Jaworski, On the connectivity of a random interval graph, On the connectivity of a random interval graph 9 (1996), no. 1-2, 137-161. 10.1002/(SICI)1098-2418(199608/09)9:1/2 · Zbl 0864.05080
[11] Piyush Gupta and Panganmala R. Kumar, Stochastic analysis, control, optimization and applications: A volume in honor of W.H. Fleming, ch. Critical power for asymptotic connectivity in wireless networks, Birkhaüser, Boston, 1998.
[12] Piyush Gupta and Panganmala R. Kumar, The capacity of wireless networks, IEEE Transactions on Information Theory IT-46 (2000), no. 2, 388-404. 10.1109/18.825799 · Zbl 0991.90511
[13] Piyush Gupta and Panganmala R. Kumar, Internets in the sky: The capacity of three dimensional wireless networks, Communications in Information and Systems 1 (2001), no. 1, 33-50. · Zbl 1044.94520
[14] Erhard Godehardt, Graphs as structural models: the applications of graphs and multigraphs in cluster analysis, Vieweg, Brunswick, 1990.
[15] Alexander Holroyd and Yuval Peres, Trees and matchings from point processes, Electronic communications in probability 8 (2003), no. Paper 3, 17-27. · Zbl 1060.60048
[16] Jordi Petit i Silvestre, Layout problems, Ph.D. thesis, Universitat Politècnica de Catalunya, Barcelona, 25 May 2001.
[17] Svante Janson, Tomasz Luczak, and Andrzej Ruciński, Random graphs, Wiley-Interscience series in discrete mathematics and optimization, Wiley-Interscience, New York, 2000. · Zbl 0968.05003
[18] Bhaskar Krishnamachari, Stephen Wicker, Ramon Bejar, and Marc Pearlman, Communications, information and network security, ch. Critical Density Thresholds in Distributed Wireless Networks, Kluwer, December 2002.
[19] Tom Leighton and Peter W. Shor, Tight bounds for minimax grid matching, with applications to the average case analysis of algorithms, Combinatorica 9 (1989), 161-87. · Zbl 0686.68039
[20] Gregory McColm, Threshold functions for random graphs on a line segment, Submitted for publication. · Zbl 1048.05076
[21] S. Muthukrishnan and Gopal Pandurangan, The bin-covering technique for thresholding random geometric graph properties, Tech. Report 2003-39, DIMACS, November 2003.
[22] Valery B. Nevzorov, Records: A mathematical theory, Translations of Mathematical Monographs, vol. 194, American Mathematical Society, Providence, Rhode Island, 2001.
[23] Mathew D. Penrose, The longest edge of the random minimal spanning tree, The Annals of Applied Probability 7 (1997), no. 2, 340-61. · Zbl 0884.60042
[24] Mathew D. Penrose, Random geometric graphs, Oxford Studies in Probability, vol. 5, Oxford University Press, May 2003. · Zbl 1029.60007
[25] S. Shakkottai, R. Srikant, and N. B. Shroff, Unreliable sensor grids: coverage, connectivity and diameter, Proc. Infocom 2003. 10.1016/j.adhoc.2004.02.001
[26] Peter W. Shor and Joseph E. Yukich, Minmax grid matching and empirical measures, The Annals of Probability 19 (1991), no. 3, 1338-48. · Zbl 0734.60005
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