Abstract
Let P be a realization of a homogeneous Poisson point process in ℝd with density 1. We prove that there exists a constant k d , 1<k d <∞, such that the k-nearest neighborhood graph of P has an infinite connected component with probability 1 when k≥k d . In particular, we prove that k 2≤213. Our analysis establishes and exploits a close connection between the k-nearest neighborhood graphs of a Poisson point set and classical percolation theory. We give simulation results which suggest k 2=3. We also obtain similar results for finite random point sets.
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Aizenman, M., Chayes, J.T., Chayes, L., Frohlich, J., Russo, L.: On a sharp transition from area law to perimeter law in a system of random surfaces. Commun. Math. Phys. 92, 19–69 (1983)
Bern, M., Eppstein, D., Yao, F.: The expected extremes in a Delaunay triangulation. Int. J. Comput. Geom. Appl. 1(1), 79–91 (1991)
Chayes, J.T., Chayes, L.: Bulk transport properties and exponent inequalities for random resistor and flow networks. Commun. Math. Phys. 105, 133–152 (1986)
Grimmett, G.: Percolation. Springer, New York (1989)
Hammersley, J.M.: A generalization of McDiarmid’s theorem for mixed Bernoulli percolation. Math. Proc. Camb. Philos. Soc. 88, 167–170 (1980)
Karp, R.M., Steele, J.M.: Probabilistic analysis of heuristics. In: Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B. (eds.) The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, New York (1985)
Kesten, H.: Percolation Theory for Mathematicians. Birkhäuser, Basel (1982)
Meester, R., Roy, R.: Continuum Percolation. Cambridge University Press, Cambridge (1996)
Menshikov, M.V., Rybnikov, K.A., Volkov, S.E.: The loss of tension in an infinite membrane with holes distributed according to a Poisson law. Adv. Appl. Probab. 34(2), 292–312 (2002)
Miles, R.E.: On the homogeneous planar Poisson point process. Math. Biosci. 6, 85–127 (1970)
Miller, G.L., Teng, S.-H., Vavasis, S.A.: An unified geometric approach to graph separators. In: 32nd Annual Symposium on Foundations of Computer Science, pp. 538–547. IEEE (1991)
Eppstein, D., Paterson, M.S., Yao, F.: On nearest-neighbor graphs. Discret. Comput. Geom. 17, 263–282 (1997)
Teng, S.-H.: Points, spheres, and separators: a unified geometric approach to graph partitioning. Ph.D. Thesis, Carnegie Mellon University, CMU-CS-91-184 (1991)
Wagon, S.: A space filling curve. In: Mathematica in Action, pp. 196–209. W.H. Freeman, New York (1991), Chapter 6.3
Wan, P.-J., Yi, C.-W.: Coverage by randomly deployed wireless sensor networks. IEEE/ACM Trans. Netw. 14, 2658–2669 (2006)
Wan, P.-J., Yi, C.-W., Yao, F., Jia, X.: Asymptotic critical transmission radius for greedy forward routing in wireless ad hoc networks, In: 7th ACM International Symposium on Mobile Ad Hoc Networking & Computing (ACM MOBIHOC), Florence, Italy, 22–25 May 2006, pp. 25–36
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Part of the work was done while S.-H. Teng was at Xerox Palo Alto Research Center and MIT.
The work of F.F. Yao was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No. CityU 1165/04E].
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Teng, SH., Yao, F.F. k-Nearest-Neighbor Clustering and Percolation Theory. Algorithmica 49, 192–211 (2007). https://doi.org/10.1007/s00453-007-9040-7
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DOI: https://doi.org/10.1007/s00453-007-9040-7