Limit theory for the random on-line nearest-neighbor graph
In the on‐line nearest‐neighbor graph (ONG), each point after the first in a sequence of points in ℝd is joined by an edge to its nearest neighbor amongst those points that precede it in the sequence. We study the large‐sample asymptotic behavior of the total power‐weighted length of the ONG on unif...
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| Vydané v: | Random structures & algorithms Ročník 32; číslo 2; s. 125 - 156 |
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| ISSN: | 1042-9832, 1098-2418 |
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| Abstract | In the on‐line nearest‐neighbor graph (ONG), each point after the first in a sequence of points in ℝd is joined by an edge to its nearest neighbor amongst those points that precede it in the sequence. We study the large‐sample asymptotic behavior of the total power‐weighted length of the ONG on uniform random points in (0,1)d. In particular, for d = 1 and weight exponent α > 1/2, the limiting distribution of the centered total weight is characterized by a distributional fixed‐point equation. As an ancillary result, we give exact expressions for the expectation and variance of the standard nearest‐neighbor (directed) graph on uniform random points in the unit interval. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008 |
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| AbstractList | In the on-line nearest-neighbor graph (ONG), each point after the first in a sequence of points in d is joined by an edge to its nearest neighbor amongst those points that precede it in the sequence. We study the large-sample asymptotic behavior of the total power-weighted length of the ONG on uniform random points in (0,1)d. In particular, for d = 1 and weight exponent > 1/2, the limiting distribution of the centered total weight is characterized by a distributional fixed-point equation. As an ancillary result, we give exact expressions for the expectation and variance of the standard nearest-neighbor (directed) graph on uniform random points in the unit interval. In the on‐line nearest‐neighbor graph (ONG), each point after the first in a sequence of points in ℝd is joined by an edge to its nearest neighbor amongst those points that precede it in the sequence. We study the large‐sample asymptotic behavior of the total power‐weighted length of the ONG on uniform random points in (0,1)d. In particular, for d = 1 and weight exponent α > 1/2, the limiting distribution of the centered total weight is characterized by a distributional fixed‐point equation. As an ancillary result, we give exact expressions for the expectation and variance of the standard nearest‐neighbor (directed) graph on uniform random points in the unit interval. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008 In the on‐line nearest‐neighbor graph (ONG), each point after the first in a sequence of points in ℝ d is joined by an edge to its nearest neighbor amongst those points that precede it in the sequence. We study the large‐sample asymptotic behavior of the total power‐weighted length of the ONG on uniform random points in (0,1) d . In particular, for d = 1 and weight exponent α > 1/2, the limiting distribution of the centered total weight is characterized by a distributional fixed‐point equation. As an ancillary result, we give exact expressions for the expectation and variance of the standard nearest‐neighbor (directed) graph on uniform random points in the unit interval. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008 |
| Author | Penrose, Mathew D. Wade, Andrew R. |
| Author_xml | – sequence: 1 givenname: Mathew D. surname: Penrose fullname: Penrose, Mathew D. email: m.d.penrose@bath.ac.uk organization: Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, England – sequence: 2 givenname: Andrew R. surname: Wade fullname: Wade, Andrew R. email: andrew.wade@bris.ac.uk organization: Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, England |
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| References_xml | – reference: M. D. Penrose, Multivariate spatial central limit theorems with applications to percolation and spatial graphs, Ann Probab 33 (2005), 1945-1991. – reference: P. Billingsley, Convergence of probability measures, 2nd edition, Wiley, New York, 1999. – reference: U. Rösler, L. Rüschendorf, The contraction method for recursive algorithms, Algorithmica 29 (2001), 3-33. – reference: K. Huang, Statistical mechanics, 2nd edition, Wiley, New York, 1987. – reference: S. N. Dorogovstev, J. F. F. Medes, Evolution of networks, Adv Phys 51 (2002), 1079-1187. – reference: J. Bertoin, A. Gnedin, Asymptotic laws for nonconservative selfsimilar fragmentations, Electr J Probab 9 (2004), 575-593. – reference: D. J. Aldous, A. Bandyopadhyay, A survey of max-type recursive distributional equations, Ann Appl Probab 15 (2005), 1047-1110. – reference: S. T. Rachev, Probability metrics and the stability of stochastic models, Wiley, Chichester, 1991. – reference: M. Penrose, Random geometric graphs (Oxford studies in probability, Vol. 6), Clarendon Press, Oxford, 2003. – reference: M. E. J. Newman, The structure and function of complex networks, SIAM Rev 45 (2003), 167-256. – reference: R. Neininger, L. Rüschendorf, A general limit theorem for recursive algorithms and combinatorial structures, Ann Appl Probab 14 (2004), 378-418. – reference: J. E. Yukich, Probability theory of classical Euclidean optimization problems (Lecture notes in mathematics, Vol. 1675), Springer, Berlin, 1998. – reference: H. Kesten, S. Lee, The central limit theorem for weighted minimal spanning trees on random points, Ann Appl Probab 6 (1996), 495-527. – reference: U. Rösler, A fixed point theorem for distributions, Stoch Process Appl 42 (1992), 195-214. – reference: J. M. Steele, Probability theory and combinatorial optimization, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997. – reference: M. D. Penrose, J. E. Yukich, Central limit theorems for some graphs in computational geometry, Ann Appl Probab 11 (2001), 1005-1041. – reference: M. Abramowitz, 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. – reference: M. D. Penrose, J. E. Yukich, Weak laws of large numbers in geometric probability, Ann Appl Probab 13 (2003), 277-303. – reference: R. Pyke, Spacings, J Royal Stat Soc Ser B 27 (1965), 395-449. – reference: D. A. Darling, On a class of problems related to the random division of an interval, Ann Math Stats 24 (1953), 239-253. – reference: M. D. Penrose, A. R. Wade, On the total length of the random minimal directed spanning tree, Adv Appl Probab 38 (2006), 336-372. – reference: A. R. Wade, Explicit laws of large numbers for random nearest-neighbour type graphs, Adv Appl Probab 39 (2007), 326-342. – start-page: 725 year: 2003 end-page: 738 – volume: 39 start-page: 326 year: 2007 end-page: 342 article-title: Explicit laws of large numbers for random nearest‐neighbour type graphs publication-title: Adv Appl Probab – year: 2005 – volume: 33 start-page: 1945 year: 2005 end-page: 1991 article-title: Multivariate spatial central limit theorems with applications to percolation and spatial graphs publication-title: Ann Probab – year: 1987 – volume: 14 start-page: 378 year: 2004 end-page: 418 article-title: A general limit theorem for recursive algorithms and combinatorial structures publication-title: Ann Appl Probab – year: 2003 – volume: 42 start-page: 195 year: 1992 end-page: 214 article-title: A fixed point theorem for distributions publication-title: Stoch Process Appl – volume: 27 start-page: 395 year: 1965 end-page: 449 article-title: Spacings publication-title: J Royal Stat Soc Ser B – volume: 51 start-page: 1079 year: 2002 end-page: 1187 article-title: Evolution of networks publication-title: Adv Phys – volume: 45 start-page: 167 year: 2003 end-page: 256 article-title: The structure and function of complex networks publication-title: SIAM Rev – start-page: 37 year: 2005 end-page: 58 – volume: 13 start-page: 277 year: 2003 end-page: 303 article-title: Weak laws of large numbers in geometric probability publication-title: Ann Appl Probab – volume: 55 year: 1965 – year: 1998 – volume: 29 start-page: 3 year: 2001 end-page: 33 article-title: The contraction method for recursive algorithms publication-title: Algorithmica – volume: 6 start-page: 495 year: 1996 end-page: 527 article-title: The central limit theorem for weighted minimal spanning trees on random points publication-title: Ann Appl Probab – volume: 24 start-page: 239 year: 1953 end-page: 253 article-title: On a class of problems related to the random division of an interval publication-title: Ann Math Stats – start-page: 110 year: 2002 end-page: 122 – year: 1997 – volume: 15 start-page: 1047 year: 2005 end-page: 1110 article-title: A survey of max‐type recursive distributional equations publication-title: Ann Appl Probab – volume: 9 start-page: 575 year: 2004 end-page: 593 article-title: Asymptotic laws for nonconservative selfsimilar fragmentations publication-title: Electr J Probab – volume: 38 start-page: 336 year: 2006 end-page: 372 article-title: On the total length of the random minimal directed spanning tree publication-title: Adv Appl Probab – start-page: 1 year: 2003 end-page: 34 – year: 1991 – volume: 11 start-page: 1005 year: 2001 end-page: 1041 article-title: Central limit theorems for some graphs in computational geometry publication-title: Ann Appl Probab – year: 1999 – ident: e_1_2_1_6_2 doi: 10.1002/9780470316962 – ident: e_1_2_1_12_2 doi: 10.1214/aoap/1034968141 – ident: e_1_2_1_23_2 doi: 10.1016/0304-4149(92)90035-O – ident: e_1_2_1_24_2 doi: 10.1007/BF02679611 – ident: e_1_2_1_29_2 doi: 10.1007/BFb0093472 – ident: e_1_2_1_8_2 doi: 10.1214/aoms/1177729030 – ident: e_1_2_1_18_2 doi: 10.1214/aoap/1015345393 – ident: e_1_2_1_13_2 doi: 10.1214/aoap/1075828056 – ident: e_1_2_1_27_2 doi: 10.1239/aap/1183667613 – volume: 27 start-page: 395 year: 1965 ident: e_1_2_1_21_2 article-title: Spacings publication-title: J Royal Stat Soc Ser B doi: 10.1111/j.2517-6161.1965.tb00602.x – ident: e_1_2_1_28_2 – volume-title: Handbook of mathematical functions, National Bureau of Standards year: 1965 ident: e_1_2_1_2_2 – ident: e_1_2_1_14_2 doi: 10.1137/S003614450342480 – ident: e_1_2_1_3_2 doi: 10.1214/105051605000000142 – volume-title: Statistical mechanics year: 1987 ident: e_1_2_1_11_2 – ident: e_1_2_1_9_2 doi: 10.1080/00018730110112519 – ident: e_1_2_1_4_2 doi: 10.1007/3-540-45061-0_57 – ident: e_1_2_1_15_2 doi: 10.1093/acprof:oso/9780198506263.001.0001 – ident: e_1_2_1_25_2 doi: 10.1137/1.9781611970029 – ident: e_1_2_1_19_2 doi: 10.1214/aoap/1042765669 – ident: e_1_2_1_26_2 – volume-title: Probability metrics and the stability of stochastic models year: 1991 ident: e_1_2_1_22_2 – ident: e_1_2_1_10_2 doi: 10.1007/3-540-45465-9_11 – volume: 38 start-page: 336 year: 2006 ident: e_1_2_1_17_2 article-title: On the total length of the random minimal directed spanning tree publication-title: Adv Appl Probab doi: 10.1239/aap/1151337075 – ident: e_1_2_1_20_2 doi: 10.1142/9789812567673_0003 – ident: e_1_2_1_16_2 doi: 10.1214/009117905000000206 – volume: 9 start-page: 575 year: 2004 ident: e_1_2_1_5_2 article-title: Asymptotic laws for nonconservative selfsimilar fragmentations publication-title: Electr J Probab – start-page: 1 volume-title: Handbook of graphs and networks year: 2003 ident: e_1_2_1_7_2 |
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| Snippet | In the on‐line nearest‐neighbor graph (ONG), each point after the first in a sequence of points in ℝd is joined by an edge to its nearest neighbor amongst... In the on‐line nearest‐neighbor graph (ONG), each point after the first in a sequence of points in ℝ d is joined by an edge to its nearest neighbor amongst... In the on-line nearest-neighbor graph (ONG), each point after the first in a sequence of points in d is joined by an edge to its nearest neighbor amongst those... |
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| Title | Limit theory for the random on-line nearest-neighbor graph |
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