Connected spatial networks over random points and a route-length statistic

Statistical Science

Volumn 25, Issue 3, 2010, Pages 275-288

  Aldous, David J ^a   Shun, Julian ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b Carnegie Mellon University   (United States)

Author keywords

Geometric graph; Proximity graph; Random graph; Spatial network

Indexed keywords

EID: 78751624415     PISSN: 08834237     EISSN: None     Source Type: Journal    
DOI: 10.1214/10-STS335     Document Type: Article

References (47)

1. ALDOUS, D. J. (2008). Spatial transportation networks with transfer costs: Asymptotic optimality of hub and spoke models. Math. Proc. Cambridge Philos. Soc. 145 471-487.
2. ALDOUS, D. J. (2008). Optimal spatial transportation networks where link-costs are sublinear in link-capacity. J. Stat. Mech. 2008 P03006.
3. ALDOUS, D. J. (2009). Which connected spatial networks on random points have linear route-lengths?
4. ALDOUS, D. J. (2009). The shape theorem for route-lengths in connected spatial networks on random points. Available at arXiv:0911.5301v1.
5. ALDOUS, D. J. (2010). Scale-invariant random spatial networks. To appear.
6. ALDOUS, D. J. and DIACONIS, P. (1995). Hammersley's interacting particle process and longest increasing subsequences. Probab. Theory Related Fields 103 199-213.
7. ALDOUS, D. J. and KENDALL, W. S. (2008). Short-length routes in low-cost networks via Poisson line patterns. Adv. in Appl. Probab. 40 1-21.
8. ALDOUS, D. J., BHAMIDI, S. and LANDO, T. (2010). The stretch-length tradeoff in geometric networks: Worst-case and average-case study. To appear.
9. ALDOUS, D. J. and STEELE, J. M. (1992). Asymptotics for Euclidean minimal spanning trees on random points. Probab. Theory Related Fields 92 247-258.
10. ALDOUS, D. J. and CHOI, A. (2009). A route-length efficiency statistic for road networks. Unpublished manuscript. Available at www.stat.berkeley.edu/~aldous/Spatial/paper.pdf.
11. AVRAM, F. and BERTSIMAS, D. (1992). The minimum spanning tree constant in geometric probability and under the independent model: A unified approach. Ann. Appl. Probab. 2 113-130.
12. AVRAM, F. and BERTSIMAS, D. (1993). On central limit theorems in geometrical probability. Ann. Appl. Probab. 3 1033-1046.
13. BACCELLI, F., TCHOUMATCHENKO, K. and ZUYEV, S. (2000). Markov paths on the Poisson-Delaunay graph with applications to routeing in mobile networks. Adv. in Appl. Probab. 32 1-18.
14. BARTHÉLEMY, M. and FLAMMINI, A. (2006). Optimal traffic networks. J. Stat. Mech. Theory Exp. 2006 L07002.
15. BERGER, N., BORGS, C., CHAYES, J. T. and SABERI, A. (2005). On the spread of viruses on the internet. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms 301-310 (electronic). ACM, New York.
16. BERN, M., EPPSTEIN, D. and YAO, F. (1991). The expected extremes in a Delaunay triangulation. Internat. J. Comput. Geom. Appl. 1 79-91.
17. BEZUIDENHOUT, C., GRIMMETT, G. and LÖFFLER, A. (1998). Percolation and minimal spanning trees. J. Stat. Phys. 92 1-34.
18. BOSE, P., DEVROYE, L., EVANS, W. and KIRKPATRICK, D. (2006). On the spanning ratio of Gabriel graphs and β-skeletons. SIAM J. Discrete Math. 20 412-427 (electronic).
19. CORTINA-BORJA, M. and ROBINSON, T. (2000). Estimating the asymptotic constants of the total length of Euclidean minimal spanning trees with power-weighted edges. Statist. Probab. Lett. 47 125-128.
20. DEVROYE, L. (1988). The expected size of some graphs in computational geometry. Comput. Math. Appl. 15 53-64.
21. DEVROYE, L., GUDMUNDSSON, J. and MORIN, P. (2009). On the expected maximum degree of Gabriel and Yao graphs. Adv. in Appl. Probab. 41 1123-1140.
22. GASTNER, M. T. and NEWMAN, M. E. J. (2006). Shape and efficiency in spatial distribution networks. J. Stat. Mech. Theory Exp. 2006 P01015 (electronic).
23. GEORGE, P.-L. and BOROUCHAKI, H. (1998). Delaunay Triangulation and Meshing. Editions Hermès, Paris.
24. GRAHAM, R. L. and HELL, P. (1985). On the history of the minimum spanning tree problem. IEEE Ann. History Comput. 07 43-57.
25. HEINRICH, L. (1994). Normal approximation for some mean-value estimates of absolutely regular tessellations. Math. Methods Statist. 3 1-24.
26. HOLROYD, A. E., PEMANTLE, R., PERES, Y. and SCHRAMM, O. (2009). Poisson matching. Ann. Inst. H. Poincaré Probab. Statist. 45 266-287.
27. HOLROYD, A. E. and PERES, Y. (2003). Trees and matchings from point processes. Electron. Comm. Probab. 8 17-27 (electronic).
28. JAROMCZYK, J. W. and TOUSSAINT, G. T. (1992). Relative neighborhood graphs and their relatives. Proc. IEEE 80 1502-1517.
29. KESTEN, H. and LEE, S. (1996). The central limit theorem for weighted minimal spanning trees on random points. Ann. Appl. Probab. 6 495-527.
30. KIRKPATRICK, D. G. and RADKE, J. D. (1985). A framework for computational morphology. In Computational Geometry (G. T. Toussaint, ed.) 217-248. Elsevier, Amsterdam.
31. LEE, S. (1997). The central limit theorem for Euclidean minimal spanning trees. I. Ann. Appl. Probab. 7 996-1020.
32. LEE, S. (1999). The central limit theorem for Euclidean minimal spanning trees. II. Adv. in Appl. Probab. 31 969-984.
33. LLOYD, E. L. (1977). On triangulations of a set of points in the plane. In 18th Annual Symposium on Foundations of Computer Science (Providence, RI, 1977) 228-240. IEEE Comput. Soc., Long Beach, CA.
34. MATULA, D. W. and SOKAL, R. R. (1980). Properties of Gabriel graphs relevant to geographical variation research and the clustering of points in the plane. Geog. Anal. 12 205-222.
35. MILES, R. E. (1970). On the homogeneous planar Poisson point process. Math. Biosci. 6 85-127.
36. MØLLER, J. (1994). Lectures on Random Voronǐo Tessellations. Lecture Notes in Statistics 87. Springer, New York.
37. NARASIMHAN, G. and SMID, M. (2007). Geometric Spanner Networks. Cambridge Univ. Press, Cambridge.
38. NARAYANASWAMY, S., KAWADIA, V., SREENIVAS, R. S. and KUMAR, P. R. (2002). Power control in ad-hoc networks: Theory, architecture, algorithm and implementation of the COMPOW protocol. In Proc. European Wireless Conference, Florence, Italy.
39. NEWMAN, M. E. J. (2003). The structure and function of complex networks. SIAM Rev. 45 167-256.
40. PENROSE, M. D. (2003). Random Geometric Graphs. Oxford Univ. Press, Oxford.
41. PENROSE, M. D. (1993). On the spread-out limit for bond and continuum percolation. Ann. Appl. Probab. 3 253-276.
42. PENROSE, M. D. and YUKICH, J. E. (2001). Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11 1005-1041.
43. PENROSE, M. D. and YUKICH, J. E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 277-303.
44. STEELE, J. M. (1997). Probability Theory and Combinatorial Optimization. CBMS-NSF Regional Conference Series in Applied Math 69. SIAM, Philadelphia, PA.
45. STOYAN, D., KENDALL, W. S. and MECKE, J. (1995). Stochastic Geometry and Its Applications, 2nd ed. Wiley, Chichester.
46. TALAGRAND, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. 81 73-205.
47. YUKICH, J. E. (1998). Probability Theory of Classical Euclidean Optimization Problems. Lecture Notes in Math. 1675. Springer, Berlin.