### Monotone properties of random geometric graphs have sharp thresholds

Ashish Goel, Sanatan Rai, and Bhaskar Krishnamachari
Source: Ann. Appl. Probab. Volume 15, Number 4 (2005), 2535-2552.

#### Abstract

Random geometric graphs result from taking n uniformly distributed points in the unit cube, [0,1]d, and connecting two points if their Euclidean distance is at most r, for some prescribed r. We show that monotone properties for this class of graphs have sharp thresholds by reducing the problem to bounding the bottleneck matching on two sets of n points distributed uniformly in [0,1]d. We present upper bounds on the threshold width, and show that our bound is sharp for d=1 and at most a sublogarithmic factor away for d≥2. Interestingly, the threshold width is much sharper for random geometric graphs than for Bernoulli random graphs. Further, a random geometric graph is shown to be a subgraph, with high probability, of another independently drawn random geometric graph with a slightly larger radius; this property is shown to have no analogue for Bernoulli random graphs.

First Page:
Primary Subjects: 60D05
Secondary Subjects: 5C80, 90B10
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.aoap/1133965771
Digital Object Identifier: doi:10.1214/105051605000000575
Mathematical Reviews number (MathSciNet): MR2187303
Zentralblatt MATH identifier: 05039567

### References

Appel, M. J. B. and Russo, R. P. (2002). The connectivity of a graph on uniform points on $[0,1]^d$. Statist. Probab. Lett. 60 351--357.
Mathematical Reviews (MathSciNet): MR1947174
Bollobás, B. (2001). Random Graphs. Cambridge Univ. Press, New York.
Mathematical Reviews (MathSciNet): MR1864966
Booth, L., Bruck, J., Franceschetti, M. and Meester, R. (2003). Covering algorithms, continuum percolation and the geometry of wireless networks. Ann. Appl. Probab. 13 722--741.
Mathematical Reviews (MathSciNet): MR1970284
Digital Object Identifier: doi:10.1214/aoap/1050689601
Project Euclid: euclid.aoap/1050689601
Zentralblatt MATH: 1029.60077
Bourgain, J. and Kalai, G. (1997). Influences of variables and threshold intervals under group symmetries. Geom. Funct. Anal. 7 438--461.
Mathematical Reviews (MathSciNet): MR1466334
Digital Object Identifier: doi:10.1007/s000390050015
Zentralblatt MATH: 0982.20004
Díaz, J., Penrose, M. D., Petit, J. and Sema, M. (1998). Approximating layout problems on geometric random graphs. Technical Report TR-417-98.
Durrett, R. (1996). Probability: Theory and Examples, 2nd ed. Duxbury Press, Pacific Grove, CA.
Mathematical Reviews (MathSciNet): MR1609153
Erdős, P. and Renyi, A. (1959). On random graphs I. Publ. Math. Debrecen 6 290--291.
Mathematical Reviews (MathSciNet): MR120167
Erdős, P. and Renyi, A. (1960). On the evolution of random graphs. Magyar Tud. Akad. Mat. Kut. Int. Kozl. 5 17--61.
Mathematical Reviews (MathSciNet): MR125031
Friedgut, E. and Kalai, G. (1996). Every monotone graph property has a sharp threshold. Proc. Amer. Math. Soc. 124 2993--3002.
Mathematical Reviews (MathSciNet): MR1371123
Digital Object Identifier: doi:10.1090/S0002-9939-96-03732-X
Zentralblatt MATH: 0864.05078
Godehardt, E. (1990). Graphs as Structural Models: The Applications of Graphs and Multigraphs in Cluster Analysis. Vieweg, Brunswick.
Mathematical Reviews (MathSciNet): MR1128322
Godehardt, E. and Jaworski, J. (1996). On the connectivity of a random interval graph. Random Structures Algorithms 9 137--161.
Mathematical Reviews (MathSciNet): MR1611752
Gupta, P. and Kumar, P. R. (1998). Critical power for asymptotic connectivity in wireless networks. In Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W. H. Fleming 547--566. Birkhäuser, Boston.
Mathematical Reviews (MathSciNet): MR1702981
Zentralblatt MATH: 0916.90101
Gupta, P. and Kumar, P. R. (2000). The capacity of wireless networks. IEEE Trans. Inform. Theory IT-46 388--404.
Mathematical Reviews (MathSciNet): MR1748976
Digital Object Identifier: doi:10.1109/18.825799
Zentralblatt MATH: 0991.90511
Gupta, P. and Kumar, P. R. (2001). Internets in the sky: The capacity of three dimensional wireless networks. Commun. Inf. Syst. 1 33--50.
Mathematical Reviews (MathSciNet): MR1907507
Holroyd, A. and Peres, Y. (2003). Trees and matchings from point processes. Electron. Comm. Probab. 8 17--27.
Mathematical Reviews (MathSciNet): MR1961286
i Silvestre, J. P. (2001). Layout problems. Ph.D. thesis, Universitat Politècnica de Catalunya, Barcelona.
Janson, S., Łuczak, T. and Ruciński, A. (2000). Random Graphs. Wiley, New York.
Mathematical Reviews (MathSciNet): MR1782847
Krishnamachari, B., Wicker, S., Bejar, R. and Pearlman, M. (2002). Critical density thresholds in distributed wireless networks. In Communications, Information and Network Security 1--15. Kluwer, Dordrecht.
Leighton, F. T. and Shor, P. W. (1989). Tight bounds for minimax grid matching, with applications to the average case analysis of algorithms. Combinatorica 9 161--187.
Mathematical Reviews (MathSciNet): MR1030371
Digital Object Identifier: doi:10.1007/BF02124678
Zentralblatt MATH: 0686.68039
McColm, G. (2004). Threshold functions for random graphs on a line segment. Combin. Probab. Comput. 13 373--387.
Mathematical Reviews (MathSciNet): MR2056406
Digital Object Identifier: doi:10.1017/S0963548304006121
Zentralblatt MATH: 1048.05076
Muthukrishnan, S. and Pandurangan, G. (2003). The bin-covering technique for thresholding random geometric graph properties. Technical Report 2003-39, DIMACS.
Nevzorov, V. B. (2001). Records: A Mathematical Theory. Amer. Math. Soc., Providence, RI.
Mathematical Reviews (MathSciNet): MR1843029
Zentralblatt MATH: 0989.62029
Penrose, M. D. (1997). The longest edge of the random minimal spanning tree. Ann. Appl. Probab. 7 340--361.
Mathematical Reviews (MathSciNet): MR1442317
Digital Object Identifier: doi:10.1214/aoap/1034625335
Project Euclid: euclid.aoap/1034625335
Zentralblatt MATH: 0884.60042
Penrose, M. D. (2003). Random Geometric Graphs. Oxford Univ. Press.
Mathematical Reviews (MathSciNet): MR1986198
Zentralblatt MATH: 1029.60007
Shakkottai, S., Srikant, R. and Shroff, N. B. (2003). Unreliable sensor grids: Coverage, connectivity and diameter. In Proc. of IEEE Infocom 2 1073--1083. San Francisco, CA.
Shor, P. W. and Yukich, J. E. (1991). Minmax grid matching and empirical measures. Ann. Probab. 19 1338--1348.
Mathematical Reviews (MathSciNet): MR1112419