Sharp thresholds For monotone properties in random geometric graphs

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.

[1]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[2]  Erhard Godehardt,et al.  On the connectivity of a random interval graph , 1996 .

[3]  G. Kalai,et al.  Every monotone graph property has a sharp threshold , 1996 .

[4]  Panganamala Ramana Kumar,et al.  RHEINISCH-WESTFÄLISCHE TECHNISCHE HOCHSCHULE AACHEN , 2001 .

[5]  M. J. Appel,et al.  The connectivity of a graph on uniform points on [0,1]d , 2002 .

[6]  W. T. Gowers,et al.  RANDOM GRAPHS (Wiley Interscience Series in Discrete Mathematics and Optimization) , 2001 .

[7]  E. Friedgut,et al.  Sharp thresholds of graph properties, and the -sat problem , 1999 .

[8]  R. Durrett.  Probability: Theory and Examples , 1993 .

[9]  M. Penrose.  The longest edge of the random minimal spanning tree , 1997 .

[10]  S. Muthukrishnan,et al.  The bin-covering technique for thresholding random geometric graph properties , 2005, SODA '05.

[11]  Erhard Godehardt,et al.  Graphs as Structural Models , 1988 .

[12]  Jehoshua Bruck,et al.  Covering Algorithms, Continuum Percolation, and the Geometry of Wireless Networks. , 2003 .

[13]  E. Godehardt.  Graphs as Structural Models: The Application of Graphs and Multigraphs in Cluster Analysis , 1988 .

[14]  Alexander E. Holroyd,et al.  Trees and Matchings from Point Processes , 2002, math/0211455.

[15]  R. Srikant,et al.  Unreliable sensor grids: coverage, connectivity and diameter , 2005, Ad Hoc Networks.

[16]  A. Rbnyi.  ON THE EVOLUTION OF RANDOM GRAPHS , 2001 .

[17]  Gregory L. McColm,et al.  Threshold Functions for Random Graphs on a Line Segment , 2004, Combinatorics, Probability and Computing.

[18]  J. Yukich,et al.  Minimax Grid Matching and Empirical Measures , 1991 .

[19]  Frank Thomson Leighton,et al.  Tight bounds for minimax grid matching with applications to the average case analysis of algorithms , 1989, Comb..

[20]  Mathew D. Penrose,et al.  Random Geometric Graphs , 2003 .

[21]  Maria J. Serna,et al.  Approximating layout problems on random graphs , 2001, Discret. Math..

[22]  Piyush Gupta,et al.  Critical Power for Asymptotic Connectivity in Wireless Networks , 1999 .

[23]  J. Bourgain,et al.  Influences of Variables and Threshold Intervals under Group Symmetries , 1997 .

[24]  B. Bollobás.  The evolution of random graphs , 1984 .

[25]  M. R. Pearlman,et al.  Critical Density Thresholds in Distributed Wireless Networks , 2003 .

[26]  P. R. Kumar,et al.  Internets in the sky: The capacity of three-dimensional wireless networks , 2001, Commun. Inf. Syst..

[27]  Paul Malliavin,et al.  Stochastic Analysis , 1997, Nature.

[28]  Valery B. Nevzorov,et al.  Records: Mathematical Theory , 2000 .

[29]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[30]  Sergey V. Buldyrev,et al.  The puzzling statistical physics of liquid water , 1998 .