One-dimensional geometric random graphs with nonvanishing densities: part I: a strong zero-one law for connectivity

Authors:
Guang Han;Armand M. Makowski
Affiliations:
Networks Advanced Technologies, Motorola, Inc., Arlington Heights, IL and Department of Electrical and Computer Engineering, and the Institute for Systems Research, University of Maryland, College ...;Department of Electrical and Computer Engineering and Institute for Systems Research, University of Maryland, College Park, MD
Venue:
IEEE Transactions on Information Theory
Year:
2009

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Abstract

We consider a collection of n independent points which are distributed on the unit interval [0, 1] according to some probability distribution function F. Two nodes are said to be adjacent if their distance is less than some given threshold value. When F admits a nonvanishing density f, we show under a weak continuity assumption on f that the property of graph connectivity for the induced geometric random graph exhibits a strong zero-one law, and we identify the corresponding critical scaling. This is achieved by generalizing to nonuniform distributions a limit result obtained by Lévy for maximal spacings under the uniform distribution.