One-Dimensional Geometric Random Graphs With Nonvanishing Densities-Part I: A Strong Zero-One Law for Connectivity
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 Levy for maximal spacings under the uniform distribution.
geometric random graphs nonvanishing densities zero-one laws connectivity critical scalings nonuniform node placement