One-Dimensional Geometric Random Graphs With Nonvanishing Densities—Part I: A Strong Zero-One Law for Connectivity

摘要

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.