Title :
One-Dimensional Geometric Random Graphs With Nonvanishing Densities—Part I: A Strong Zero-One Law for Connectivity
Author :
Han, Guang ; Makowski, Armand M.
Author_Institution :
Dept. of Electr. & Comput. Eng., Univ. of Maryland, College Park, MD, USA
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 Levy for maximal spacings under the uniform distribution.
Keywords :
geometry; graph theory; radio networks; random processes; statistical distributions; adjacent node; critical scaling; maximal spacing; nonuniform distribution; nonvanishing density; one-dimensional geometric random graph connectivity; probability distribution function; strong zero-one law; uniform distribution; unit interval; weak continuity assumption; wireless network; Collaborative work; Electronic mail; Government; Probability distribution; Random variables; Solid modeling; Statistical distributions; Wireless networks; Connectivity; critical scalings; geometric random graphs; nonuniform node placement; nonvanishing densities; zero-one laws;
Journal_Title :
Information Theory, IEEE Transactions on
DOI :
10.1109/TIT.2009.2032799
