A strong zero-one law for connectivity in one-dimensional geometric random graphs with non-vanishing densities

Abstract

We consider the geometric random graph where n points are distributed independently on the unit interval [0,1] according to some probability distribution function F. Two nodes communicate with each other if their distance is less than some transmission range. When F admits a continuous density f which is strictly positive on [0,1], we show that the property of graph connectivity exhibits a strong critical threshold and we identify it. This is achieved by generalizing a limit result on maximal spacings due to Levy for the uniform distribution.