One-dimensional geometric random graphs with nonvanishing densities II: a very 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 communicate with each other if their distance is less than some given threshold value. When F admits a density f which is strictly positive on [0,1], we give conditions on f under which the property of graph connectivity for the induced geometric random graph obeys a very strong zero-one law when the transmission range is scaled appropriately with n large. The very strong critical threshold is identified. This is done by applying a version of the method of first and second moments.