Sensitivity of critical transmission ranges to node placement distributions

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 with density f. Two nodes are adjacent (i.e., communicate with each other) if their distance is less than some transmission range. We survey results, some classical and some recently obtained by the authors, concerning the existence of zero-one laws for graph connectivity, the type of zero-one laws under the specific assumptions made, the form of its critical scaling and its dependence on the density f. We also present results and conjectures concerning the width of the corresponding phase transition. Engineering implications are discussed for power allocation.