On the Sensitivity of the Critical Transmission Range: Lessons from the Lonely Dimension

We consider geometric random graphs where n points are distributed independently on the unit interval [0, 1] according to some probability distribution function F with density function f. Two nodes communicate with each other if their distance is less than some transmission range. For this class of random graphs, we survey results concerning the existence of zero-one laws for graph connectivity, the type of the zero-one law obtained under specific assumptions on the density function f, the form of its critical scaling and its dependence on f, and the width of the corresponding phase transitions. This is motivated by the desire to understand how node distribution affects the critical transmission range as specified by the disk model. Engineering implications are discussed for power allocation.