Probabilistic connectivity threshold for directional antenna widths

Consider the task of maintaining connectivity in a wireless network where the network nodes are equipped with directional antennas. Nodes correspond to points on the unit disk and each uses a directional antenna covering a sector of a given angle α.The width required for a connectivity problem is to find out the necessary and sufficient conditions of α that guarantee connectivity when an antenna's location is uniformly distributed and the orientation of the antenna's sector is either random or fixed.We show that when the number of network nodes is big enough, the required αˇ approaches zero. Specifically, on the unit disk, assuming uniform orientation, it holds with high probability that the threshold for connectivity is αˇ=Θ(log⁡nn4). This is shown by the use of Poisson approximation and geometrical considerations. Moreover, when the model is relaxed, assuming that the antenna's orientation is directed towards the center of the disk, we demonstrate that αˇ=Θ(log⁡nn) is a necessary and sufficient condition.