Probabilistic connectivity threshold for directional antenna widths

In this work, we have studied the problem of arranging randomly scattered wireless sensor antennas in a way that guarantees the connectivity of the induced communication graph. The main challenge here is to minimize energy consumption while preserving node connectivity [2]. In order to save power, increase transmission capacity and reduce interference, antennas should communicate along a wedge-shaped area, that is, an angular and practically infinite section of a certain angle α whose apex is the antenna (see Figure 1). The smaller the angle is, the better it is in terms of energy saving. We show that when the number of network nodes is big enough, the required α approaches zero. Our results show that if one can adjust the direction of the antennas, then in order to guarantee the network connectivity, the angle should be Θ ( log(n) n ) . In contrast, if the direction of the antenna is a random variable, then the angle should be