We study mesh networks formed by nodes equipped with directional antennas in a high node-density setting. To do so we create a random geometric graph with n nodes placed uniformly at random. The antenna at each node chooses a direction of orientation at random and edges are placed between pairs of nodes based on their distance from each other and their directions of orientation according to the gain function of the antennas. To model the directionality of the antennas we consider a realistic gain function where the signal fades away from the direction of orientation. We also consider an idealised function that concentrates the gain uniformly in a sector of angle 2θ centred around the direction of orientation. In this setting we show theoretically that with probability tending to 1 the optimal power required for achieving connectivity is significantly lower than that needed for connectivity in an omnidirectional setting. We capture mathematically the relationship between this optimal power level and the maximum gain of the antenna, showing that as the directionality of the antenna increases the power needed for connectivity decreases. However this optimal power level is also inversely proportional to the probability of connectivity of two randomly placed nodes, which decreases as directionality increases. We validate these results through simulation
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