# Probabilistic connectivity threshold for directional antenna widths Academic Article

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### abstract

• 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, where the orientation of the sector is either random or not. 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 direction is either random or not. We prove basic and fundamental results about this (reformulated) problem. We show that when the number of network nodes is big enough, the required $$\check{\alpha}$$ approaches zero. Specifically, on the unit disk it holds with high probability that the threshold for connectivity is $$\check{\alpha}=\Theta(\sqrt[4]{\frac{\log n}{n}})$$. This is shown by the use of Poisson approximation and geometrical considerations. Moreover, when the model is relaxed to allow orientation towards the center of the area, we demonstrate that $$\check{\alpha}=\Theta(\frac{\log n}{n})$$ is a necessary and sufficient condition.

### publication date

• January 1, 2015