On edges and connectivity in ad hoc networks

In this paper we derive an unbiased estimator for the number of direct connections (edges or neighbors) in 2 and 3 dimensional ad hoc networks. We show how this estimator is based on a minimal number of assumptions regarding the topology of the network and is a good estimator for both sparse and dense networks. We then develop a relationship between the number of edges in the network and the probability of that network being strongly connected. We show for realistically sized networks, that if the nodes have, on average, just under 10 neighbors that the network is completely connected with high probability. This is a so called "magic number" which is the subject of some dispute in the literature at present. We go on to develop a simple algorithm that uses the prior results to construct strongly connected sensor or smart-dust networks. This algorithm is attractive as it is very robust whilst requiring only communication between adjacent nodes. We compare the performance of our algorithm against the shortest path algorithm.