The Critical Transmitting Range for Connectivity in Sparse Wireless Ad Hoc Networks

In this paper, we analyze the critical transmitting range for connectivity in wireless ad hoc networks. More specifically, we consider the following problem: Assume n nodes, each capable of communicating with nodes within a radius of r, are randomly and uniformly distributed in a d-dimensional region with a side of length l; how large must the transmitting range r be to ensure that the resulting network is connected with high probability? First, we consider this problem for stationary networks, and we provide tight upper and lower bounds on the critical transmitting range for one-dimensional networks and nontight bounds for two and three-dimensional networks. Due to the presence of the geometric parameter l in the model, our results can be applied to dense as well as sparse ad hoc networks, contrary to existing theoretical results that apply only to dense networks. We also investigate several related questions through extensive simulations. First, we evaluate the relationship between the critical transmitting range and the minimum transmitting range that ensures formation of a connected component containing a large fraction (e.g., 90 percent) of the nodes. Then, we consider the mobile version of the problem, in which nodes are allowed to move during a time interval and the value of r ensuring connectedness for a given fraction of the interval must be determined. These results yield insight into how mobility affects connectivity and they also reveal useful trade offs between communication capability and energy consumption.