Han, GuangConnectivity is one of the most fundamental properties of wireless ad-hoc networks as most network functions are predicated upon the network being connected. Although increasing node transmission power will improve network connectivity, too large a power level is not feasible as energy is a scarce resource in wireless ad-hoc networks. Thus, it is crucial to identify the minimum node transmission power that will ensure network connectivity with high probability. It is known that there exists a critical level transmission power such that a suitably larger power will ensure network connectivity with high probability. A small variation across this threshold level will lead to a sharp transition of the probability that the network is connected. Thus, in order to precisely estimate the minimum node transmission power, not only do we need to identify this critical threshold, but also how fast this transition takes place. To characterize the sharpness of transition, we define weak, strong and very strong critical thresholds associated with increasing transition speeds. In this dissertation, we seek to estimate the minimum node transmission power for large scale one-dimensional wireless ad-hoc networks under the Geometric Random Graph (GRG) models. Unlike in previous works where nodes are taken to be uniformly distributed, we assume a more general node distribution. Using the methods of first and second moments, we theoretically prove the existence of a very strong critical threshold when the density function is everywhere positive. On the other hand, only weak thresholds are shown to exist when the density function contains vanishing densities. We also study the connectivity of two-dimensional wireless ad-hoc networks under the random connection model, which accounts for statistical channel variations. With the help of the Stein-Chen method, we derive a closed form formula for the limiting probability that there are no isolated nodes under a very general assumption of channel variations. The node transmission power to ensure the absence of isolated nodes provides a tight lower bound on the transmission power needed to ensure network connectivity.en-USDissertationEngineering, Electronics and Electrical