Asymptotic Critical Total Power for $k$ -Connectivity of Wireless Networks

An important issue in wireless ad hoc networks is to reduce the transmission power subject to certain connectivity requirement. In this paper, we study the fundamental scaling law of the minimum total power (termed as critical total power) required to ensure k -connectivity in wireless networks. Contrary to several previous results that assume all nodes use a (minimum) common power, we allow nodes to choose different levels of transmission power. We show that under the assumption that wireless nodes form a homogeneous Poisson point process with density lambda in a unit square region [0, 1]², the critical total power required to maintain k-connectivity is Theta((Gamma(c/2 + k)/(k - 1)!) lambda^1-c/2) with probability approaching one as lambda goes to infinity, where c is the path loss exponent. If k also goes to infinity, the expected critical total power is of the order of k^c/2 lambda^1-c/2. Compared with the results that all nodes use a common critical transmission power for maintaining k-connectivity, we show that the critical total power can be reduced by an order of (log lambda)^c/2 by allowing nodes to optimally choose different levels of transmission power. This result is not subject to any specific power/topology control algorithm, but rather a fundamental property of wireless networks.