On the Critical Total Power for Asymptotic $k$-Connectivity in Wireless Networks

In this paper, we investigate the minimum total power (termed as {\em critical total power}) required to ensure asymptotic k-connectivity in heterogeneous wireless networks where nodes may transmit using different levels of power. We show that under the assumption that wireless nodes form a homogeneous Poisson point process with density \lambda on a unit square region [0,1]^2 and the Toroidal model \cite{penrose97}, the critical total power required for maintaining k-connectivity is \Theta({\Gamma(c/2+k) \over (k-1)!}\lambda^{1-c/2}) with probability approaching one as \lambda goes to infinity, where c is the path loss exponent. Compared with the result that all nodes use a {\em common} critical transmission power for maintaining k-connectivity \cite{penrose99,wan04}, we show that the critical total power can be reduced by an order of (\log \lambda)^{c/2} by allowing node to optimally choose different levels of transmission power. These results are not subject to any specific power/topology control algorithm, but rather a fundamental property in wireless networks