On the critical total power for asymptotic k-connectivity in wireless networks

In this paper, we investigate the minimum total power (termed as 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 λ on a unit square region [0, 1]² and the Toroidal model [M.D. Penrose, 1997], the critical total power required for maintaining k-connectivity is θ((Γ(e/2+k))/((k-1)l)λ^1-e2/) with probability approaching one as λ goes to infinity, where e is the path loss exponent. Compared with the results that all nodes use a common critical transmission power for maintaining k-connectivity [M.D. Penrose, 1999], [P.-J. Wan and C. Yi, 2004], we show that the critical total power can be reduced by an order of (log λ)e/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 in wireless networks.