Connectivity of sensor networks with power control

In this paper, we consider a sensor network with an average of n nodes randomly placed over a region of unit area. We assume that each node is equipped with a wireless transceiver, and are interested in the minimum transmit power required for maintaining connectivity of the network when power control is employed (i.e., each node can choose a power level for transmission independent of any other node). We show that the average power gain per node (the ratio of the transmit power required without and with power control) increases with the (average) number of nodes n as (logn)^α2/, where α is the path loss exponent.