Critical Power for Asymptotic Connectivity in Wireless Networks

In wireless data networks each transmitter’s power needs to be high enough to reach the intended receivers, while generating minimum interference on other receivers sharing the same channel. In particular, if the nodes in the network are assumed to cooperate in routing each others’ packets, as is the case in ad hoc wireless networks, each node should transmit with just enough power to guarantee connectivity in the network. Towards this end, we derive the critical power a node in the network needs to transmit in order to ensure that the network is connected with probability one as the number of nodes in the network goes to infinity. It is shown that if n nodes are placed in a disc of unit area in ℜ2 and each node transmits at a power level so as to cover an area of πr2 = (log n + c(n))/n, then the resulting network is asymptotically connected with probability one if and only if c(n) → +∞.