Minimum number of neighbors for fully connected uniform ad hoc wireless networks

Determining the minimum number of neighboring nodes required to guarantee full connectivity, i.e., to ensure that a node can reach, through multiple hops, any other node in the network, is an important problem in ad hoc wireless networks. In this paper, we consider reservation-based wireless networks with stationary and uniform (on average) node spatial distribution. Assuming that any communication route is a sequence of minimum length hops, we show that, in an ideal case without inter-node interference (INI) and on the basis of a suitable definition of transmission range, the minimum number of neighbors required for full connectivity is, on average, π. Full connectivity is guaranteed if the transmitted power (in the case of fixed node spatial density) or, equivalently, the node spatial density (in the case of fixed transmitted power) are larger than critical minimum values. In a realistic case with INI, we prove that there are situations where full connectivity cannot be guaranteed, regardless of the number of neighbors or the transmitted power.