Asymptotic critical transmission radius for k-connectivity in wireless ad hoc networks

A range assignment to the nodes in a wireless ad hoc network induces a topology in which there is an edge between two nodes if and only if both of them are within each other’s transmission range. The critical transmission radius for -connectivity is the smallest such that if all nodes have the transmission radius , the induced topology is -connected. In this paper, we study the asymptotic critical transmission radius for -connectivity in a wireless ad hoc network whose nodes are uniformly and independently distributed in a unit-area square or disk. We provide a precise asymptotic distribution of the critical transmission radius for -connectivity. In addition, the critical neighbor number for -connectivity is the smallest integer such that if every node sets its transmission radius equal to the distance between itself and its -th nearest neighbor, the induced (symmetric) topology is -connected. Applying the critical transmission radius for -connectivity, we can obtain an asymptotic almost sure upper bound on the critical neighbor number for -connectivity.