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 k-connectivity is the smallest r such that if all nodes have the transmission radius r, the induced topology is k -connected. In this paper, we study the asymptotic critical transmission radius for k -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 k -connectivity. In addition, the critical neighbor number for k -connectivity is the smallest integer l such that if every node sets its transmission radius equal to the distance between itself and its l-th nearest neighbor, the induced (symmetric) topology is k-connected. Applying the critical transmission radius for k-connectivity, we can obtain an asymptotic almost sure upper bound on the critical neighbor number for k-connectivity.