Asymptotic Critical Transmission Radius for k-Connectivity in Wireless Ad Hoc Networks

A range assignment to the nodes in a wirelessad hocnetwork 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 fork-connectivity is the smallestrsuch that if all nodes have the transmission radiusr, the induced topology isk-connected. In this paper, we study the asymptotic critical transmission radius fork-connectivity in a wirelessad hocnetwork 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 fork-connectivity. In addition, the critical neighbor number fork-connectivity is the smallest integerlsuch that if every node sets its transmission radius equal to the distance between itself and itsl-th nearest neighbor, the induced (symmetric) topology isk-connected. Applying the critical transmission radius fork-connectivity, we can obtain an asymptotic almost sure upper bound on the critical neighbor number fork-connectivity.