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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 induce topology is k-connected. 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 an its l-th nearest neighbor, the induce topology is k-connecte. In this paper, we study the asymptotic critical transmission radius for k-connectivity an asymptotic critical neighbor number for k-connectivity in a wireless ad hoc network whose nodes are uniformly an independently distribute in a unit-area square or disk. We provide a precise asymptotic distribution of the critical transmission radius for k-connectivity and an improve asymptotic almost sure upper bound on the critical neighbor number for k-connectivity.

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