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

Abstract

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 kconnectivity is the smallest r such that if all nodes have the transmission radius r, the induced 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 and its l-th nearest neighbor, the induced topology is k-connected. In this paper, we study the asymptotic critical transmission radius for k-connectivity and asymptotic critical neighbor number 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 and an improved asymptotic almost sure upper bound on the critical neighbor number for k-connectivity.