This talk will investigate the average-case behavior of certain algorithms designed for connecting mobile agents in the two- or three-dimensional space. The general model is the following: Let X be a set of points in the d-dimensional Euclidean space Ed, d . 2; r be a function that associates each element of x C with a positive real number r(x). A graph G(X, r) is an oriented graph with the vertex set X, in which (x, y) is an edge if and only if r( x, y) £ r(x), where r( x, y) denotes the Euclidean distance in the space Ed. Given a set X, the goal is to find a function r so that graph G(X, r) is strongly connected (note that the graph G(X, r) need not be symmetric). Given a random set of points, the function r computed by the proposed algorithm is such that, for any constant d, the average value of r(x)d (the average transmitter power) is almost surely constant.