Low degree connectivity in ad-hoc networks

The aim of the paper is to investigate the average case behavior of certain algorithms that are 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 E-d, d >= 2; r be a function that associates each element of x is an element of X 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 p(x, y) <= r(x), where p(x, y) denotes the Euclidean distance in the space Ed. Given a set X, the goal is to find a function r so that the 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 algorithm of the present paper is such that, for any constant delta, the average value of r(x)(6) (the average transmitter power) is almost surely constant.