Closed Form Solutions for Connectivity of Fixed Radius Random Graphs in One-Dimensional Space

We consider the connectivity of a class of random graph models called graphs defined by the fixed radius model . The model has a clear relationship with state-of-the-art wireless communication networks, often called ad-hoc networks. In this model, each random graph is defined by nodes placed in a Euclidean plane randomly according to some distribution; each pair of nodes is connected by an edge if and only if the distance between the nodes is within the common radius . Hence the model can be naturally interpreted as a mathematical model of wireless communication networks in which every mobile node can communicate with other nodes within the distance In this paper, we present some analytical results concerning the probability that such random graphs are connected (i.e., there is a path between every pair of nodes), assuming that the fixed number of nodes are distributed in one-dimensional space according to the uniform distribution. Related results have been obtained in our previous paper only implicitly in the form of recursive equations. On the other hand, the results of this paper are significant in that they are closed form solutions of the recursive equations thus present the probability explicitly.