Connectivity in Two-Dimensional Lattice Networks

Connectivity has been extensively studied in ad hoc networks, most recently with the application of percolation theory in two-dimensional square lattices. Given a message source and the bond probability to connect neighbor vertexes on the lattice, percolation theory tries to determine the critical bond probability above which there exists an infinite connected giant component with high probability. This paper studies a related but different problem: what is the connectivity from the source to any vertex on the square lattice following certain directions? The original directed percolation problem has been studied in statistical physics for more than half a century, with only simulation results available. In this paper, by using a recursive decomposition approach, we have obtained the analytical expressions for directed connectivity. The results can be widely used in wireless and mobile ad hoc networks, including vehicular ad hoc networks.