The connectivity of ad hoc networks has been
extensively studied in the literature. Most recently, researchers model ad hoc
networks with two-dimensional lattices and apply percolation theory for
connectivity study. On the lattice, given a message source and the bond
probability to connect any two neighbor vertices, percolation theory tries to
determine the critical bond probability above which a giant connected component
appears. This paper studies related but different problem, directed
connectivity: what is the exact probability of the connection from the source
to any vertex following certain directions? The existing studies in math and
physics only provide approximation or numerical results. In this paper, by
proposing a recursive decomposition approach, we can obtain a closed-form
polynomial expression of the directed connectivity of square lattice networks
as a function of the bond probability. Based on the exact expression, we have
explored the impacts of the bond probability and lattice size and ratio on the
lattice connectivity, and determined the complexity of our algorithm. Further,
we have studied a realistic ad hoc network scenario, i.e., an urban VANET,
where we show the capability of our approach on both homogeneous and
heterogeneous lattices and how related applications can benefit from our
results
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