Percolation in directed random geometric graphs

The connectivity graph of wireless networks, under many models as well as in practice, may contain unidirectional links. The simplifying assumption that such links are useless is often made, mainly because most wireless protocols use per-hop acknowledgments. However, two-way communication between a pair of nodes can be established as soon as there exists paths in both directions between them. Therefore, instead of discarding unidirectional links, one might be interested in studying the strongly connected components of the connectivity graph. In this paper, we look at the percolation phenomenon in some directed random geometric graphs that can be used to model wireless networks. We show that among the nodes that can be reached from the origin, a non-zero fraction can also reach the origin. In other words, the percolation threshold for strong connectivity is equal to the threshold for one-way connectivity.