Decentralized Routing in Nonhomogeneous Poisson Networks

In his seminal work, Jon Kleinberg considers a small-world network model consisting of a k-dimensional lattice augmented with shortcuts. Under the assumption that the probability of a shortcut being present between two nodes u and v decays as a power, d(u,v) -alpha, of the distance d(u,v) between them, Kleinberg shows that decentralized routing scheme such as greedy geographic routing is efficient if alpha=k and that there is no efficient decentralized routing algorithm if alphaneq k. The results are extended to a continuum model recently, wherein the nodes are distributed as a homogeneous Poisson point process by Franceschetti and Meester, Draief and Ganesh. In our work, we extend the result further to a more realistic model constructed from a nonhomogeneous Poisson point process, wherein each node is connected to all its neighbors within some fixed radius, as well as possessing random shortcuts to more distant nodes. More importantly, we show that in nonhomogeneous cases, the necessary and sufficient condition for greedy geographic routing to be efficient is that the probability of a shortcut being present from node u to v should be inversely proportional to the number of nodes which are closer to u than v is. We also demonstrate some applications of our results to wireless networks.