Highway Preferential Attachment Models for Geographic Routing

In the 1960s, the world-renowned social psychologist Stanley Milgram conducted experiments that showed that not only do there exist ``short chains'' of acquaintances between any two arbitrary people, but that these arbitrary strangers are able to find these short chains. This phenomenon, known as the \emph{small-world phenomenon}, is explained in part by any model that has a low diameter, such as the Barab\'asi and Albert's \emph{preferential attachment} model, but these models do not display the same efficient routing that Milgram's experiments showed. In the year 2000, Kleinberg proposed a model with an efficient $\mathcal{O}(\log^2{n})$ greedy routing algorithm. In 2004, Martel and Nguyen showed that Kleinberg's analysis was tight, while also showing that Kleinberg's model had an expected diameter of only $\Theta(\log{n})$ -- a much smaller value than the greedy routing algorithm's path lengths. In 2022, Goodrich and Ozel proposed the \emph{neighborhood preferential attachment} model (NPA), combining elements from Barab\'asi and Albert's model with Kleinberg's model, and experimentally showed that the resulting model outperformed Kleinberg's greedy routing performance on U.S. road networks. While they displayed impressive empirical results, they did not provide any theoretical analysis of their model. In this paper, we first provide a theoretical analysis of a generalization of Kleinberg's original model and show that it can achieve expected $\mathcal{O}(\log{n})$ routing, a much better result than Kleinberg's model. We then propose a new model, \emph{windowed NPA}, that is similar to the neighborhood preferential attachment model but has provable theoretical guarantees w.h.p. We show that this model is able to achieve $\mathcal{O}(\log^{1 + \epsilon}{n})$ greedy routing for any $\epsilon > 0$.