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 $\epsilon > 0$.

翻译：20世纪60年代，世界著名社会心理学家斯坦利·米尔格拉姆通过实验表明，任意两人之间不仅存在"短链"熟人关系，而且这些素不相识的陌生人能够找到这些短链。这一被称为"小世界现象"的现象，部分可通过低直径模型（如巴拉巴西和阿尔伯特的优先连接模型）解释，但这些模型未能体现米尔格拉姆实验中的高效路由特性。2000年，克莱因贝格提出一种具有高效贪心路由算法（时间复杂度为$\mathcal{O}(\log^2{n})$）的模型。2004年，马特尔和阮证明克莱因贝格的分析是紧的，同时指出其模型预期直径仅为$\Theta(\log{n})$——远小于贪心路由算法的路径长度。2022年，古德里奇和奥泽尔提出邻域优先连接模型（NPA），融合了巴拉巴西-阿尔伯特模型与克莱因贝格模型的特点，实验表明该模型在美国道路网络上的贪心路由性能优于克莱因贝格模型。尽管他们展示了令人印象深刻的实证结果，但未提供模型的理论分析。本文首先对克莱因贝格原始模型的泛化形式进行理论分析，证明其能达到预期$\mathcal{O}(\log{n})$的路由性能，显著优于原模型；随后提出新型"窗口化NPA"模型，该模型与邻域优先连接模型相似，但具有可证的高概率理论保证。我们证明，对于任意$\epsilon > 0$，该模型能实现$\mathcal{O}(\log^{1 + \epsilon}{n})$的贪心路由复杂度。