AI 中文总结
针对随机几何图，通过划分空间、拼接重边、多级缩放与权重分段的方法，估计满足约束的两点间最大权重路径，证明其近最优性并给出实例。
AI 中文摘要
本文研究由n个顶点均匀分布在平面单位正方形S上构成的随机几何图（RGG）G，为G的每条边赋予独立权重。假设顶点间的邻接距离大于连通阈值，利用满足长度和权重约束的边，估计连接S中两个固定点O₁和O₂的路径的最大权重。策略为：先将O₁与O₂之间的空间划分为小正方形，识别包含重边的“良好正方形”；再采用迭代拼接过程连接这些重边，估计所得路径P的权重；最后结合多级缩放过程与权重分段，建立O₁和O₂间任意路径最大权重的上界，从而证明P的近最优性。还通过边权重满足幂律和指数衰减的示例说明结果。
英文摘要
In this paper, we consider a random geometric graph (RGG)~\(G\) formed by~\(n\) vertices distributed uniformly in the unit square~\(S\) on the plane and equip each edge of~\(G\) with an independent weight. We assume that the adjacency distance between vertices is larger than the connectivity threshold and estimate the maximum weight of a path connecting two fixed points~\(O_1\) and~\(O_2\) in~\(S,\) using edges of~\(G\) that satisfy length and weight constraints. Our strategy is to first divide the space between~\(O_1\) and~\(O_2\) into small squares and identify nice squares containing heavy edges. We then use an iterative stitching procedure to connect these heavy edges and estimate the weight of the resulting path~\(P.\) Finally, we invoke a multi-level scaling procedure along with weight segmentation to establish an upper bound for the maximum weight of \emph{any} path between~\(O_1\) and~\(O_2\) and thereby demonstrate the near-optimality of~\(P.\) We also illustrate our results using examples involving edge weights satisfying power law and exponential decay.
