大型图中的命中时间和通勤时间通常会产生误导
Hitting and commute times in large graphs are often misleading
摘要：
除了最短路径距离，图中顶点之间的第二个最流行的距离函数是通勤距离（阻力距离）。对于两个顶点u和v，命中时间H_{uv}是从u到v随机游走所需的预期时间。通勤时间是其对称化版本C_{uv} = H_{uv} + H_{vu}。本文研究了当图的顶点数n很大时，图的可换距离和击中时间的性质。我们证明，当n收敛到无穷大时，击中时间和通勤距离收敛到完全不考虑图的全局结构的表达式。也就是说，击中时间H_{uv}收敛于1/d_v，而交换时间收敛于1/d_u + 1/d_v，其中d_u和d_v表示顶点u和v的度。在这些情况下，击中时间和交换时间在某种意义上是误导的，因为它们不提供关于图的结构的信息。我们主要研究两类随机图：随机几何图（k-最近邻图、ε-图、高斯相似图）和具有给定期望度的随机图（特别是具有和不具有种植分区的Erdos-Renyi图）
英文摘要：
Next to the shortest path distance, the second most popular distance function between vertices in a graph is the commute distance (resistance distance). For two vertices u and v, the hitting time H_{uv} is the expected time it takes a random walk to travel from u to v. The commute time is its symmetrized version C_{uv} = H_{uv} + H_{vu}. In our paper we study the behavior of hitting times and commute distances when the number n of vertices in the graph is very large. We prove that as n converges to infinty, hitting times and commute distances converge to expressions that do not take into account the global structure of the graph at all. Namely, the hitting time H_{uv} converges to 1/d_v and the commute time to 1/d_u + 1/d_v where d_u and d_v denote the degrees of vertices u and v. In these cases, the hitting and commute times are misleading in the sense that they do not provide information about the structure of the graph. We focus on two major classes of random graphs: random geometric graphs (k-nearest neighbor graphs, epsilon-graphs, Gaussian similarity graphs) and random graphs with given expected degrees (in particular, Erdos-Renyi graphs with and without planted partitions)
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
