基于命中概率的有向图与马尔可夫链度量
A metric on directed graphs and Markov chains based on hitting probabilities
AI总结：
本文提出了一种基于命中概率的度量，适用于遍历有限状态时齐马尔可夫链和有向图，能在$O(n^3)$时间内计算，对最短和平均游走距离不敏感，并展示了其在社区结构弱恢复、可视化和多尺度聚类等任务中的实用性。
摘要：
无向图上的最短路径、通勤时间和扩散距离已被广泛应用于降维、链接预测和行程规划等应用中。人们对利用源自马尔可夫链和有向图的数据的不对称结构越来越感兴趣，但很少有度量专门适用于此任务。我们引入了一种度量，适用于任何遍历、有限状态、时齐马尔可夫链的状态空间，特别是适用于任何源自有向图的马尔可夫链。我们的构造基于命中概率，度量空间中的接近度与平稳状态下随机游走者从一个节点转移到另一个节点有关。值得注意的是，我们的度量对最短和平均游走距离不敏感，因此与现有度量相比提供了新信息。我们利用度量中可能的退化性来发展有向图的有趣结构理论，并探索了相关的商运算过程。该度量可在$O(n^3)$时间内计算，其中$n$是状态数，在示例中，我们在台式计算机上扩展到$n=10,000$个节点和$\approx 38M$条边。在几个示例中，我们探索了该度量的性质，将其与替代方法进行比较，并展示了其在稠密图社区结构的弱恢复、可视化、结构恢复、动力学探索和多尺度聚类检测中的实用性。
英文摘要：
The shortest-path, commute time, and diffusion distances on undirected graphs have been widely employed in applications such as dimensionality reduction, link prediction, and trip planning. Increasingly, there is interest in using asymmetric structure of data derived from Markov chains and directed graphs, but few metrics are specifically adapted to this task. We introduce a metric on the state space of any ergodic, finite-state, time-homogeneous Markov chain and, in particular, on any Markov chain derived from a directed graph. Our construction is based on hitting probabilities, with nearness in the metric space related to the transfer of random walkers from one node to another at stationarity. Notably, our metric is insensitive to shortest and average walk distances, thus giving new information compared to existing metrics. We use possible degeneracies in the metric to develop an interesting structural theory of directed graphs and explore a related quotienting procedure. Our metric can be computed in $O(n^3)$ time, where $n$ is the number of states, and in examples we scale up to $n=10,000$ nodes and $\approx 38M$ edges on a desktop computer. In several examples, we explore the nature of the metric, compare it to alternative methods, and demonstrate its utility for weak recovery of community structure in dense graphs, visualization, structure recovering, dynamics exploration, and multiscale cluster detection.
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。
