基于随机游动抽样的最短路径长度分布估计
Estimating Shortest Path Length Distributions via Random Walk Sampling
摘要：
在网络中，节点之间的最短路径非常重要，因为它们允许节点之间最快和最强的交互。然而，测量大型网络中所有节点之间的最短路径在计算上是昂贵的。本文提出了一种利用随机游动抽样估计网络最短路长分布的方法。为了处理样本中二对（节点对）包含概率不等的问题，我们推广了Hansen-Hurwitz估计和Horvitz-Thompson估计（及其比值形式）的用法，并将它们应用于样本二对中。基于马尔可夫链的理论，证明了二元系统的选择概率正比于两个节点的度的乘积。为了近似一个二元体的实际SPL，我们使用在具有大度可变性的网络的诱导子图中观察到的SPL，即，标准差至少是平均值的两倍，对于度变化较小的网络，使用度变化较小的网络的界标估计SPL。通过对实际网络的仿真研究和应用，我们发现：1）对于大型网络，采用单个或多个总步数至少等于网络中20%的随机游动都可以获得较高的估计精度：2）估计性能随网络规模的增大而提高，但当网络规模足够大时趋于稳定; 3）单次随机游动的性能与多次随机游动的性能一样好; 4）Horvitz-Thompson比估计在四种估计中性能最好。
英文摘要：
In a network, the shortest paths between nodes are of great importance as they allow the fastest and strongest interaction between nodes. However measuring the shortest paths between all nodes in a large network is computationally expensive. In this paper we propose a method to estimate the shortest path length (SPL) distribution of a network by random walk sampling. To deal with the unequal inclusion probabilities of dyads (pairs of nodes) in the sample, we generalize the usage of Hansen-Hurwitz estimator and Horvitz-Thompson estimator (and their ratio forms) and apply them to the sampled dyads. Based on theory of Markov chains we prove that the selection probability of a dyad is proportional to the product of the degrees of the two nodes. To approximate the actual SPL for a dyad, we use the observed SPL in the induced subgraph for networks with large degree variability, i.e., the standard deviation is at least two times of the mean, and for networks with small degree variability, estimate the SPL using landmarks for networks with small degree variability. By simulation studies and applications to real networks, we find that 1) for large networks, high estimation accuracy can be achieved by using a single random or multiple random walks with total number of steps equal to at least 20% of the nodes in the network; 2) the estimation performance increases as the network size increases but tends to stabilize when the network is large enough; 3) a single random walk performs as well as multiple random walks; 4) the Horvitz-Thompson ratio estimator performs best among the four estimators.
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。
