序列分层再生：大状态空间的MCMC及其在子图计数估计中的应用
Sequential Stratified Regeneration: MCMC for Large State Spaces with an Application to Subgraph Count Estimation
摘要：
这项工作认为，估计一个有界函数的图形的边缘的总和的一般任务，给定的邻域查询访问和访问整个网络是非常昂贵的。为了估计这个总和，先前的工作提出了马尔可夫链蒙特卡罗（MCMC）方法，该方法使用从某个种子顶点开始的随机游走，并且其平衡分布是所有边缘上的均匀分布，从而消除了对所有边缘的遍历。不幸的是，这些现有的估计是不可扩展到大规模的现实世界的图形。在本文中，我们介绍了Ripple，MCMC为基础的估计，实现了前所未有的可扩展性分层的马尔可夫链状态空间到有序层与一种新的技术，我们表示{ \em顺序分层再生}。我们证明了Ripple估计量是一致的，高度并行化的，并且扩展良好。我们通过将Ripple应用于估计给定一些输入图的连接，诱导子图计数的任务来经验性地评估我们的方法。其中，我们证明了Ripple是准确的，可以估计多达$12$节点子图的计数，这是一个被认为是不可达到的规模的任务，不仅通过先前的基于MCMC的方法，而且通过其他采样方法。例如，在这个目标应用程序中，我们给出了马尔可夫链状态空间与$10^{43}$一样大的结果，Ripple平均在不到$4$小时的时间内计算估计值。
英文摘要：
This work considers the general task of estimating the sum of a bounded function over the edges of a graph, given neighborhood query access and where access to the entire network is prohibitively expensive. To estimate this sum, prior work proposes Markov chain Monte Carlo (MCMC) methods that use random walks started at some seed vertex and whose equilibrium distribution is the uniform distribution over all edges, eliminating the need to iterate over all edges. Unfortunately, these existing estimators are not scalable to massive real-world graphs. In this paper, we introduce Ripple, an MCMC-based estimator that achieves unprecedented scalability by stratifying the Markov chain state space into ordered strata with a new technique that we denote {\em sequential stratified regenerations}. We show that the Ripple estimator is consistent, highly parallelizable, and scales well. We empirically evaluate our method by applying Ripple to the task of estimating connected, induced subgraph counts given some input graph. Therein, we demonstrate that Ripple is accurate and can estimate counts of up to $12$-node subgraphs, which is a task at a scale that has been considered unreachable, not only by prior MCMC-based methods but also by other sampling approaches. For instance, in this target application, we present results in which the Markov chain state space is as large as $10^{43}$, for which Ripple computes estimates in less than $4$ hours, on average.
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