PageRank算法的连续极限
A continuum limit for the PageRank algorithm
摘要：
半监督和无监督机器学习方法通常依赖于图来建模数据，这促使人们研究如何在学习问题中利用图上运算符的理论属性。虽然大多数现有的文献集中在无向图，但有向图在实践中非常重要，可以为物理，生物或运输网络以及许多其他应用提供模型。在本文中，我们提出了一个新的框架，严格研究连续极限的学习算法上的有向图。我们使用新的框架来研究PageRank算法，并展示它如何可以被解释为一个数值计划的有向图涉及一种类型的规范化图拉普拉斯算子。我们表明，相应的连续极限问题，这是作为网页的数量增长到无穷大，是一个二阶，可能退化，椭圆型方程，其中包含反应，扩散和对流项。我们证明了该数值格式是一致的和稳定的，并计算了离散解到连续极限PDE解的显式收敛速度。我们给出了应用程序来证明PageRank向量的稳定性和渐近正则性。最后，我们用数值实验说明了我们的结果，并探索了数据深度的应用。
英文摘要：
Semi-supervised and unsupervised machine learning methods often rely on graphs to model data, prompting research on how theoretical properties of operators on graphs are leveraged in learning problems. While most of the existing literature focuses on undirected graphs, directed graphs are very important in practice, giving models for physical, biological, or transportation networks, among many other applications. In this paper, we propose a new framework for rigorously studying continuum limits of learning algorithms on directed graphs. We use the new framework to study the PageRank algorithm, and show how it can be interpreted as a numerical scheme on a directed graph involving a type of normalized graph Laplacian. We show that the corresponding continuum limit problem, which is taken as the number of webpages grows to infinity, is a second-order, possibly degenerate, elliptic equation that contains reaction, diffusion, and advection terms. We prove that the numerical scheme is consistent and stable and compute explicit rates of convergence of the discrete solution to the solution of the continuum limit PDE. We give applications to proving stability and asymptotic regularity of the PageRank vector. Finally, we illustrate our results with numerical experiments and explore an application to data depth.
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