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 normalised 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 partial differential equation. 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.

半监督和无监督机器学习方法通​​常依赖图来建模数据，这促使人们研究如何在学习问题中利用图上算子的理论属性。虽然大多数现有文献都关注无向图，但有向图在实践中非常重要，它为物理、生物或交通网络以及许多其他应用提供模型。在本文中，我们提出了一个新框架，用于严格研究有向图学习算法的连续极限。我们使用新框架来研究 PageRank 算法并展示如何将其解释为涉及一种归一化图拉普拉斯算子的有向图上的数值方案. 我们表明，随着网页数量增长到无穷大，相应的连续极限问题是一个包含反应、扩散和平流项的二阶、可能退化的椭圆方程。我们证明了数值格式是一致的和稳定的，并计算了离散解的显式收敛速度连续极限偏微分方程的解。我们给出了证明 PageRank 向量的稳定性和渐近规律性的应用程序。最后，我们通过数值实验来说明我们的结果，并探索数据深度的应用。