  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.
