A continuum limit for the PageRank algorithm

Semi-supervised and unsupervised machine learning methods often rely on graphs to model data, prompting research how theoretical properties of operators are leveraged in problems. While most the existing literature focuses undirected graphs, directed very important practice, giving models for physical, biological or transportation networks, among many other applications. In this paper, we propose a new framework rigorously studying continuum limits algorithms graphs. We use study PageRank algorithm show it can be interpreted as numerical scheme graph involving type normalised Laplacian . that corresponding limit problem, which is taken number webpages grows infinity, second-order, possibly degenerate, elliptic equation contains reaction, diffusion advection terms. prove consistent stable compute explicit rates convergence discrete solution partial differential equation. give applications proving stability asymptotic regularity vector. Finally, illustrate our results with experiments explore an application data depth.