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A continuum limit for the PageRank algorithm

Part of: Elliptic equations and systems Representations of solutions

Published online by Cambridge University Press:  27 April 2021

A. YUAN Open the ORCID record for A. YUAN [Opens in a new window] ,
J. CALDER Open the ORCID record for J. CALDER [Opens in a new window]  and
B. OSTING Open the ORCID record for B. OSTING [Opens in a new window]
A. YUAN
Affiliation:
Department of Mathematics, University of Minnesota, Minneapolis, MN55455, USA emails: [email protected]; [email protected]
J. CALDER
Affiliation:
Department of Mathematics, University of Minnesota, Minneapolis, MN55455, USA emails: [email protected]; [email protected]
B. OSTING
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT84112, USA email: [email protected]
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Abstract

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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.

Keywords

Partial differential equations on graphs and networkssecond-order elliptic equationsviscosity solutions

MSC classification

Primary: 35J15: Second-order elliptic equations
Secondary: 35D40: Viscosity solutions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 3 , June 2022 , pp. 472 - 504
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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