Optimal accuracy for linear sets of equations with the graph Laplacian

TL;DR

This paper investigates the conditions under which certain graph Laplacian linear systems, relevant to PageRank and Markov chains, achieve optimal accuracy, and characterizes their behavior based on graph properties.

Contribution

It establishes the optimal accuracy conditions for graph Laplacian systems and relates these to parameters like PageRank teleportation and Markov chain discount.

Findings

Optimal accuracy is guaranteed when the relative error is bounded by the residual norm.

Systems can be categorized based on their asymptotic optimality related to the graph size.

Numerical experiments support the theoretical results.

Abstract

We show that certain Graph Laplacian linear sets of equations exhibit optimal accuracy, guaranteeing that the relative error is no larger than the norm of the relative residual and that optimality occurs for carefully chosen right-hand sides. Such sets of equations arise in PageRank and Markov chain theory. We establish new relationships among the PageRank teleportation parameter, the Markov chain discount, and approximations to linear sets of equations. The set of optimally accurate systems can be separated into two groups for an undirected graph -- those that achieve optimality asymptotically with the graph size and those that do not -- determined by the angle between the right-hand side of the linear system and the vector of all ones. We provide supporting numerical experiments.