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Optimal accuracy for linear sets of equations with the graph Laplacian

Richard B. Lehoucq, Michael Weylandt, Jonathan W. Berry

TL;DR

This work develops accuracy guarantees for solving linear systems with the α-discounted graph Laplacian $L = I - \alpha D^{-1}A$, establishing size-independent bounds on the inverse and condition numbers and introducing a data-dependent condition number $\kappa_D(L,\mathbf{b})$ that tightens the relation between relative error and relative residual via the geometry to the all-ones vector. It shows that for undirected graphs the SPD reformulation yields uniform bounds, and it connects these results to discrete potential theory and PageRank, providing conditions under which optimal accuracy is achieved, including bounds that scale with the angle to $\mathbf{1}$. The paper unifies PageRank, Markov chain concepts, and discrete potential theory, deriving theoretically tight bounds and validating them with numerical experiments that illustrate when high-accuracy solutions are guaranteed and how the rhs choice affects conditioning and rankings. Practical implications include improved certificates of solution quality for PageRank variants and centrality measures, with performance guarantees that are robust to graph size.

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.

Optimal accuracy for linear sets of equations with the graph Laplacian

TL;DR

This work develops accuracy guarantees for solving linear systems with the α-discounted graph Laplacian , establishing size-independent bounds on the inverse and condition numbers and introducing a data-dependent condition number that tightens the relation between relative error and relative residual via the geometry to the all-ones vector. It shows that for undirected graphs the SPD reformulation yields uniform bounds, and it connects these results to discrete potential theory and PageRank, providing conditions under which optimal accuracy is achieved, including bounds that scale with the angle to . The paper unifies PageRank, Markov chain concepts, and discrete potential theory, deriving theoretically tight bounds and validating them with numerical experiments that illustrate when high-accuracy solutions are guaranteed and how the rhs choice affects conditioning and rankings. Practical implications include improved certificates of solution quality for PageRank variants and centrality measures, with performance guarantees that are robust to graph size.

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.
Paper Structure (5 sections, 3 theorems, 14 equations, 1 figure)

This paper contains 5 sections, 3 theorems, 14 equations, 1 figure.

Table of Contents

  1. Graph Condition Number Bounds via Discounting
  2. Uniform Optimality
  3. Discrete potential theory
  4. PageRank
  5. Experiments

Key Result

Proposition 1

Consider the symmetric positive definite system spd corresponding to an undirected graph where $\bm{b}$ is not collinear with $\bm{1}$, i.e., $\cos_{\bm{D}}\angle(\bm{b}, \bm{1}) < 1$ and the discount $\alpha$ satisfies $0 < \alpha < 1$. Then euc-2s improves to where Moreover, if $\sin_{\bm{D}} \angle(\bm{x}, \bm{1}) \leqslant 1-\alpha$ then $\kappa_{\bm{D}}(L,b) \leqslant \sqrt{1+(1+\alpha)^2}

Figures (1)

  • Figure 1: Three plots of the ratio of relative error to the norm of the relative residual over three choices of discount of $\alpha$ on a sample of BTER undirected graphs with the choice of $\bm{b}=\bm{1}_{\Omega}$ (MHT) and $b= \bm{1}_{\widetilde{\Omega}}$ (PPR) where $\widetilde{\Omega}$ contains a vertex of the graph selected randomly and $\bm{1}_{\widetilde{\Omega}}+\bm{1}_{\Omega} = \bm{1}$. The solution $x$ was computed via the conjugate gradient (CG) iteration using a relative residual tolerance of $10^{-12}$. The approximations $\hat{\bm{x}}$ were computed via the CG iteration terminated when the relative residual tolerance satisfied $10^{-3}$ was achieved or a maximum of $40$, $120$, and $480$ CG iterations were performed.

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 2
  • Proposition 3