[Paper Review] A Note on the Eigenvalues of the Google Matrix

Lars Eldén|ArXiv.org|Jan 15, 2004

Algorithms and Data Compression11 references17 citations

TL;DR

This paper provides a concise, alternative proof showing that the eigenvalues of the Google matrix A = αP + (1−α)ve^T are {1, αλ₂, αλ₃, ..., αλₙ}, where P is column-stochastic with eigenvalues {1, λ₂, ..., λₙ}. The key result is that the second largest eigenvalue in magnitude is always α, directly determining the convergence rate of the power method used in PageRank computation.

ABSTRACT

The Google matrix is a positive, column-stochastic matrix that is used to compute the pagerank of all the web pages on the Internet: the eigenvector corresponding to the eigenvalue 1 is the pagerank vector. Due to its huge dimension, of the order of billions, the (presently) only viable method to compute the eigenvector is the power method. For the convergence of the iteration, it is essential to know the eigenvalue distribution of the matrix. A theorem concerning the eigenvalues was recently proved by Langville and Meyer. In this note another proof is given.

Motivation & Objective

• To provide a simple, alternative proof of the eigenvalue structure of the Google matrix A = αP + (1−α)ve^T.
• To clarify the spectral properties of the Google matrix, particularly the magnitude of the second largest eigenvalue.
• To support the theoretical foundation of the PageRank algorithm by rigorously establishing the eigenvalue distribution.
• To reinforce the understanding of convergence behavior in the power method for large-scale web graph computations.

Proposed method

• Use orthogonal transformation via U = [ê, U₁], where ê is the normalized all-ones vector, to decompose the matrix P into block form.
• Exploit the column-stochastic property e^T P = e^T to show that P transforms under U^T into a block matrix with identity in the (1,1) position and zero in the (1,2) block.
• Express the transformed matrix U^T A U as a block matrix combining α times the transformed P and (1−α) times the outer product ve^T in the transformed basis.
• Derive the block structure of U^T A U = [1, 0; w₁, αT], where T contains the eigenvalues λ₂, ..., λₙ of P.
• Leverage similarity transformation to preserve eigenvalues and conclude that the eigenvalues of A are {1, αλ₂, ..., αλₙ}.
• Use the normalization of e to e^T v = 1 and the Euclidean normalization of ê to maintain consistency in the transformation.

Experimental results

Research questions

• RQ1 What are the exact eigenvalues of the Google matrix A = αP + (1−α)ve^T, given that P is column-stochastic and v is a probability vector?
• RQ2 How does the second largest eigenvalue of A depend on the parameter α, and what is its magnitude?
• RQ3 Does the spectral structure of A remain stable under perturbations from the rank-one correction (1−α)ve^T?
• RQ4 Can the eigenvalue distribution of A be derived using orthogonal transformation and block matrix decomposition?
• RQ5 What is the theoretical justification for the convergence rate of the power method in PageRank being governed by α?

Key findings

• The eigenvalues of the Google matrix A are exactly {1, αλ₂, αλ₃, ..., αλₙ}, where λ₂, ..., λₙ are the eigenvalues of the original column-stochastic matrix P.
• The second largest eigenvalue in magnitude of A is always equal to α, regardless of the structure of P.
• Even if P has multiple eigenvalues equal to 1, the spectral gap of A is still determined by α, ensuring predictable convergence.
• The power method's convergence rate for computing PageRank is governed by α, as the spectral radius of the second eigenvalue is α.
• The transformation using orthogonal matrix U preserves eigenvalues and allows a clean block decomposition that reveals the eigenvalue scaling.
• The result confirms that the PageRank computation converges at a rate proportional to α, which is critical for large-scale web indexing.