The Second Eigenvalue of the Google Matrix

Abstract

We determine analytically the modulus of the second eigenvalue for the web hyperlink matrix used by Google for computing PageRank. Specifically, we prove the following statement: ``For any matrix $A=[cP + (1-c)E]^T$, where $P$ is an $n \times n$ row-stochastic matrix, $E$ is a strictly positive $n \times n$ rank-one row-stochastic matrix, and $0 \leq c \leq 1$, the second eigenvalue of $A$ has modulus $|\lambda_2| \leq c$. Furthermore, if $P$ has at least two irreducible closed subsets, the second eigenvalue $\lambda_2 = c$.'' This statement has implications for the convergence rate of the standard PageRank algorithm as the web scales, for the stability of PageRank to perturbations to the link structure of the web, for the detection of Google spammers, and for the design of algorithms to speed up PageRank.