A General Setting for the Parametric Google Matrix

The spectral and Jordan structures of the web hyperlink matrix G(c) = cG + (1 − c)ev have been analyzed when G is the basic (stochastic) Google matrix, c is a real parameter such that 0 < c < 1, v is a nonnegative probability vector, and e is the all-ones vector. Typical studies have relied heavily on special properties of nonnegative, positive, and stochastic matrices. There is a unique nonnegative vector y(c) such that y(c)G(c) = y(c) and y(c) e = 1. This PageRank vector y(c) can be computed effectively by the power method. We consider a square complex matrix A and nonzero complex vectors x and v such that Ax = λx and v∗x = 1. We use standard matrix analytic tools to determine the eigenvalues, the Jordan blocks, and a distinguished left λ-eigenvector of A(c) = cA+ (1− c)λxv∗ as a function of a complex variable c. If λ is a semisimple eigenvalue of A, there is a uniquely determined projection P such that limc→1 y(c) = Pv for every v such that v∗x = 1; this limit may fail to exist for some v if λ is not semisimple. As a special case of our results, we obtain a complex analog of PageRank for the web hyperlink matrix G(c) with a complex parameter c. We study regularity, limits, expansions, and conditioning of y(c), and we propose a complex extrapolation algorithm that may provide an efficient way to compute PageRank.