Google PageRanking problem: The model and the analysis

The spectral and Jordan structures of the Web hyperlink matrix G ( c ) = c G + ( 1 − c ) e v T 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 ) T G ( c ) = y ( c ) T and y ( c ) T e = 1 . This PageRank vector y ( c ) can be computed effectively by the power method. sider a square complex matrix A and nonzero complex vectors x and v such that A x = λ x 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 ) = c A + ( 1 − c ) λ x v ∗ as a function of a complex variable c . If λ is a semisimple eigenvalue of A , there is a uniquely determined projection N such that lim c → 1 y ( c ) = N v for all v ; 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 algorithms (e.g., complex extrapolation, power method on a modified matrix etc.) that may provide an efficient way to compute PageRank also with c close or equal to 1. An interpretation of the limit vector N v and a related critical discussion on the model, on its adherence to reality, and possible ways for its improvement, represent the contribution of the paper on modeling issues.