Stochastic matrices: ergodicity coefficients and applications to ranking

Abstract

We present two different views of (row) stochastic matrices, which are nonnegative matrices with row sums equal to one. For applications to ranking, we examine the computation of a dominant left eigenvector of a stochastic matrix. The stochastic matrix of interest is called the Google matrix and contains information about how pages of the Internet are linked to one another. The dominant left eigenvector of the Google matrix yields a ranking for each Web page, which helps to determine the order in which search results are returned. These results are presented in Chapter 1.

Chapter 2 is concerned with coefficients of ergodicity, which measure the rate at which products of stochastic matrices, especially products whose number of factors is unbounded, converge to a matrix of rank one. Ergodicity arises in the context of Markov chains and signals the tendency of the rows of such products to equalize. We present unified notation and definitions for coefficients of ergodicity applied to stochastic matrices, extend the definitions to general complex matrices, and illustrate the connection between ergodicity coefficients and inclusion regions for eigenvalues and singular values.