Random walks associated with symmetric M-matrices

Abstract

In this paper we generalize the transition probability matrix for a random walk on a finite network by defining the transition probabilities through a symmetric M-matrix. Usually, the walker jumps from a vertex to a neighbor according to the probabilities given by the adjacency matrix. Moreover, we can find in the literature the relation between random walks and the normalized laplacian or the combinatorial laplacian that are singular and symmetric M-matrices. Our model takes into consideration not only the probability of transitioning given by the adjacency matrix but also some added probability that depends on a node property. This also includes the probability of remaining in each node, when the M-matrix is not singular. The nodes importance is taking into account by considering the lower eigenvalue and its associated eigenfunction for the given M-matrix. We give expressions for the mean first passage time and Kemeny's constant for such a random walks in terms of 1-inverses of the considered M-matrix.