  Random walks over directed graphs are used to model activities in many
domains, such as social networks, influence propagation, and Bayesian graphical
models. They are often used to compute the importance or centrality of
individual nodes according to a variety of different criteria. Here we show how
the pseudoinverse of the "random walk" Laplacian can be used to quickly compute
measures such as the average number of visits to a given node and various
centrality and betweenness measures for individual nodes, both for the network
in general and in the case a subset of nodes is to be avoided. We show that
with a single matrix inversion it is possible to rapidly compute many such
quantities.
