Google's PageRank algorithm for ranking nodes in general networks

This paper extends the random surfer approach of Google's PageRank algorithm to general finite Markov chains that may consist of several ergodic classes and possible transient states. We will introduce the new concept of an extended ergodic projector of a Markov chain and we will show how the extended ergodic projector allows for intuitive better ranking of transient states. Numerical examples are provided to illustrate the effect of this new ranking approach.