So basically to calculate the stationary distribution (distribution over a long time of evaluation) of a markov chain, you have to calculate the eigenvectors of

So basically to calculate the stationary distribution (distribution over a long time of evaluation) of a markov chain, you have to calculate the eigenvectors of the transition matrix. And these take into account the graph structure, so I think the stationary distribution will be exactly what is calculated this way, there isn't really any space for anything beyond that. Incidentally, this is very similar to the original pagerank algorithm. The rank roughly corresponds to the probability you will be on any given page if you randomly follow links.