Indeed, the Monte Carlo post was inspired by some discussions after reading that chapter. I also borrowed and adapted a proof or two from the SVD chapter for that second post.
Your Monte Carlo post is interesting, but it doesn't consider the context of markov networks, for example reading Norvig Modern IA there are good examples. There is a subtle point that having probabilities locally you must prove that the global structure is a probability and this requires some order in the vertexes to propagate the information from the root to the leafs, also Markov models can have higher degree and then they are not like random walks. Anyway, the context of matrices and eigenvalues is interesting.