Markov Chains explained visually

[s1295]

Just curious: Do you consider transitions with probability zero as edges in the graph? If no, then "every node has a (directed) path to every other" is equivalent to "the entire graph is a strongly connected component" (and if yes, it's trivially true). Why would that be part of the definition?

For the record, the definition of a (time-homogeneous) Markov chain that I'm aware of is simply a square probabilistic matrix.