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Structural Properties of Finite Markov Processes

    
Find the communicating classes, highlighted in the graph through different colors:
Wolfram Language code: proc = DiscreteMarkovProcess[1, {{0, 1 / 3, 0, 2 / 3, 0}, {1 / 2, 0, 0, 0, 1 / 2}, {0, 0, 1 / 2, 1 / 2, 0}, {0, 0, 1 / 2, 1 / 2, 0}, {0, 0, 0, 0, 1}}];
Wolfram Language code: gr = Graph[proc, GraphLayout -> "LayeredDrawing"]
Wolfram Language code: MarkovProcessProperties[proc, "CommunicatingClasses"]
Check for irreducibility, which means there is only one communicating class:
Wolfram Language code: MarkovProcessProperties[proc, "Irreducible"]
Find the recurrent classes, which are classes that cannot be exited:
Wolfram Language code: MarkovProcessProperties[proc, "RecurrentClasses"]
Find the transient classes, which are classes that can be exited:
Wolfram Language code: MarkovProcessProperties[proc, "TransientClasses"]
Find the absorbing classes, which are recurrent classes with a single state:
Wolfram Language code: MarkovProcessProperties[proc, "AbsorbingClasses"]