I see, where you're coming from, but in that particular case, the "intuitive" explanation (walk on a graph) is far less intuitive for me than the proper explana

I see, where you're coming from, but in that particular case, the "intuitive" explanation (walk on a graph) is far less intuitive for me than the proper explanation, that a Markov process is memoryless. That said, I used MCMC in the past to do physics simulation, where the Markov property also applies to the underlying physical process. So maybe it's just me.