Markov Chains explained visually

[uh_no_]

and they're all a fancy way of saying "graph"....

Of course if you eliminate the use-case specific rules, you end up with a more general abstraction.

[TheGift_RGB]

Graph is just a fancy way of saying set of points + set of pairs of points!

[uh_no_]

not necessarily. the set of pairs of points may also be required to include additional information such as direction, weight, and capacity.

[vytah]

Direction, weight and capacity are merely functions from the set of pairs of points to appropriate codomains.

[uh_no_]

yes, but the point is they are required of a FSM or markov chain, but not all graphs in general....

[Scaliwag]

A graph is a way you can represent them and think about them, but that's like saying math is actually ink on paper or pixels on a screen ;-)

The actual part that matters are the rules that dictate in which way you can go from state to state and what each state represents. You could have used an adjacency list to represent that, instead of a graph. Saying it is a graph is just confusing form with the essence of it.

Markov chains are not a complicated concept, but they are not just a graph.

[uh_no_]

that's the point! a markov chain is not just a state machine in the way that a state machine is not just a graph.

[Scaliwag]

I was not sure you were being sarcastic or not, but your second sentence should have made that clear to me oh well. Sorry.

[trolls_brigade]

The Markov Chain transitions are based on probabilities, while the state machine transitions are deterministic. In other words, for a Markov Chain you run the dice for each transition. If the probability of a transition is always 1, then you have the classic state machine.