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=====Markov Chain Example=====

=====Markov Chain Example=====

In the literature <ref name=":0"/>, Hoel gives an example of a state transition matrix ([math]f_m[/math])of a Markov chain with 8 states, as shown in the left figure below. Among them, the first 7 states transfer with equal probability, and the last state is independent and can only transition to its own state.

In the literature <ref name=":0"/>, Hoel gives an example of a state transition matrix ([math]f_m[/math]) of a Markov chain with 8 states, as shown in the left figure below. Among them, the first 7 states transfer with equal probability, and the last state is independent and can only transition to its own state.

However, for more general Markov chains and more general state  groupings, this simple operation of averaging probabilities is not  always feasible. This occurs when the merged transition matrix does not meet Markov chain conditions, such as  row normalization or element values within [0, 1]. Refer to the “Reduction of Markov Chains” section or the “[[Coarse-graining of Markov Chains]]” entry for details on feasible macroscopic Markov chains.  

However, for more general Markov chains and more general state  groupings, this simple operation of averaging probabilities is not  always feasible. This occurs when the merged transition matrix does not meet Markov chain conditions, such as  row normalization or element values within [0, 1]. Refer to the “Reduction of Markov Chains” section or the “[[Coarse-graining of Markov Chains]]” entry for details on feasible macroscopic Markov chains.

 

=====Boolean Network Example=====

=====Boolean Network Example=====