@Walter I agree with you, and @Zubin you need to narrow down the question to a particular reference. We don't seem to understand what you want here.
I suspect you probably have a state-transition matrix, T, and with a given input conditions, A_t0, tyou can compute the output state as a matrix-vector product,
Ofcourse you need to map your cluster index to a Matrix, and find the values of the matrix T, assuming it is not time dependent, etc.
So details are key here.