We can express this as, assuming that \( T \) and \( A \) are time-independent, $$ \begin{equation*} w_i(t+1) = \sum_j \left [ w_j(t)T_{j\rightarrow i} A_{j\rightarrow i} +w_i(t)T_{i\rightarrow j}\left ( 1- A_{i\rightarrow j} \right) \right ] \,. \end{equation*} $$