For large \( n \) we require that \( {\cal w}^{(n\rightarrow \infty)}_i = p_i \), the desired probability distribution. Taking this limit, gives the balance requirement
$$ \begin{equation} \sum_j \left [p_jT_{j\rightarrow i} A_{j\rightarrow i}-p_iT_{i\rightarrow j}A_{i\rightarrow j} \right ] = 0, \tag{7} \end{equation} $$