The further constraints are $0 \le W_{ij} \le 1$ and $0 \le w_{j} \le 1$.

• We can thus write the action of $W$ as

$$\begin{equation*} w_i(t+1) = \sum_jW_{ij}w_j(t), \end{equation*}$$ or as vector-matrix relation $$\begin{equation*} \hat{w}(t+1) = \hat{W\hat{w}}(t), \end{equation*}$$ and if we have that $||\hat{w}(t+1)-\hat{w}(t)||\rightarrow 0$, we say that we have reached the most likely state of the system, the so-called steady state or equilibrium state.