\delta (c_i , c_{k_l} ) \delta (c_j, c_{m_l}) {\lambda}^{(t+1)}
\label{eqtrans}$$ when the edges $(c_{k_l}, c_{m_l})$ were traversed during the last random walk, where $0\leq \gamma \leq 1$ is a forgetting factor (damping parameter), and $\lambda$ is a non-negative reward that will increment the related *h*-value function. It follows that the forgetting factor $ \gamma $ in Eq. \[eqtrans\] can show a positive effect in speed-up learning and an adverse effect on the amount of efficiency.[^2] The simplest function for $\omega$ is for $f(h)=h$ as will be used in the current study and have used in the previous studies[^3]. In addition, the weight matrix $\omega = h^{(t)} (c_i, c_j)$ is initially unit ($h^{(1)} (c_i, c_j )=1$) for all edges.

Subsequently, learning happens by changing transition probabilities of Eq. \[eqprob\] on a given pair of $(c_i, c_j )$ by updating the rule of Eq. \[eqtrans\]. One can evaluate the efficiency of learning, $r^{(t)}$, by adding together the amounts of desirable joint probabilities $P^{(n)} (a_s^{*} , s)$ at time $t$, where $ a_s^{*} $ is a pleasing action on a given percept $ s $. Assuming that $$r^{(t)} = \sum_s P^{(t)} (s)\, P^{(t)} (a_s^{*} |s) , \quad P^{(n)} (s) = \frac{1}{\textit{number of different 