otal number of incoming density matrices in $R_{j} $ per a given $\pi _{S} $, as $$\label{ZEqnNum000000}  \left|B_{R_{j} } \left(\pi _{S} \right)\right|=\sum _{i,k}\left|B\left(R_{i} \left(\pi _{S} \right),\sigma _{k} \right)\right| ,$$ where $\left|B\left(R_{i} \left(\pi _{S} \right),\sigma _{k} \right)\right|$ refers to the number of density matrices arrive from $R_{i} $ for swapping with $\sigma _{k} $ per $\pi _{S} $.  