conclusions of Theorem \[Teo:BipApplyADMM\], except that $B_i^\top \otimes I_n$ has full column-rank, are derived straightforwardly Assumptions \[Ass:FunctionsAndSets\] and \[Ass:ProblemSolvable\] and the equivalence between  and . To see how Assumption \[Ass:Network\] implies that each $B_i^\top \otimes I_n$ has full column-rank for $i=1,2$, note that it is sufficient to prove that $B_i^\top$ has full column-rank. If we prove that $B_iB_i^\top$ has full rank, then the results follows because $\text{rank}(B_iB_i^\top) = \text{rank}(B_i^\top)$. Now note that $B_iB_i^\top$ is a diagonal matrix, where in the diagonal are the degrees of the nodes belonging to the subnetwork composed by the nodes in $\mathcal{C}_i$. Assumption \[Ass:Network\] states that the network is connected and thus no node can have degree $0$; consequently, $B_iB_i^\top$ has full rank.

[[**Algorithm for bipartite networks.**]{}]{} We now apply ADMM, i.e., - directly to . By analyzing the augmented Lagrangian , we will see that  yields $C_1$ optimization problems that can be solved in parallel. Similarly, yields $C_2$ optimization problems that can be solved in parallel. In fact, developing the squared term in  we obtain for the $x_1$-minimization : $$\label{Eq:BipMinx1}
        x_1^{k+1} \in \arg\min_{x_1 \in X_1} g_1(x_1) + \bar{\eta}_1^\top x_1 + \frac{\rho}{2} x_1^\top (A_1^\top A_1) x_1\,,$$ where $\bar{\eta}_1 :=A_1^\top \lambda^k + \rho (A_1^\top A_2)x_2^k$. Using 