finding its estimate $x_p^{k+1}$; it then broadcasts $x_p^{k+1}$ to its neighbors. After having received all the estimates from its neighbors, node $p$ can update $\gamma_p$ as in step \[SubAlg:ADMMBipDualVarUp\].

Algorithm \[Alg:ADMMBipartite\] only applies to bipartite networks and its convergence is guaranteed by Theorems \[Teo:ConvergenceADMM\] and \[Teo:BipApplyADMM\]. Next, we generalize it for any type of network.

[[**Algorithm for arbitrary networks.**]{}]{} So far we assumed the network was bipartite, i.e., it could be colored with just two colors: $\chi(\mathcal{G})=2$. Now, suppose the network is colored with $C\geq 2$ colors. We derived problem  from problem  by partitioning the node-arc incidence matrix $B$ into $\begin{bmatrix}B_1^\top & B_2^\top\end{bmatrix}^\top$, where $B_1$ and $B_2$ contain the rows corresponding to nodes in $\mathcal{C}_1$ and $\mathcal{C}_2$, respectively. For convenience, we had assumed the nodes were ordered such that $\mathcal{C}_1 = \{1,2,\ldots,C_1\}$ and $\mathcal{C}_2 = \{C_1+1,C_1+2,\ldots,C\}$. Without loss of generality, assume a similar ordering for the nodes in the current case, i.e., $\mathcal{C}_1 = \{1,2,\ldots,C_1\}$, $\mathcal{C}_2 = \{C_1+1,\ldots,C_1+C_2\}$, …, $\mathcal{C}_C = \{\sum_{c=1}^{C-1}C_c+1,\ldots,C\}$. This induces a natural partition of $B$ as $\begin{bmatrix}B_1^\top & B_2^\top & \cdots & B_C^\top\end{bmatrix}^\top$. The counterpart of  for this case is then $$\label{Eq:GenNetCloned}
            \begin{array}{ll}
                \underset{\bar{x}_1,\ldots,\bar{x}_C}{\text{minimize}} & \sum_{c=1}^C \sum_{p \in \mathcal{C}_p} f_p(x_p) \\
                \text{subject to} & \bar{x}_c \in 