, the quadratic term in  becomes $(\rho/2)\bar{x}_1^\top(B_1B_1^\top \otimes I_n) \bar{x}_1$. As we had seen before, $B_1B_1^\top$ is a diagonal matrix with the degree of node $p$, $D_p$, in the $(pp)$th entry. This quadratic term can then be rewritten as $(\rho/2)\sum_{p \in \mathcal{C}_1}D_p\|x_p\|^2$. Regarding the linear term, $$\begin{aligned}
          \bar{\eta}_1^\top \bar{x}_1 
        &=
          \Bigl((B_1\otimes I_n)\lambda^k  + \rho(B_1B_2^\top \otimes I_n)\bar{x}_2^k\Bigr)^\top \bar{x}_1
        \notag
        \\
        &=
          \sum_{p \in \mathcal{C}_1} 
                \Bigl(
                    \sum_{j \in \mathcal{N}_p}\text{sign}(j-p) \lambda_{\{p,j\}}^k - \rho\sum_{j \in \mathcal{N}_p}x_j^k
                \Bigr)^\top x_p\,,
        \label{Eq:BipGamma}
    \end{aligned}$$ where $\text{sign}(w) = 1$ if $w \geq 0$ and $\text{sign}(w) = -1$ otherwise. We decomposed the dual variable $\lambda$ as $(\ldots,\lambda_{\{i,j\}},\ldots)$, where $\lambda_{\{i,j\}}= \lambda_{\{j,i\}}$ is associated with the edge $\{i,j\} \in \mathcal{E}$. To see why  holds, note that the $(ij)$th entry of $B_1B_2^\top$ is $-1$ if $\{i,j\} \in \mathcal{E}$ and $0$ otherwise. In conclusion, using  on , we get $C_1$ optimization problems that can be solved in parallel: $$x_p^{k+1} \in \arg\min_{x_p \in X_p} f_p(x_p) + ({\gamma_p^k}^\top -\rho\sum_{j \in \mathcal{N}_p}x_j^k)^\top x_p + \frac{\rho D_p}{2}\|x_p\|^2\,,$$ where $\gamma_p^k := \sum_{j \in \mathcal{N}_p}\text{sign}(j-p) \lambda_{\{p,j\}}^k$, for $p \in \mathcal{C}_1$, i.e., all nodes with color $1$. A similar analysis for  yields the same problem for each node in $\mathcal{C}_2$, but using the neighbors’ estimates at $k+1$.

Note that the optimization 