region* which we define as follows.

**Entropic region-** the entropic region $\Gamma_{N}^{*}\subset \mathbb{R}_{+}^{2^{N}-1}$ is the set of all entropic vectors of order $N$, i.e. the set of all possible entropic vectors that can correspond to the information possessed by $N$ agents. Thus, if a vector $\overrightarrow{{\bf H}}$ is entropic, then $\overrightarrow{{\bf H}}\in\Gamma_{N}^{*}$ [@ref20].

We denote by $\tilde{\mathcal{H}}$ the set of entropic vectors having $H(X_{1},X_{2},...,X_{N})=\sum_{i=1}^{N}H(X_{i})$, where $\tilde{\mathcal{H}}\subset\Gamma_{N}^{*}$. The set of entropic vectors in $\tilde{\mathcal{H}}$ is simply a hyperplane in $\Gamma_{N}^{*}$ that correspond to all entropic vectors with no information redundancies, which captures the aggregation models in [@ref12] [@ref13] [@ref141].

The entropic vector can be constructed as follows. Given the set of agents $\mathcal{N}$ and a corresponding set of random variables $\mathcal{X}=\{X_{1},X_{2},...,X_{N}\}$, we construct the set $\mathcal{V}=\mathcal{P}(\mathcal{X})/\{\phi\}$, where $\mathcal{P}(\mathcal{X})$ is the power set of $\mathcal{X}$. If $\mathcal{V}=\{v_{1},v_{2},...,v_{|\mathcal{V}|}\}$, then the entropic vector is given by $\overrightarrow{{\bf H}}=\left(H(X_{v_{i}})\right)_{i=1}^{|\mathcal{V}|}$, where $|\mathcal{V}|=2^{N}-1$, and $H(X_{v_{i}})$ is the joint entropy between all random variables in the set $v_{i}$. For instance, if we have 3 agents in the network, then $\mathcal{V}=\{\{1\},\{2\},\{3\},\{1,2\},\{2,3\},\{1,3\},\{1,2,3\}\}$, and the entropic vector $\overrightarrow{{\bf H}}$ is given by $\left(H(X_{1}),H(X_{2}),H(X_{3}),H(X_{1,2}),H(X_{1,3}),H(X_{2,3}),\right.$ $\left.H(X_{1,2,3})\right)^{T}$, where $H(X_{1,2})=H(X_{1},X_{2})$. We denote a single element in the entropic vector as $\overrightarrow{{\bf H}}(v)=H(X_{v})$. The mutual information 