well. However, for the second family of entropic vector, since there is more heterogeneity in the amount of information possessed by the agents, no single agents monopolizes the information at any value of the KL divergence, thus the $\mathcal{K}_{M}$ region does not vanish for the second vector for any value of $\mbox{KL}(\mathcal{X})$.

While Lemma 1 focuses on the impact of link cost on the connectivity of the network, it does not provide a complete characterization for an NE network. In the next Theorem, we give the necessary and sufficient conditions for the emergence of an arbitrary CIN topology in NE.

![Impact of link cost and information redundancy on the network’s connectivity.[]{data-label="fig_pase3"}](Pasetr3.eps){width="3in"}

A network in which the components are precisely $\{\mathcal{C}_{1},\mathcal{C}_{2},...,\mathcal{C}_{K}\}$ can be supported in a NE if and only if the following relationships between the cost and the value of information are satisfied

1.  $f(H(X_{\mathcal{C}_{i}}))-\min\{f(H(X_{\mathcal{C}_{i}/\{j\}})),f(H(X_{j}))\}\geq c$, $\forall i\in\{1,2,...,K\},j\in\mathcal{C}_{i}$.

2.  $f(H(X_{\mathcal{C}_{i}\cup\mathcal{C}_{j}}))-f(H(X_{\mathcal{C}_{i}}))\leq c,\,\forall i,j\in\{1,2,...,K\}$.

See Appendix D.

From Theorem 2 we know that, at NE, the network is generally composed of multiple components and each component is minimally connected. Each component possesses a set of random variables that are jointly highly correlated to the joint random variables possessed by other components. Condition 