two agents can break the link formed and get a strictly higher utility.

Characterization of the NE for $\mathcal{G}^{N}$
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In this subsection, we present a generic characterization for the NE of $\mathcal{G}^{N}$.

(Network minimality) *In every NE, all network components are minimally connected.*

See Appendix B.

Proposition 1 implies that agents in each component will form the minimal number of links possible to gather the maximum amount of information. This results from indirect information sharing within each network component, i.e. if there exists a path to an agent then there is no extra benefit in making a direct link to that agent since all the information from that agent is already accessible.

Next, we characterize the connectivity of the network as a function of the link cost in the following Lemma.

[*(Network connectivity regions)*]{}

(i) If $c\leq c_{l}$, with $c_{l}=f\left(H(\mathcal{X})\right)-f(\min_{i}H(X_{-i}))$, then, at every NE (a) the network is minimally connected (the network has one component) and (b) the amount of information possessed by each agent is $H(\mathcal{X})$ (all information is shared).

(ii) If $c\geq c_{u}$, where $c_{u}=f\left(H(\mathcal{X})\right)-f(\min_{i}H(X_{i}))$, then there is a unique NE which is strict. At this equilibrium, the network is fully disconnected and the amount of information 