<p>Well-known theoretical results for the “connections" model of network formation emphasize the tension between stability and efficiency. We set aside stability in order to characterize efficient networks as generally as possible. Our most general model allows for both (i) general link formation technology, whereby links are costly to form and bounded in the unit interval; and (ii) arbitrary heterogeneity in the value derived from linking to other agents. We provide necessary conditions for efficiency in the general model. We then fully characterize efficient networks in two restricted cases, both of which are generalizations of the seminal models, and we show that well-known results can be derived as special cases of both. In contrast to existing works, we provide alternative, more direct proofs, while also fully characterizing the boundary cases.</p>