Characterization of efficient networks in a generalized connections model with endogenous link strength

We consider a natural generalization of Jackson and Wolinsky's (J Econ Theory 71:44- 74, 1996) connections model where the quality or strength of a link depends on the amount invested in it and is determined by a non-decreasing function of that amount. The information that the nodes receive through the network is the revenue from investments in links. We prove that in this most general version of the connections model, the only possibly non-empty efficient networks, in the sense of maximizing the aggregate profit, are still the all-encompassing star and the complete network, with the sole and rare exception of a highly particular case where there is a draw between the allencompassing star, the complete network and a whole range of a particular type of nested split graph structures intermediate between them.