A hierarchical network formation model

We present a network formation model based on a particularly interesting class of networks in social settings, where individuals' positions are determined according to a topic-based or hierarchical taxonomy. In this game-theoretic model, players are located in the leaves of a complete b-ary tree as the seed network with the objective of minimizing their collective distances to others in the network. In the grid-based model of Even-Dar and Kearns [Even-Dar, E., and M. Kearns, A small world threshold for economic network formation, Advances in Neural Information Processing Systems 19 (2007), 385–392], they demonstrate the existence of small diameter networks with the threshold of α=2 where the cost of a new link depends on the distance between the two endpoints to the power of α. We show the appearance of small diameter equilibrium networks with the threshold of α=1/4 in the hierarchical or tree-based networks. Moreover, the general set of equilibrium networks in our model are guaranteed to exist and they are pairwise Nash stable with transfers [Bloch, F., and M. O. Jackson Definitions of equilibrium in network formation games, Int J Game Theory 34 (3) (2006), 305–318].