Network Formation Under Heterogeneous Costs: The Multiple Group Model

It is widely recognized that the shape of networks influences both individual and aggregate behavior. This raises the question which types of networks are likely to arise. In this paper we investigate a model of network formation, where players are divided into groups and the costs of a link between any pair of players are increasing in the distance between the groups that these players belong to. We give a full characterization of the networks induced by a minimal curb set for any number of groups. To do so, we show that in our multiple group model each minimal curb set is a so-called super-tight curb set, that is a minimal curb set satisfying the condition that in each state of the set every player has the same best reply. From the proof it follows that every recurrent class of an unperturbed best reply dynamics is a minimal (super-tight) curb set and reversely, which yields the characterization result. We show that in case of multiple groups networks in minimal curb sets may have features that can not occur in networks with at most two groups. Nevertheless, local centrality and center-sponsorship are still important characteristics of the networks in minimal curb sets.