Integration in Multi-Community Networks

There are many social and economic situations where two or more communities need to be integrated in an efficient and stable way that facilitates overall resource access throughout the network. We study structures for efficient integration of multi-community networks where building bridges across communities incur an additional link cost compared to links within a community. Building on the connections models with direct link cost and direct and indirect benefits, we show that the efficient structure for homogeneous cost and benefit parameters, and for communities of arbitrary size, always has a diameter no greater than 3. We further show that if the internal cost is not small enough to justify a full graph for each community, integration always follows one of these three structures: Single star, two hub-connected stars, and a new structure we introduce in this paper as parallel hyperstar, which is a special multi-core/periphery structure with parallel bridges that connect the core nodes of different communities and includes a wide range of efficiently integrated structures. Then we investigate stability conditions of these structures, using two different definitions: The standard pairwise stability, as well as a new stability notion we introduce in this paper as post transfer pairwise stability, which allows for bilateral utility transfers. We show that while the parallel hyperstar structure can never be both efficient and pairwise stable, once post transfer pairwise stability is used, efficiency guarantees stability. Furthermore, we show that all possible efficient structures can simultaneously be post transfer pairwise stable. In the end, we discuss future extensions of this model to multiple communities.