Not all bridges connect: integration in multi-community networks

This paper studies structures for efficient and stable integration of multi-community networks where establishing bridges across communities incur additional link cost compared to those within communities. Building on the connections models with 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 for non-trivial cases, 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 core/periphery structure with parallel bridges that connect the core nodes of different communities. Then we investigate stability conditions of these structures, using the standard pairwise stability, as well as post-transfer pairwise stability, a new stability notion we introduce in this paper, 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 be simultaneously post-transfer pairwise stable. In the end, we provide some numerical results and discussion of empirical evidences.