A Unified Framework for Weighted Hypergraphic Networks and Fractional Matching

Network formation theory studies how agents create and maintain relationships, and the stability of those relationships with respect to individual incentives. A central stability concept in this literature is pairwise stability, introduced by Jackson and Wolinsky (1996) for unweighted networks (agents are either connected or not) and later extended by Bich and Morhaim (2020) to weighted networks (connections can have different intensities). In this paper, we pursue two main objectives. First, we extend the notion of stability to networks defined on hypergraphs, where relationships may involve more than two agents simultaneously and where agents face budget constraints on the sum of the intensity of all their connections. We introduce a stability concept that preserves the core intuition of pairwise stability while generalizing it to relationships involving more than two agents, and that accounts for budget constraints. Second, we propose a stronger notion that we call full stability, inspired by stability concepts from matching theory, in which agents are allowed to adjust multiple connections simultaneously rather than through single-link deviations. We give existence results for both stability notions under various assumptions, as well as explicit solutions or algorithms, and provide counter-examples for most cases that do not satisfy those assumptions, establishing an almost complete theory. Our framework provides a unified approach to constrained network formation in hypergraphic settings and builds a conceptual bridge between the theories of weighted network formation and fractional matching.