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.

翻译：网络形成理论研究个体如何建立并维持关系，以及这些关系相对于个体激励的稳定性。该领域的核心稳定性概念是Jackson与Wolinsky（1996）针对无权网络（个体间要么连接要么不连接）提出的成对稳定性，后由Bich与Morhaim（2020）扩展至加权网络（连接可具有不同强度）。本文致力于实现两个主要目标：首先，将稳定性概念拓展至基于超图定义的网络，其中关系可同时涉及两个以上个体，且个体面临其所有连接强度总和的预算约束。我们提出一种稳定性概念，既保留了成对稳定性的核心直观思想，又将其推广至涉及多个个体的关系，同时纳入预算约束的考量。其次，受匹配理论中稳定性概念的启发，我们提出一种更强的“完全稳定性”概念，允许个体同时调整多个连接而非仅通过单边偏离实现调整。我们在不同假设条件下给出两种稳定性概念的存在性结果，并提供显式解或算法，同时针对不满足这些假设的大多数情形构造反例，从而建立起近乎完整的理论体系。本框架为超图场景下的约束网络形成提供了统一方法，并在加权网络形成理论与分数匹配理论之间架起了概念桥梁。