A Unified Framework for Weighted Hypergraphic Networks and Fractional Matching

TL;DR

The paper addresses how to endogenize constrained network formation on hypergraphs by extending pairwise stability to coalitions with budget and capacity constraints and by introducing full stability inspired by matching theory. It develops existence results for stable networks under continuity and quasiconcavity, and delineates limits via counterexamples; it then provides positive results for full stability in unconstrained hypergraphs with positive externalities and in bipartite, budget-constrained settings using the Generalized Deferred Acceptance algorithm. The work unifies weighted network formation with fractional matching, showing that stable fractional matchings correspond to fully stable networks in many cases and recovering classical discrete matching results as special cases. It also offers algorithmic approaches and explicit characterizations in key cases, clarifying when full stability can be achieved and how it relates to externalities, separability, and concavity. Overall, the framework advances a unified theory of constrained, hypergraphic network formation with broad implications for economics, finance, and coordination problems where group interactions and budgets matter.

Abstract

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.

Figures (3)

• Figure 1: Summary of the existing work and of our contributions. Every entry indicates a proof of existence of a stable (or fully stable) network, in the context (type of graph, constraints or not) defined by the row and column labels, except for Theorems \ref{['thm:counter']} and \ref{['thm:counter_2']} which are counter-examples. The arrows indicate the nestedness of concepts: for instance, an existence result for hypergraphs in a given column implies existence in graphs, and thus in bipartite graphs, in the same column. For counter-examples, the implications go in the opposite direction. Some results are only true under some assumptions on the payoff functions, indicated in parenthesis: L = linear, S = additively separable, C = concave, NE = no externalities, PE = positive externalities. All results require at least continuous and quasiconcave payoff functions; those assumptions are thus omitted from the table.
• Figure 2: Illustration of the limitation of the notion of stability in the presence of constraints. Left: network $x = (1, 0, 0)$, right: network $x' = (0, 1, 0)$. Both agents $1$ and $3$ are better off under $x'$, nonetheless $x$ is stable according to the classical definition of pairwise stability.
• Figure 3: Graphs of the two functions that compose the payoff of agents in the counter-example used in the proof of \ref{['thm:counter_2']}.

Theorems & Definitions (32)

• Definition 2.1: Network game
• Definition 2.2: Pairwise stable network
• Theorem : Bich--Morhaim bich_morhaim_2020
• Definition 3.1: Constrained network game
• Definition 3.2: Stable network
• Theorem 3.3
• proof : Sketch of proof.
• Definition 4.1: Fully stable network
• Definition 4.2: Matching problem
• Definition 4.3: Stable matching