Clearing in Liability Networks via Sheaves on Directed Hypergraphs

2026-05-15

AI Summary

This study addresses the problem of efficient and consistent debt clearing in complex liability networks. By modeling liability relationships as layers over a directed hypergraph and interpreting clearing configurations as global sections, the work introduces sheaf-theoretic and categorical finite-limit methods for the first time to unify the characterization of clearing processes through the equalizer structure of distribution and aggregation operators. Within this framework, a clearing invariance theorem across data categories is established, proving existence, uniqueness, and iterative convergence of clearing solutions under conditions such as complete lattices, acyclic graphs, or metric contraction. The approach subsumes classical models like Eisenberg–Noe and reveals a unified structural nature underlying clearing solutions.

Abstract

We associate to a decorated liability network a liability sheaf on a directed hypergraph whose hyperedges separate the distribution of payments from the collection of receipts. Clearing configurations are precisely the global sections of this sheaf, and the global-section object is canonically the equalizer of the identity and a clearing operator $Φ=A\circ D$ factored into collective distribution $D$ and aggregation $A$; an institution-edge duality identifies it equivalently with the equalizer of the dual operator $D\circ A$ on the edge side. This identifies liability clearing as a finite-limit construction in the ambient data category. The construction is functorial under change of coefficient category: a Clearing Invariance Theorem shows that a finite-limit-preserving functor compatible with constraint subobjects induces a canonical isomorphism on global-section objects, enabling uniform comparison of clearing problems across categories of payment data. Existence, uniqueness, and iterative computation of clearing sections are organized by the structure carried on payment objects: Tarski's theorem yields existence and a complete-lattice structure under complete-lattice global elements; Scott continuity refines this to convergent Kleene iteration; an acyclic underlying graph admits a unique clearing section in finitely many steps with no order or metric hypothesis; and Banach's theorem on global elements yields uniqueness under metric contraction. The Eisenberg--Noe model and lattice liability networks arise as special cases.

Problem

• liability networks
• clearing
• directed hypergraphs
• sheaves
• global sections

Innovation

• liability sheaf
• directed hypergraph
• clearing operator
• finite-limit construction
• Clearing Invariance Theorem