Clearing in Liability Networks via Sheaves on Directed Hypergraphs

TL;DR

This paper models liability networks using sheaves on directed hypergraphs, providing a unified, functorial framework for analyzing clearing configurations with various existence and uniqueness results.

Contribution

It introduces a sheaf-theoretic, finite-limit construction for liability clearing, unifying existing models and enabling comparison across different data categories.

Findings

Provides a functorial invariance theorem for clearing solutions.

Establishes existence and uniqueness conditions using lattice and metric theories.

Special cases include Eisenberg--Noe and lattice liability networks.

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 $\Phi=A\circ D$ 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.