Paper: arXiv 2503.17836

Authors: Robert Ghrist, Julian Gould, Miguel Lopez, Hans Riess

Abstract

Modern financial networks involve complex obligations that transcend simple monetary debts: multiple currencies, prioritized claims, supply chain dependencies, and more. We present a mathematical framework that unifies and extends these scenarios by recasting the classical Eisenberg-Noe model of financial clearing in terms of lattice liability networks. Each node in the network carries a complete lattice of possible states, while edges encode nominal liabilities. Our framework generalizes the scalar-valued clearing vectors of the classical model to lattice-valued clearing sections, preserving the elegant fixed-point structure while dramatically expanding its descriptive power. Our main theorem establishes that such networks possess clearing sections that themselves form a complete lattice under the product order. This structure theorem enables tractable analysis of equilibria in diverse domains, including multi-currency financial systems, decentralized finance with automated market makers, supply chains with resource transformation, and permission networks with complex authorization structures. We further extend our framework to chain-complete lattices for term structure models and multivalued mappings for complex negotiation systems. Our results demonstrate how lattice theory provides a natural language for understanding complex network dynamics across multiple domains, creating a unified mathematical foundation for analyzing systemic risk, resource allocation, and network stability.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly novel and mathematically rigorous generalization of the Eisenberg-Noe financial clearing model using lattice theory and order-theoretic fixed-point theorems. However, it is purely theoretical with no empirical data analysis or backtesting, placing it squarely in the ‘Lab Rats’ quadrant.

Research Flowchart

  flowchart TD
    A[Research Goal: Generalize Eisenberg-Noe clearing model to complex multi-asset networks] --> B[Inputs: Lattice Liability Networks <br><i>Lattice-valued states & nominal liabilities</i>]
    B --> C[Methodology: Recast clearing vectors as lattice-valued clearing sections]
    C --> D[Mathematical Analysis: Tarski's Fixed-Point Theorem & Order Theory]
    D --> E[Outcome: Existence of clearing sections forming a Complete Lattice]
    E --> F[Applications: DeFi AMMs, Multi-Currency Systems, Supply Chains & Term Structure]

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0 of 58 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.