Paper: arXiv 2506.12546

Authors: Marc Homs-Dones, Robert S. MacKay, Bazil Sansom, Yijie Zhou

Abstract

We introduce the Circular Directional Flow Decomposition (CDFD), a new framework for analyzing circularity in weighted directed networks. CDFD separates flow into two components: a circular (divergence-free) component and an acyclic component that carries all nett directional flow. This yields a normalized circularity index between 0 (fully acyclic) and 1 (for networks formed solely by the superposition of cycles), with the complement measuring directionality. This index captures the proportion of flow involved in cycles, and admits a range of interpretations - such as system closure, feedback, weighted strong connectivity, structural redundancy, or inefficiency. Although the decomposition is generally non-unique, we show that the set of all decompositions forms a well-structured geometric space with favourable topological properties. Within this space, we highlight two benchmark decompositions aligned with distinct analytical goals: the maximum circularity solution, which minimizes nett flow, and the Balanced Flow Forwarding (BFF) solution, a unique, locally computable decomposition that distributes circular flow across all feasible cycles in proportion to the original network structure. We demonstrate the interpretive value and computational tractability of both decompositions on synthetic and empirical networks. They outperform existing circularity metrics in detecting meaningful structural variation. The decomposition also enables structural analysis - such as mapping the distribution of cyclic flow - and supports practical applications that require explicit flow allocation or routing, including multilateral netting and efficient transport.

Complexity vs Empirical Score

  • Math Complexity: 6.5/10
  • Empirical Rigor: 5.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a rigorous theoretical framework for Circular Directional Flow Decomposition (CDFD) with clear mathematical proofs and well-structured topological properties. It applies these concepts to synthetic and real-world networks across domains like multilateral netting and trade flow, providing both theoretical novelty and empirical applicability.

Research Flowchart

  flowchart TD
    A[Research Goal: Quantify & decompose circular vs. directional flow in weighted directed networks] --> B[Input: Weighted Directed Network Flow Data]
    B --> C[Method: Circular Directional Flow Decomposition - CDFD Framework]
    C --> D[Compute Benchmark Solutions: Max Circularity & Balanced Flow Forwarding - BFF]
    D --> E[Output: Normalized Circularity Index 0 to 1 & Flow Allocation Maps]
    E --> F[Applications & Findings: Superior structural detection, multilateral netting & efficient routing]