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Decorated Cospans at Work: Coupling Heterogeneous Dynamical Systems via Pushouts and Particle Filters

Wesley Phoa

Abstract

Decorated cospans provide a categorical framework for composing open systems along shared interfaces. This paper is a computational proof of concept: we show that the framework produces a working coupled dynamical system when the decorations are quantitative models from different mathematical traditions. Specifically, we couple a linearised New Keynesian DSGE, a stochastic compartmental epidemic (multi-strain SEIR), and a nonlinear vaccine adoption model with hysteresis into a single sequential Monte Carlo sampler. Each model is a decorated cospan -- interior dynamics as decoration, exposed variables as interfaces. The composite system is the pushout along variable identifications, with coupling functions encoded as factor graph constraints. The coupled system produces a rejection bifurcation: some trajectories escape via vaccination, others enter a self-reinforcing cycle of mandate backlash, vaccine refusal, sustained infection, and recession. This is a structural property of the coupling, not an input assumption. Coupling shifts the output gap by 0.78 pp and rejection by 22 pp relative to the uncoupled system. A fourth narrative -- fiscal/political dynamics, calibrated to the US COVID fiscal response -- attaches via a second pushout and introduces the first positive coupling channel. With pandemic-scale spending parameters, 14% of trajectories overshoot into positive output gap territory; the bearish bias shrinks, but persists. A computable bias decomposition separates three sources of this asymmetry -- sampling, structural, and observational -- and localises the structural component to specific coupling functions whose directional asymmetry can be tested against historical analogues.

Decorated Cospans at Work: Coupling Heterogeneous Dynamical Systems via Pushouts and Particle Filters

Abstract

Decorated cospans provide a categorical framework for composing open systems along shared interfaces. This paper is a computational proof of concept: we show that the framework produces a working coupled dynamical system when the decorations are quantitative models from different mathematical traditions. Specifically, we couple a linearised New Keynesian DSGE, a stochastic compartmental epidemic (multi-strain SEIR), and a nonlinear vaccine adoption model with hysteresis into a single sequential Monte Carlo sampler. Each model is a decorated cospan -- interior dynamics as decoration, exposed variables as interfaces. The composite system is the pushout along variable identifications, with coupling functions encoded as factor graph constraints. The coupled system produces a rejection bifurcation: some trajectories escape via vaccination, others enter a self-reinforcing cycle of mandate backlash, vaccine refusal, sustained infection, and recession. This is a structural property of the coupling, not an input assumption. Coupling shifts the output gap by 0.78 pp and rejection by 22 pp relative to the uncoupled system. A fourth narrative -- fiscal/political dynamics, calibrated to the US COVID fiscal response -- attaches via a second pushout and introduces the first positive coupling channel. With pandemic-scale spending parameters, 14% of trajectories overshoot into positive output gap territory; the bearish bias shrinks, but persists. A computable bias decomposition separates three sources of this asymmetry -- sampling, structural, and observational -- and localises the structural component to specific coupling functions whose directional asymmetry can be tested against historical analogues.
Paper Structure (42 sections, 6 equations, 9 figures, 11 tables)

This paper contains 42 sections, 6 equations, 9 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Framework
  3. Particle filters for single narratives
  4. Factor graphs for multiple narratives
  5. Decorated cospans for heterogeneous composition
  6. The Three Narratives
  7. Economic cycle (N$_1$): New Keynesian DSGE
  8. Epidemic (N$_2$): SEIR with strain emergence
  9. Vaccine dynamics (N$_3$): innovation, adoption, rejection
  10. Cross-Narrative Coupling
  11. Variable identification: $N \equiv L$
  12. Expectations and Information Structure
  13. Probability and Salience
  14. Algorithm
  15. Results
  16. ...and 27 more sections

Figures (9)

  • Figure 1: Factor graph for the coupled narrative system. Variable nodes are coloured by narrative; factor nodes (black squares) encode cross-narrative coupling. The dashed arc indicates the variable identification $N \equiv L$ (population $\equiv$ labour force). Factors $f_1$ and $f_2$ transmit epidemic shocks to the economy; $f_4$ transmits vaccine protection back to the epidemic; $f_5$ and $f_6$ transmit economic conditions to vaccine dynamics. Factor $f_3$ (outbreak $\to$ uptake demand) is embedded in the vaccine model's target function and does not appear as a separate node.
  • Figure 2: Fan charts for (a) output gap, (b) infection prevalence, and (c) vaccine rejection. Shaded bands: 5--95th and 25--75th percentiles. Solid line: median. Dashed lines: five archetypal trajectories (Particles A--E).
  • Figure 3: Cross-narrative correlations over time. Colour encodes the Pearson correlation coefficient between variable pairs.
  • Figure 4: Final rejection vs. cumulative deaths, coloured by cluster. Stars mark weighted-average archetypes (Particles A--E). The distribution separates into low-rejection and high-rejection regimes.
  • Figure 5: Terminal distributions of key variables under coupled (blue) and uncoupled (grey) systems. Coupling shifts the output gap leftward and rejection rightward.
  • ...and 4 more figures

Theorems & Definitions (1)

  • Remark 1