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Complementary Characterization of Agent-Based Models via Computational Mechanics and Diffusion Models

Roberto Garrone

TL;DR

ABMs produce both temporal trajectories and high-dimensional snapshots, challenging unified characterization. The authors propose a dual framework that combines ε-machine–based computational mechanics with diffusion-based density estimation to yield a two-axis representation: temporal predictability and distributional geometry. They formalize this complementarity with mappings between process and distribution domains and validate the approach on an elder–caregiver ABM, illustrating joint insights into regime structure and population-level heterogeneity. The work offers a principled pipeline for integrating intrinsic computation with density-based analysis, with broad implications for validation, scenario comparison, and synthetic data generation in complex adaptive systems.

Abstract

This article extends the preprint "Characterizing Agent-Based Model Dynamics via $ε$-Machines and Kolmogorov-Style Complexity" by introducing diffusion models as orthogonal and complementary tools for characterizing the output of agent-based models (ABMs). Where $ε$-machines capture the predictive temporal structure and intrinsic computation of ABM-generated time series, diffusion models characterize high-dimensional cross-sectional distributions, learn underlying data manifolds, and enable synthetic generation of plausible population-level outcomes. We provide a formal analysis demonstrating that the two approaches operate on distinct mathematical domains -- processes vs. distributions -- and show that their combination yields a two-axis representation of ABM behavior based on temporal organization and distributional geometry. To our knowledge, this is the first framework to integrate computational mechanics with score-based generative modeling for the structural analysis of ABM outputs, thereby situating ABM characterization within the broader landscape of modern machine-learning methods for density estimation and intrinsic computation. The framework is validated using the same elder-caregiver ABM dataset introduced in the companion paper, and we provide precise definitions and propositions formalizing the mathematical complementarity between $ε$-machines and diffusion models. This establishes a principled methodology for jointly analyzing temporal predictability and high-dimensional distributional structure in complex simulation models.

Complementary Characterization of Agent-Based Models via Computational Mechanics and Diffusion Models

TL;DR

ABMs produce both temporal trajectories and high-dimensional snapshots, challenging unified characterization. The authors propose a dual framework that combines ε-machine–based computational mechanics with diffusion-based density estimation to yield a two-axis representation: temporal predictability and distributional geometry. They formalize this complementarity with mappings between process and distribution domains and validate the approach on an elder–caregiver ABM, illustrating joint insights into regime structure and population-level heterogeneity. The work offers a principled pipeline for integrating intrinsic computation with density-based analysis, with broad implications for validation, scenario comparison, and synthetic data generation in complex adaptive systems.

Abstract

This article extends the preprint "Characterizing Agent-Based Model Dynamics via -Machines and Kolmogorov-Style Complexity" by introducing diffusion models as orthogonal and complementary tools for characterizing the output of agent-based models (ABMs). Where -machines capture the predictive temporal structure and intrinsic computation of ABM-generated time series, diffusion models characterize high-dimensional cross-sectional distributions, learn underlying data manifolds, and enable synthetic generation of plausible population-level outcomes. We provide a formal analysis demonstrating that the two approaches operate on distinct mathematical domains -- processes vs. distributions -- and show that their combination yields a two-axis representation of ABM behavior based on temporal organization and distributional geometry. To our knowledge, this is the first framework to integrate computational mechanics with score-based generative modeling for the structural analysis of ABM outputs, thereby situating ABM characterization within the broader landscape of modern machine-learning methods for density estimation and intrinsic computation. The framework is validated using the same elder-caregiver ABM dataset introduced in the companion paper, and we provide precise definitions and propositions formalizing the mathematical complementarity between -machines and diffusion models. This establishes a principled methodology for jointly analyzing temporal predictability and high-dimensional distributional structure in complex simulation models.

Paper Structure

This paper contains 26 sections, 1 theorem, 10 equations.

Table of Contents

  1. Introduction
  2. Background
  3. ABM Outputs: Temporal and Distributional Components
  4. $\epsilon$-Machines (Computational Mechanics)
  5. Diffusion Models
  6. Mathematical Complementarity
  7. Processes vs. Distributions
  8. Assumptions.
  9. Orthogonality of Extracted Information
  10. Proposition 2 (Complementary Representational Axes).
  11. Analytical Extensions for Parameter-Space Structure
  12. Regime Analysis.
  13. Qualitative Pattern Mapping.
  14. Behavioral Clustering.
  15. Complementarity with Sensitivity Analysis.
  16. ...and 11 more sections

Key Result

Proposition 1

$\mathcal{P}$ and $\mathcal{D}$ are mathematically distinct: $\mathcal{P}$ defines laws over infinite sequences, while $\mathcal{D}$ defines laws over static vectors. Hence $\Phi$ and $\Psi$ operate on orthogonal domains.

Theorems & Definitions (5)

  • Definition 1: Process Domain
  • Definition 2: Distribution Domain
  • Proposition 1: Disjoint Analytical Domains
  • proof : Sketch
  • proof : Sketch