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

This article extends the preprint "Characterizing Agent-Based Model Dynamics via $\epsilon$-Machines and Kolmogorov-Style Complexity" by introducing diffusion models as orthogonal and complementary tools for characterizing the output of agent-based models (ABMs). Where $\epsilon$-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 $\epsilon$-machines and diffusion models. This establishes a principled methodology for jointly analyzing temporal predictability and high-dimensional distributional structure in complex simulation models.