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.

翻译：本文通过引入扩散模型作为正交且互补的工具来表征智能体模型（ABMs）的输出，从而扩展了预印本《通过ε-机与柯尔莫哥洛夫式复杂性表征智能体模型动力学》的研究。其中，ε-机捕捉ABM生成时间序列的预测性时间结构与内在计算，而扩散模型则刻画高维截面分布、学习底层数据流形，并能够合成生成合理的群体水平结果。我们提供了形式化分析，证明这两种方法作用于不同的数学领域——过程与分布——并表明它们的结合基于时间组织与分布几何产生了ABM行为的双轴表示。据我们所知，这是首个将计算力学与基于分数的生成建模相结合用于ABM输出结构分析的框架，从而将ABM表征置于更广泛的现代机器学习方法（用于密度估计与内在计算）的背景下。该框架使用配套论文中引入的同一老年护理-护理者ABM数据集进行了验证，并提供了精确的定义与命题，形式化地阐述了ε-机与扩散模型之间的数学互补性。这为联合分析复杂仿真模型中的时间可预测性与高维分布结构建立了一种原则性方法论。