Characterizing Agent-Based Model Dynamics via $ε$-Machines and Kolmogorov-Style Complexity

TL;DR

This work develops a two-level information-theoretic framework to characterize ABM dynamics within Complex Adaptive Systems, using a global pooled ε-machine as a system-wide reference and per-dyad ε-machines with Kolmogorov-style proxies (LZ78 and compression-bps) for fine-grained analyses. It introduces a rigorous reconstruction pipeline that alternates symbolization, causal-state estimation, unifilar transitions, and entropy/compression metrics to separate semantic predictive structure from syntactic description length. Empirical results on a caregiver–elder ABM show that walkability exhibits bounded temporal memory and recurrent micro-patterns under ordinal encodings, while efforts and hours-not-cared are largely memoryless at the dyad level; compression metrics corroborate distinct regimes across variables and symbolizations. The study demonstrates that coupling ε-machine analysis with compression-based diagnostics yields a coherent, multi-scale picture of where predictive information resides and how emergence manifests across system layers, with implications for model design, policy testing, and cross-domain applicability in empirical complex systems.

Abstract

We propose a two-level information-theoretic framework for characterizing the informational organization of Agent-Based Model (ABM) dynamics within the broader paradigm of Complex Adaptive Systems (CAS). At the macro level, a pooled $\varepsilon$-machine is reconstructed as a reference model summarizing the system-wide informational regime. At the micro level, $\varepsilon$-machines are reconstructed for each caregiver--elder dyad and variable, complemented by algorithm-agnostic Kolmogorov-style measures, including normalized LZ78 complexity and bits per symbol from lossless compression. The resulting feature set, $\{h_μ, C_μ, E, \mathrm{LZ78}, \mathrm{bps}\}$, enables distributional analysis, stratified comparisons, and unsupervised clustering across agents and scenarios. Empirical results show that coupling $\varepsilon$-machines with compression diagnostics yields a coherent picture of where predictive information resides in the caregiving ABM. Global reconstructions provide a memoryless baseline ($L{=}0$ under coarse symbolizations), whereas per-dyad models reveal localized structure, particularly for walkability under ordinal encodings ($m{=}3$). Compression metrics corroborate these patterns: dictionary compressors agree on algorithmic redundancy, while normalized LZ78 captures statistical novelty. Socioeconomic variables display cross-sectional heterogeneity and near-memoryless dynamics, whereas spatial interaction induces bounded temporal memory and recurrent regimes. The framework thus distinguishes semantic organization (predictive causation and memory) from syntactic simplicity (description length) and clarifies how emergence manifests at different system layers. It is demonstrated on a caregiver--elder case study with dyad-level $\varepsilon$-machine reconstructions and compression-based diagnostics.

Figures (3)

• Figure 1: Structural signatures. Left: State--transition probability matrix. Heatmap of transition probabilities between causal states (rows = current $s$, columns = next $s'$). Bright diagonal elements indicate persistence within the same state, whereas off-diagonal clusters denote systematic switching among subsets of states. The resulting pattern highlights modular organization and localized predictability within the global $\varepsilon$-machine. Emission and transition mass distributions. Middle: Fraction of total emission probability associated with each ordinal symbol ($a_0$--$a_5$). Middle-range motifs ($a_2$, $a_3$) dominate, suggesting gradual rather than abrupt changes in relative walkability. Right: Most probable state-to-state transitions (top 25 by mass). The concentration of probability mass among a few edges (e.g., $S_5 \!\to\! S_7$, $S_5 \!\to\! S_5$) indicates limited recurrent micro-patterns and short-range stability.
• Figure 2: Global $\varepsilon$-machine for walkability (ordinal $m=3$). Directed graph representation of the reconstructed $\varepsilon$-machine. Nodes correspond to causal states ($S_0$--$S_{17}$) annotated with their stationary probability ($\pi$) and local emission entropy ($H$). Edges denote predictive transitions labeled by emitted ordinal symbols ($a_0$--$a_5$). The layout reveals a moderately connected topology with clusters of recurrent states and several self-loops, indicating locally stable yet globally fragmented dynamics.
• Figure 3: Left: normalized LZ78 complexity vs. LZMA bits per symbol (statistical vs. algorithmic compressibility). Right: violin plots of normalized LZ78 by variable showing heterogeneity and burstiness.