[Paper Review] Causal Emergence in Discrete and Continuous Dynamical Systems

Thomas F. Varley|arXiv (Cornell University)|Mar 29, 2020

Cellular Automata and Applications18 references4 citations

TL;DR

This study applies causal emergence (CE) analysis—using information-theoretic measures on state-transition networks—to discrete elementary cellular automata and continuous Rossler systems, revealing that determinism, degeneracy, and emergence vary dramatically across dynamical regimes. Notably, causal emergence drops sharply at the onset of period-doubling chaos, challenging the 'edge of chaos' hypothesis.

ABSTRACT

Emergence, the phenomena where a system's micro-scale dynamics facilitate the development of non-trivial, informative higher scales, has become a foundational concept in modern sciences, tying together fields as diverse as physics, biology, economics, and ecology. Despite it's apparent universality and the considerable interest, historically researchers have struggled to provide a rigorous, formal definition of emergence that is applicable across fields. Recent theoretical work using information theory and network science to formalize emergence in state-transition networks (causal emergence) has provided a promising way forward, however the relationship between this new framework and other well-studied system dynamics is unknown. In this study, we apply causal emergence analysis to two well-described dynamical systems: the 88 unique elementary cellular automata and the continuous Rossler system in periodic, critical, and chaotic regimes. We find that emergence, as well as its component elements (determinism, degeneracy, and effectiveness) vary dramatically in different dynamical regimes in sometimes unexpected ways. We conclude that the causal emergence framework provides a rich new area of research to explore both to theoreticians and natural scientists in many fields.

Motivation & Objective

• To investigate how causal emergence (CE) behaves in well-defined discrete and continuous dynamical systems.
• To examine whether the CE framework reveals meaningful relationships between macro-scale information and micro-scale dynamics in systems with known behaviors.
• To test if causal emergence peaks at critical or chaotic regimes, as hypothesized in some complexity theories.
• To explore how determinism, degeneracy, and efficacy—key components of CE—respond to dynamical phase transitions.
• To validate the CE framework's sensitivity to dynamical regime changes in systems with known bifurcation structures.

Proposed method

• Applied causal emergence analysis to 88 unique elementary cellular automata (ECA), using brute-force enumeration of all state transitions to construct discrete state-transition networks.
• Used spectral clustering (Griebenow et al., 2019) to identify macro-scale partitions from state-transition networks for both ECA and Rossler systems.
• For the continuous Rossler system, constructed ordinal partition networks (OPNs) to discretize continuous time-series into finite states and transition probabilities.
• Calculated determinism, degeneracy, and efficacy as core components of causal emergence using information-theoretic measures on the state-transition networks.
• Swept the Rossler system’s parameter (a) across 0.37–0.43 to observe transitions through periodic, critical, and chaotic regimes.
• Quantified causal emergence as the difference between macro-scale and micro-scale information, with higher values indicating stronger emergence.

Experimental results

Research questions

• RQ1 How does causal emergence vary across different dynamical regimes in elementary cellular automata?
• RQ2 How do determinism, degeneracy, and efficacy change during phase transitions in the Rossler system?
• RQ3 Does causal emergence peak at the 'edge of chaos' in the Rossler system, as commonly hypothesized?
• RQ4 Can ordinal partition networks (OPNs) effectively represent continuous dynamical systems for causal emergence analysis?
• RQ5 To what extent do degeneracy and determinism anticipate or respond to dynamical transitions in continuous systems?

Key findings

• Causal emergence dropped significantly at the onset of the period-doubling cascade in the Rossler system, contradicting the expectation that emergence peaks at the edge of chaos.
• Determinism and degeneracy both increased during periodic regimes and collapsed upon transition to chaos, with degeneracy showing anticipatory rises before deterministic plateaus.
• The effectiveness measure (difference between determinism and degeneracy) spiked at the onset of period doubling, indicating a temporary increase in causal structure quality.
• Micro-scale determinism decreased sharply after the first bifurcation and then rose again before collapsing during the period-doubling cascade.
• The drop in causal emergence at the transition to chaos was robust and not an artifact, confirmed by higher-resolution sampling around the critical parameter value (a ≈ 0.384).
• In ECA, the highest emergence values were not consistently associated with the most visually complex or dynamic rules, indicating emergence is not synonymous with visual complexity.