Information transfers and flows in Markov chains as dynamical causal effects.

Area of Science:

• Complex Systems Science
• Information Theory
• Causal Inference

Background:

• Markov chains are fundamental models for systems with discrete states and probabilistic transitions.
• Understanding directional (causal) couplings is crucial for analyzing complex systems.
• Existing causality measures, such as transfer entropy, offer insights but lack a unified theoretical framework.

Purpose of the Study:

• To develop a systematic, logical sequence of information-theoretic quantifiers for causal couplings in Markov chains.
• To establish a framework based on dynamical causal effects (DCEs) for generating these quantifiers.
• To reveal the interrelationships between various causality measures within this framework.

Main Methods:

• Generation of dynamical causal effects (DCEs) from simplest to complex forms.
• Systematic construction of 11 information-theoretic quantifiers based on DCEs.
• Rigorous and numerical analysis of relationships, particularly between transfer entropy and Liang-Kleeman information flow in coupled two-state Markov chains.

Main Results:

• A comprehensive system of 11 information-theoretic quantifiers for causal couplings in Markov chains was generated.
• The framework elucidates logical relationships between existing causality measures (e.g., transfer entropy, Liang-Kleeman information flow) and newly derived ones.
• Specific quantitative relationships between transfer entropy and Liang-Kleeman information flow were identified for coupled two-state Markov chains.

Conclusions:

• The dynamical causal effects (DCEs) framework provides a unified and logical approach to quantifying causal information flow in Markov chains.
• This systematic generation clarifies the landscape of causality measures, revealing their interconnections.
• The findings offer a deeper understanding of information transfer mechanisms in complex systems.