Emergence and Causality in Complex Systems: A Survey on Causal Emergence and Related Quantitative Studies

TL;DR

The paper surveys quantitative approaches linking emergence and causality in complex systems, foregrounding two core frameworks: Erik Hoel's causal emergence based on effective information ($EI$) and Fernando Rosas's information-decomposition approach ($\phi$ID). It traces developments from computational mechanics and Granger-based notions of emergence to network-level analyses and continuous-system extensions, then connects these theories to machine learning via Neural Information Squeezer (NIS) and NIS+ for automatic coarse-graining and macro-dynamics discovery. The work also discusses the relation to causal representation learning, the ontological/epistemological debate, and practical implications for out-of-distribution generalization and robust modeling of multi-scale dynamics. Overall, the survey highlights how causality-inspired measures can quantify emergence, guide data-driven identification of emergent macro-variables, and inform applications across biology, neuroscience, and artificial intelligence. It also identifies methodological challenges, such as coarse-graining non-uniqueness, reliance on mutual information, and the need for scalable, automatic coarse-graining strategies. The framework offers a roadmap for integrating causal emergence into multi-scale modeling and AI systems, with potential to improve generalization, interpretability, and understanding of complex adaptive phenomena.

Abstract

Emergence and causality are two fundamental concepts for understanding complex systems. They are interconnected. On one hand, emergence refers to the phenomenon where macroscopic properties cannot be solely attributed to the cause of individual properties. On the other hand, causality can exhibit emergence, meaning that new causal laws may arise as we increase the level of abstraction. Causal emergence theory aims to bridge these two concepts and even employs measures of causality to quantify emergence. This paper provides a comprehensive review of recent advancements in quantitative theories and applications of causal emergence. Two key problems are addressed: quantifying causal emergence and identifying it in data. Addressing the latter requires the use of machine learning techniques, thus establishing a connection between causal emergence and artificial intelligence. We highlighted that the architectures used for identifying causal emergence are shared by causal representation learning, causal model abstraction, and world model-based reinforcement learning. Consequently, progress in any of these areas can benefit the others. Potential applications and future perspectives are also discussed in the final section of the review.

Figures (17)

• Figure S1: The causal models (left) column and the corresponding causal hierarchy (right) column.
• Figure S2: The various types of causation that exist within multi-scale complex systems. In this figure, solid arrows represent common causation, which is generally accepted without much controversy. Dotted arrowed lines indicate a form of "causation" driven by supervenience, whereas dashed arrowed lines represent emergent causation, occurring either at an intra-level or downward.
• Figure S3: G-causality and G-autonomous.
• Figure S4: G-autonomous.
• Figure S5: The basic idea of Erik Hoel's causal emergence theory. The colored circles represent micro-states, whereas the colored squares represent macro-states.