Causal Emergence 2.0: Quantifying emergent complexity (2503.13395v3)

Abstract: Complex systems can be described at myriad different scales, and their causal workings often have multiscale structure (e.g., a computer can be described at the microscale of its hardware circuitry, the mesoscale of its machine code, and the macroscale of its operating system). While scientists study and model systems across the full hierarchy of their scales, from microphysics to macroeconomics, there is debate about what the macroscales of systems can possibly add beyond mere compression. To resolve this longstanding issue, here a new theory of emergence is introduced wherein the different scales of a system are treated like slices of a higher-dimensional object. The theory can distinguish which of these scales possess unique causal contributions, and which are not causally relevant. Constructed from an axiomatic notion of causation, the theory's application is demonstrated in coarse-grains of Markov chains. It identifies all cases of macroscale causation: instances where reduction to a microscale is possible, yet lossy about causation. Furthermore, the theory posits a causal apportioning schema that calculates the causal contribution of each scale, showing what each uniquely adds. Finally, it reveals a novel measure of emergent complexity: how widely distributed a system's causal workings are across its hierarchy of scales.

• The paper presents an axiomatic framework for multiscale causal analysis that quantifies emergent complexity via normalized causal contributions and entropy-based measures.
• It introduces a causal apportioning schema that robustly distinguishes scale-specific causal content by assessing sufficiency, necessity, and their information-theoretic counterparts.
• The methodology is validated on coarse-grained Markov chains, revealing hidden macroscale causal structures overlooked by traditional single-scale approaches.

Overview

The paper "Causal Emergence 2.0: Quantifying emergent complexity" (2503.13395) extends the framework of causal emergence by developing an axiomatic approach for multiscale analysis of complex systems. Rather than reducing a system to a single effective macroscale, the authors treat the different scales as slices of a higher-dimensional object, thereby capturing the full spectrum of causal contributions. The work introduces a causal apportioning schema that provides both a qualitative and quantitative basis for assessing the multiscale causal structure in systems such as coarse-grained Markov chains.

Axiomatic Reconstruction of Multiscale Causation

The theory is founded on a set of axioms built on causal primitives such as sufficiency and necessity, extrapolated to encompass their information-theoretic counterparts—determinism and degeneracy. The axiomatic formulation permits an unambiguous prescription of causal contributions at different scales, addressing limitations of previous approaches (e.g., effective information-based frameworks).

Key claims include:

• Scale Invariance with Unique Identification: By treating every scale as a legitimate contributor and by evaluating the causal primitives, the methodology distinguishes scales with non-redundant causal content versus those that are merely compressive.
• Robustness to Background Assumptions: The axiomatic structure minimizes dependency on arbitrary background assumptions, enhancing the precision in identifying macroscale phenomena even when microscale reductions are formally available.

Quantitative Measure of Emergent Complexity

A core contribution is the introduction of a numerical measure for emergent complexity (EC). This measure derives from analyzing the distribution of causal contributions along a defined micro-to-macro path, i.e., a sequential series of valid dimension reductions. At each reduction step, the change in causal primitives ($\Delta CP$) quantifies the causal contribution of that intermediary scale.

The emergent complexity is computed using an entropy-based formula over the distribution of normalized causal contributions:

p_i = \frac{\Delta CP_i}{\sum_{j} \Delta CP_j}

EC = \log_2(L) - \sum_{i=1}^{L} p_i \log_2(p_i)

Here, $L$ denotes the length of the micro-to-macro path. Systems exhibiting widely distributed contributions (i.e., a larger entropy in the distribution $\{p_1, p_2, \ldots, p_L\}$) are characterized by higher emergent complexity. This formulation allows for a nuanced distinction between systems with “top-heavy” causation, where most causal contribution is confined to the highest level, and those with significant mesoscale contributions.

Demonstration on Coarse-Grained Markov Chains

The paper provides an application of the framework through coarse-graining of Markov chains. The approach is structured as follows:

• Definition of Macroscales: The trajectories of Markov chains are coarse-grained to generate valid macroscales that maintain consistency in causal dynamics.
• Micro-to-Macro Reduction Path: A sequence of coarse-graining steps is formulated, each step corresponding to a valid scale reduction where causal primitives are consistently altered.
• Computation of $\Delta CP$: At each level, the change in causal primitives ($\Delta CP$) is computed, forming the basis for the subsequent normalization and entropy calculation.
• Revealing Undetected Macroscale Causation: The approach uncovers cases of macroscale causation that were not captured by earlier effective information measures, thus highlighting the granularity and uniqueness of causal contributions across scales.

The thorough treatment of Markov chains serves as a proof-of-concept, demonstrating that a multiscale treatment can both detect and quantify emergent causation in systems where reduction to a single scale loses critical causal information.

Implications and Applications

This work provides a robust framework for practitioners and theoreticians alike. In practical terms, the multiscale causal apportioning schema can be applied to a broad array of complex systems where hierarchical structure is intrinsic. For example:

• Neuroscience: In neural networks, different layers or regions might be assessed for their causal contributions, potentially guiding interventions or adaptations in brain-inspired architectures.
• Systems Biology: The framework could be used to discern the unique causal contributions of molecular versus cellular processes in regulatory networks.
• Economic Systems: A refined analysis of macroeconomic vs. microeconomic factors might be achieved by mapping the emergent complexity of agent-based models.

Moreover, the quantitative measure of emergent complexity offers a diagnostic tool that can be leveraged in both simulation-based studies and real-world data analyses, assisting in the detection of conditions where emergent properties are not merely epiphenomenal but are integral to the system’s causal dynamics.

Concluding Remarks

"Causal Emergence 2.0" significantly refines the conceptual and computational perspectives on emergence by moving beyond the single-scale paradigm. Through its axiomatic basis, robust quantification using an entropy-based measure, and comprehensive demonstration on Markovian dynamics, the theory provides a coherent framework for assessing, quantifying, and ultimately leveraging multiscale causation in complex systems. Its numerical formulation and emphasis on scale-specific causal contributions establish a promising direction for research and application in various fields where understanding multiscale interactions is crucial.