# Quantifying emergent complexity

## TL;DR

The paper introduces a new theory to determine how different scales of complex systems contribute to their causal workings, offering a way to measure emergent complexity.

## Contribution

Causal Emergence 2.0 is introduced as a mathematical framework to quantify how causation is distributed across scales of complex systems.

## Key findings

- Causal Emergence 2.0 identifies which scales irreducibly contribute to a system’s causal workings.
- The theory measures emergent complexity by how widely distributed causation is across system scales.
- The framework provides a principled way to compare and choose modeling scales in scientific analysis.

## Abstract

Complex systems can be described at myriad different scales, and their causal workings often have a 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 long-standing issue, here, a new theory of emergence is introduced that can distinguish which scales irreducibly contribute to a system’s causal workings. The theory’s application is demonstrated in coarse grains of Markov chains, revealing a novel measure of emergent complexity: how widely distributed a system’s causal contributions are across its hierarchy of scales.

•A new theory reveals the hidden multiscale structure of complex systems•Identifies the subset of scales that irreducibly causally contribute•Measures emergent complexity based on how spread out causation is across scales

A new theory reveals the hidden multiscale structure of complex systems

Identifies the subset of scales that irreducibly causally contribute

Measures emergent complexity based on how spread out causation is across scales

Scientists routinely choose a particular scale at which to study a system. A neuroscientist might track the individual calcium imaging within synapses, while another neuroscientist might aggregate brain activity across entire brain regions. The choices of how to model a system vary enormously—yet science still lacks a complete formal framework for making such choices in a principled way. The problem is not only practical but also a long-standing philosophical paradox: how can the microscale truly matter? If it does not matter causally, then the vast majority of the entities in sciences are more like useful fictions or convenient compressions than genuine causes.

Resolving this issue requires taking the details of causal analysis and tracking causal influence seriously across the scales and developing a formal framework for comparing what each macroscale can add. With a mathematical toolkit focused on error correction and probabilistic measures of causation, a theory of emergence can assist scientists in making more informed modeling choices, understanding why certain high-level explanations succeed, and uncovering hidden structure spanning system scales even in simple systems. The goal is a science of scales itself—one that finally puts the hierarchy of nature on a firm quantitative footing.

Do the macroscales of complex systems add anything beyond being convenient compressions? This paper introduces Causal Emergence 2.0, a mathematical toolkit for apportioning out the causation across the many different scales of a system.