Causal Emergence 2.0: Quantifying emergent complexity

TL;DR

CE 2.0 addresses the question of whether higher-level descriptions contribute causally beyond microscale descriptions by formulating an axiomatic multiscale framework and a path-based apportioning scheme. It defines causal primitives (sufficiency, necessity) and their determinism and degeneracy generalizations, and quantifies macroscale causation via gains in these primitives along a micro→macro path, CP = suff( e|c ) + nec( e|c ) - 1. Emergent complexity (EC) then measures how broadly these causal contributions are distributed across scales, using EC = - sum_i p_i log2 p_i with p_i = ΔCP_i / sum_j ΔCP_j, normalized by log2 L. The framework demonstrates that CE 2.0 captures all macroscale causation—unlike CE 1.0—and yields a taxonomy of causal structure (top-heavy vs mesoscale) while acknowledging practical limitations like combinatorial explosion and proposing SVD-based heuristics for scalable analysis. The work has broad implications for scientific modeling across physics, biology, neuroscience, economics, and AI interpretability, offering a principled way to quantify when macroscales meaningfully contribute to a system’s causal workings.

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.

Figures (7)

• Figure 1: A micro$\rightarrow$macro path visualized. An example 8-state Markov chain, with the probabilities of transitions represented in grayscale (for its TPM, see Fig.2A). Starting at the full partition of $(0), (1), (2), (3), (4), (5), (6), (7)$ (the microscale) states are coarse-grained together, with each further partition being a step in the path (and thus a scale in the system), ending at $(0), (1, 2, 3, 4, 5, 6, 7)$. Changes to being the same color indicate when states are coarse-grained together along the chosen path (color contagion).
• Figure 2: Causal primitives along a micro$\rightarrow$macro path. (A) The TPM of the microscale, with cells colored based on their probability ($p=1$ being a darker blue). (B) The TPM of the macroscale past which $\Delta \mathrm{CP}$ transitions abruptly to zero. (C) The same macroscale visualized as a Markov chain, with the coarse-grained macrostate labeled (its self-loop of $p=1$ is not shown). (D) The change in causal primitives across the path of increasing dimension reduction, with the total gain in CP of $0.33$ (in terms of determinism plus specificity) reflecting the degree of causal emergence.
• Figure 3: Causal contributions across scales. (A) Microscale TPM of a system with no mesoscale structure. (B) The same system visualized. (C) CE 2.0 identifies this system's causal contributions as "top-heavy," in that the last dimension reduction contributes the most. (D) Microscale TPM of an otherwise similar system with mesoscale structure. (E) The mesoscale system visualized. (F) Causal contributions are shifted toward lesser dimension reductions, indicating a predominately multiscale causal structure; it therefore possesses more emergent complexity.
• Figure 4: CE 1.0 cannot capture all macroscale causation. (A) TPMs (probabilities shown in bluescale) of a "block model" system with two macrostates over its equivalency classes at the beginning, midpoint, and end of increasing the self-loop probabilities of each state. Redistribution is performed by drawing away probability from its full set of transitions via increments of $1/steps$ until the microscale is entirely composed of states with self-loops of $p=1$. (B) CE 2.0 detects the macroscale causation and decreases sensibly as the microscale becomes more causally distinguishable during the probability redistribution, while the EI does not.
• Figure S1: Increasing uncertainty in the causal relationships of an 8-state system. Starting in a state of self-loops with $p=1$, states in the network were changed over a set number of steps equal to the size (number of states) of the system in three ways. Along the x-axis, self-loop probabilities were reduced by $1/steps$ and distributed equally to the other states (thus increasing the uncertainty of a particular effect, given a cause) until the system was an all-to-all network with random transitions. Along the y-axis, at each step a state was replaced with the transition distributions of another state (increasing the number of common causes and thus increasing the uncertainty of a cause, given an effect), until all states in the system shared the same transition. The system was also subjected to both changes at each step (the middle diagonal), ending again in an all-to-all state of random transitions.