Improved estimators of causal emergence for large systems

TL;DR

The paper tackles the challenge of measuring emergence in large complex systems by addressing double-counting of shared information in existing Shannon-invariant measures. It introduces a lattice-expansion framework that yields a family of corrected emergence estimators $\Psi^{(k,q)}$, bridging fast coarse measures and full partial information decomposition. Through Reynolds flocking and schooling fish case studies, the authors demonstrate that correcting for redundancy enhances detection of causal emergence and reveals how information is shared across interaction orders. This approach improves robustness for large systems and suggests applicability to a broader class of information-theoretic, Shannon-invariant metrics.

Abstract

A central challenge in the study of complex systems is the quantification of emergence -- understood as the ability of the system to exhibit collective behaviours that cannot be traced down to the individual components. While recent work has proposed practical measures to detect emergence, these approaches tend to double-count the contribution of shared components, which substantially hinders their capability to effectively study large systems. In this work, we introduce a family of improved information-theoretic measures of emergence that iteratively correct for double-counted terms. Our approach is computationally efficient and provides a controllable trade-off between computational load and sensitivity, leading to more accurate and versatile estimates of emergence. The benefits of the proposed approach are demonstrated by successfully detecting emergence in both simulated and real-world data related to flocking behaviour.

Figures (6)

• Figure 1: Graphical illustration of lattice expansion for a system of $n = 3$ variables. Lattice nodes representing information atoms that get added together are marked in red, while the ones subtracted are marked in blue. Intuitively, the information provided by each of the three micro variables towards the macro includes the redundant information they share. When summing these mutual information quantities into a "sum-of-parts" term that gets subtracted from the "whole", this redundancy is included three times rather than once. The first-order lattice expansion re-adds the redundancy shared between all components, while the second-order expansion corrects for redundant information among groups of two. In a system of three variables, only two orders of expansion are necessary.
• Figure 2: Phenomenonologically different instances of the Reynolds flocking model under increasing avoidance. (a) No avoidance ($a_2 = 0$): shows rigid milling behaviour, which should manifest very high redundancy. (b) Intermediary avoidance ($a_2 = 0.1$): reminds of a flock of birds flying around their nest, or fish in a tank. This behaviour should be characterised by a balance of redundant and synergetic information. (c) Higher avoidance ($a_2 = 0.2$): almost random behaviour, which should be characterised by small redundancy and small synergy.
• Figure 3: Causal emergence and first-order redundancy in the flocking boids model.$\Psi^{(1,0)}$ peaks in the intermediary state, and is negative in the low avoidance scenario. But when considering the first-order redundancy in the system in computing $\Psi^{(1,1)}$, the criterion also correctly finds the no-avoidance, redundancy-driven milling movement as emergent. Error bars show standard error across $R=20$ simulations for each parameter set.
• Figure 4: Causal emergence and redundancy in the flocking boids model. Only the redundancy in the first lattice expansion is relevant in this system, being orders of magnitude while higher expansion orders ($q=2$ to $q=10$) are negligible. Error bars represent standard error across $R=20$ simulations for each parameter set.
• Figure 5: Trajectories of 200 timesteps (24 seconds) of ten schooling fish from three different measurements, showing increasing degrees of dispersion, or decreasing degrees of aggregation, while maintaining phenomenologically similar dynamical behaviour, which reminds of the intermediary regime in the Reynolds model (Fig. \ref{['fig:phenom-reynolds']}).

Theorems & Definitions (3)

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