Engineering Emergence

TL;DR

The paper tackles how complex systems exhibit multiscale causal structure and argues that a non-reductive, non-mysterious framework is needed to identify the causally relevant scales. It extends causal emergence to CE 2.0, evaluating causal power (CP) across all coarse-grainings of a system's Markovian description and defining CP as $CP = \text{determinism}_T + \text{specificity}_T - 1$, with interventions drawn from a uniform distribution $p(C)$. By distributing CP across micro$\rightarrow$macro paths, it reveals the emergent hierarchy—the subset of scales that contribute non-zero, non-redundant causal power—leading to a taxonomy (top-heavy, bottom-heavy, mesoscale peaks) and a literal notion of scale-freeness. The work also introduces measures of emergent complexity, shows how to drive intermediate emergence via network growth and a branching greedy algorithm, and demonstrates methods to engineer pinpoint emergence, with implications for robust, controllable systems and applications in neuroscience and human–AI interfaces.

Abstract

A defining property of complex systems is that they have multiscale structure. How does this multiscale structure come about? We argue that within systems there emerges a hierarchy of scales that contribute to a system's causal workings. An intuitive example is how a computer can be described at the level of its hardware circuitry (its microscale) but also its machine code (a mesoscale) and all the way up at its operating system (its macroscale). Here we show that even simple systems possess this kind of emergent hierarchy, which usually forms over only a small subset of the super-exponentially many possible scales of description. To capture this formally, we extend the theory of causal emergence (version 2.0) so as to analyze how causal contributions span the full multiscale structure of a system. Our analysis reveals that systems can be classified along a taxonomy of emergence, such as being either top-heavy or bottom-heavy in their causal workings. From this new taxonomy of emergence, we derive a measure of complexity based on a literal notion of scale-freeness (here, when causation is spread equally across the scales of a system) and compare this to the standard network science definition of scale-freeness based on degree distribution, showing the two are closely related. Finally, we demonstrate the ability to engineer not just the degree of emergence in a system, but to control it with pinpoint precision.

Figures (6)

• Figure 1: The multiscale structure of a system is based on a partitioning of its states. Partitions can be partially ordered by refinement to construct partitions lattices. The associated Hasse diagrams are shown here for 2, 3, 4, and 8-state systems. For any two partitions $\pi_a, \pi_b$ we set $\pi_a \leq \pi_b \iff \forall b \in \pi_b :\exists a \in \pi_a \text{ s.t. } a\subseteq b$. This puts coarse partitions with few blocks (macroscales) at the top, and fine partitions with many blocks (microscales) at the bottom. The number of possible partitions of $n$ states (the number of elements in the unpartitioned microscale at the bottom) grows superexponentially with $n$, namely, as the $n$th Bell number (the lattices above contain respectively 2, 5, 15, 4140 partitions).
• Figure 2: A demonstration of calculating $\text{CP}$ at specific scales out of the full set of scales of a 6-state system (here a Markov chain described by its microscale TPM, the probabilities of which are shown in grayscale in the middle). Above the microscale TPM we show the associated Hasse diagram. Moving up the lattice traverses all coarse-grainings, proceeding from smaller groups to larger one. Left: a generic coarse-graining that reduces $\text{CP}$ (by lowering determinism) indicating that scale fails to add anything to the system's causal workings. Right: an especially informative coarse-graining that groups the six states into appropriate pairs, increasing determinism and decreasing degeneracy. This is reflected in the associated graph by the dominance of self-loops.
• Figure 3: A visualization of our method on a small system with 5 states that correspond, respectively, to a source state, two cycle states, and two sink states. The microscale TPM is shown at the bottom of the panels. Panel A shows the full partition lattice over 5 states, which consists of 52 possible coarse-grainings. Highlighted in black are the scales with a positive $\Delta\text{CP}$ score that increase the CP value beyond any of the scales below them. Panel B shows the sublattice that contains only these five scales (the emergent hierarchy) with the corresponding partitions indicated next to the nodes. In the visualization of the emergent hierarchy, the size of the node reflects its $\Delta\text{CP}$ value. Note that the intuitively “correct" scale that partitioned the system into source, cycle, and sink nodes indeed contributes the most $\Delta\text{CP}$. This is further shown by the vertical plot next to it that shows the average $\Delta\text{CP}$ value per row (per dimensionality of the coarse grained systems). Finally, panel C shows an abstracted representation of the hierarchy, which is just a symmetrized version of the vertical $\Delta\text{CP}$ plot. This representation is used in later figures to illustrate the difference between top-heavy, bottom-heavy, and complex systems.
• Figure 4: Systems can display causal emergence in many ways, here this is showcased as a “garden" of five systems (all the same size at the microscale) that have different emergent hierarchies. Top: The emergent hierarchies visualized across the eight levels of coarse-graining. We show the sublattice of the full partition lattice with positive $\Delta \mathrm{CP}$, i.e., the nodes that correspond to coarse-grainings that contribute $\text{CP}$ beyond any of its ancestors in the lattice. The size of the node is proportional to $\Delta \mathrm{CP}$. From left to right, the systems correspond to i) a system composed of two equivalence classes of states, ii) a system with two length-4 cycles (explored in more detail in Figure \ref{['fig:balloons']}), iii) a system with a mesoscale (taken from hoel2025causal), iv) a system with fully deterministic but degenerate transitions, and v) a system with three modules: three source states, a 3-cycle processing module, and 2 alternating sink states. Bottom: The width of the shaded region above each TPM corresponds to the average $\Delta \mathrm{CP}$ at that level (a particular row in the sublattice) of the emergent hierarchy shown in the panel above it (as in Figure \ref{['fig:example_system']}C) .
• Figure 5: Preferential attachment networks show different emergent hierarchies depending on the preference parameter $\alpha$, and display a phase transition around the $\alpha=1$ scale-free point. The number of connections for newly introduced nodes is set to $m=1$ throughout, in order to yield sparse and more hierarchical tree-like networks. Top: We show three measures that contribute to the system's complexity. The average negentropy of $\Delta \mathrm{CP}$ across the levels of the system (Equation \ref{['eq:row_negentropy']}) is shown on the left (capturing the differentiation of $\Delta \mathrm{CP}$ within levels). In the middle are plotted the average entropies of $\Delta \mathrm{CP}$ across 100 uniformly sampled paths on the sublattice (capturing how uniformly $\Delta \mathrm{CP}$ is spread across levels). For each $\alpha$ value, we report the mean and standard error across five sampled networks with 40 nodes each. The row negentropy sharply drops beyond $\alpha=1$, which reflects a strong decrease in the differentiation of $\Delta \mathrm{CP}$ within levels. On the right is plotted the multiplied values of both row negentropy and path entropy. Previous methods, like hoel2025causal and klein2020emergence, do not take into account parallel paths and therefore couldn't capture this horizontal heterogeneity. Bottom: We show the mean $\Delta$CP value for every dimension (in the same representation as in the bottom of Figure \ref{['fig:lattice_garden']}), averaged across five sampled networks at each $\alpha$. The distribution of $\Delta$CP can be seen to go from “bottom-heavy" at low $\alpha$ to “top-heavy" at high $\alpha$, with the transition taking place between $\alpha=1$ and $\alpha=1.5$.