Emergence Taxonomy Explained

• Emergence Taxonomy is the systematic classification of distinct types and sources of emergent phenomena using formal criteria and quantitative analysis.
• It integrates dynamical-systems theory, information theory, and hierarchy-aware machine learning to map multi-scale emergence and causal power distribution.
• Applications include engineered emergent structures in artificial communication protocols and dynamic topic taxonomies driven by neural models.

Emergence taxonomy is the systematic classification of distinct types, sources, and organizational patterns by which emergent phenomena arise within multi-level, multi-agent, or multiscale systems. Recent research provides formal criteria for distinguishing kinds of emergence, quantitative tools for mapping multiscale emergence, and empirical taxonomies for emergent structures in artificial agents and hierarchies. The field synthesizes dynamical-systems theory, information theory, hierarchy-aware machine learning, and empirical studies of coordinated symbolic and causal structure formation.

1. Formal Criteria for Emergence and Its Varieties

A foundational axis in emergence taxonomy is the dynamical relationship between levels of description—microstates and macrostates—formalized through specific maps and decompositions. Carroll & Parola (Carroll et al., 20 Oct 2024) classify emergence by the structural properties of the micro-to-macro mapping $\Phi$ and its compatibility with subsystem decompositions:

• Type-0 (“Featureless”) Emergence: There exists a many-to-one (possibly partial-domain) coarse-graining map $\Phi: A \to B$ between microstates and macrostates such that the macro evolution law $E_{\beta}$ is exactly tracked by evolving the microstates under $E_{\alpha}$ and then applying $\Phi$. No subsystem (factorized) structure is imposed.
• Type-1 (“Local”) Emergence: Both micro and macro spaces are decomposed into Cartesian subsystems. The mapping $\Phi$ respects these decompositions and is locality-preserving: each emergent subsystem $b_n$ depends only on a block of micro-components $\{a_i\}$, and $\Phi$ is either algorithmically simple (Type-1a, e.g., block-spin renormalization) or algorithmically complex (Type-1b, e.g., recognition of Game of Life gliders).
• Type-2 (“Nonlocal”) Emergence: Subsystem decompositions exist, but the emergent variables or interactions become nonlocal with respect to the micro-level description. Macro variables may be defined globally, or macro-level interactions may require nonlocal microstate information.
• Type-3 (“Augmented” or “Strong”) Emergence: Macrostates include genuinely new ontological variables $\{\bar{b}_q\}$ that do not supervene on microstates; the emergence relation $\Phi$ is a binary relation rather than a function. Microdynamics can be fundamentally incomplete without specifying macro-level variables.

Types 0–2 are forms of “weak” emergence (full micro–macro supervenience), while Type-3 formalizes “strong” emergence (ontological novelty). This schema objectively grounds distinctions often left to philosophical debate (Carroll et al., 20 Oct 2024).

2. Quantitative Multiscale Emergence Taxonomy

Emergent hierarchies are rigorously mapped by assigning quantitative measure to the distribution of causal power across all possible coarse-grainings of a system. Jansma & Hoel (Jansma et al., 3 Oct 2025) extend causal emergence theory:

• Causal primitives at each possible partition $\pi$ (coarse-graining) are measured via determinism and degeneracy, capturing the effective information contributed by each scale. They define the unique non-redundant gain

$$\Delta\mathrm{CP}(\pi) = \mathrm{CP}(\pi) - \max_{\tau\in\mathrm{Anc}(\pi)}\mathrm{CP}(\tau)$$

where $\mathrm{CP}$ is the sum of determinism and $[1-\mathrm{degeneracy}]^{-1}$, normalized.

• Emergent hierarchy: The set of partitions with $\Delta\mathrm{CP}>0$, forming a weighted sublattice within the full lattice of partitions.
• Hierarchy shape is classified using two entropy-based metrics:
  • Path entropy $H$ (vertical): Measures how many levels meaningfully contribute on any micro–macro path. High $H$ signals distributed, scale-free emergence.
  • Row entropy $S_{\mathrm{row}}$ (horizontal): Measures within-level diversity in emergent contribution.

Typologically, emergent hierarchies can be:

• Bottom-heavy: Causal contribution concentrated at microscale.
• Top-heavy: Major gain occurs at topmost scale.
• Mesoscale-peaked: An intermediate “balloon” level dominates.
• Scale-free/complex: Significant, differentiated contributions exist across several scales and partitions (Jansma et al., 3 Oct 2025).

3. Emergence Taxonomy in Communication and Symbolic Systems

The emergence of structured communication protocols, particularly written languages in artificial agents, yields empirically grounded taxonomies based on symbol–meaning mappings and context dependence (Verma et al., 2019):

• Logographic/Compositional Systems: Emergent systems in which each target entity is mapped to a unique, context-independent symbol. This mapping is highly consistent—quantified using the variance of the Laplacian (VoL) of mean symbol images. For example, in distractor-agnostic agents, the consistency score per target is high (0.019).
• Ideographic/Context-Dependent Systems: Emergent protocols in which the written symbol encodes a distinction between the target and its distractors; the same entity may yield different symbols depending on context. This results in lower consistency per target (0.007) but higher consistency for target–distractor pairs (0.015).

This taxonomic axis (compositional vs. context-dependent) maps directly onto known distinctions in human script evolution and is analytically supported by formal consistency metrics (Verma et al., 2019).

4. Emergent Taxonomies in Hierarchical Topic Models

In the dynamic expansion of topic taxonomies, the emergence of new topical structure is shaped both by local content and global hierarchy constraints. Lee et al. (Lee et al., 2022) present a data-driven taxonomy expansion paradigm:

• Taxonomy expansion framework: Operates on a rooted tree $T=(C,R)$, where $C$ are topic nodes with term clusters $P_j$ and $R$ are directed hierarchical edges.
• Emergence of novel topics: Achieved by hierarchy-aware neural models (relational GCNs and transformer decoders) that synthesize new multiword topic phrases from document streams, positioning them for maximal hierarchical consistency.
• Structural emergence: This approach supports the dynamic emergence of taxonomies whose form is not simply an accretion of frequently used terms, but reflects genuinely new concepts integrated within the existing hierarchy.
• Evaluation metrics: Term coherence, relation accuracy, and subtopic integrity quantitatively measure the quality and consistency of the expanded taxonomy. TopicExpan yields term coherence of 0.98 and relation accuracy of 0.88, far exceeding classic approaches (Lee et al., 2022).

A plausible implication is that the global structure of topic hierarchies—depth, breadth, and local connectivity—emerges from the statistical and semantic dynamics of the underlying document stream and the inductive biases of the taxonomy expansion model.

5. Influence of Generative Dynamics and Engineering of Emergent Structures

The form of emergent taxonomy is strongly contingent on the generative rules and system dynamics:

• In multi-agent systems, protocol typology (logographic vs. ideographic) can be engineered by varying agent observability (distractor-agnostic vs. distractor-aware), size of candidate sets, or reward structures—trading off symbol reuse and compositionality (Verma et al., 2019).
• In multiscale causal networks, the distribution of emergence across scales is controlled by the fine structure of the transition probability matrix—directed cycles (“balloons”) lead to mesoscale-peaked hierarchies, while altering network growth via preferential attachment ($\alpha$ parameter) continuously tunes the system from bottom-heavy to maximally complex, as measured by path and row entropy (Jansma et al., 3 Oct 2025).

These mechanisms provide actionable design principles: fine-tuning observability or network growth rules allows for deliberate placement of emergence within the system’s hierarchical structure.

6. Broader Implications and Theoretical Synthesis

Emergence taxonomy frameworks unify state-based, information-theoretic, and empirical perspectives:

• The comprehensive mapping of emergence types via dynamical commutativity, subsystem decomposition, and ontological augmentation yields an objective, implementation-agnostic foundation for classifying emergent phenomena (Carroll et al., 20 Oct 2024).
• Quantitative emergent hierarchies (via $\Delta\mathrm{CP}$, path entropy, row entropy) provide a diagnostic for the distribution and complexity of emergent causal power across all scales (Jansma et al., 3 Oct 2025).
• Structural taxonomies in communication and topic systems show how learning dynamics and task constraints select among possible organizational types and how novel typologies may be engineered or predicted (Verma et al., 2019, Lee et al., 2022).

A plausible implication is that these multiple axes—dynamical, structural, statistical—enable convergence toward a unified, formal theory of emergence that is both empirically testable and methodologically constructive for engineering and interpreting complex systems.