Emergent Capabilities in Neural Networks

Updated 16 October 2025

Emergent capabilities in neural networks are functional properties such as modularity, symbolic abstraction, and structured self-organization that naturally arise from training dynamics.
Gradient descent, combined with regularization and initial symmetry, drives networks to self-organize into specialized architectures with block sub-diagonals, skip connections, and percolative structures.
Quantitative metrics from phase transitions and percolation models provide actionable insights into network generalization, informing both theory and practical design improvements.

Emergent capabilities in neural networks refer to functional properties, behaviors, or structural features that arise through learning or system dynamics rather than being explicitly predefined in the architecture, loss function, or training procedure. These capabilities typically manifest as the neural network adapts to data via stochastic optimization, often giving rise to qualitatively new phenomena such as modularity, compositional reasoning, symbolic abstraction, phase transitions in generalization, and the spontaneous formation of structured representations or dynamic sub-networks. The paper of emergence in neural networks intersects with statistical physics, dynamical systems, information theory, neuroscience, and cognitive science, and is motivated by both the desire to understand the internal mechanisms of deep learning and the practical goal of controlling, predicting, and leveraging these capabilities.

1. Structural and Functional Emergence: From Unstructured Networks to Specialized Architectures

Several neural network architectures are capable of developing structured connectivity and functional specialization, even when initialized without such priors. The Unstructured Recursive Network (URN) exemplifies this phenomenon (Golkar, 2019). The URN is initialized as a fully dense, structureless network—each neuron is recursively updated, and no layer-by-layer structure is imposed a priori. Under gradient descent with a loss function incorporating cross-entropy for task performance and regulator terms for sparsity (e.g., $L_1$ penalties on weights and activations), the URN self-organizes into familiar network patterns:

Weight matrices condense into block sub-diagonal forms, mimicking the connectivity of multi-layer perceptrons (MLPs).
With additional regularization—such as a “synaptic length” penalty enforcing locality—URNs dynamically morph into architectures akin to locally connected or convolutional neural networks, where long-distance weights are suppressed.
Residual structures can also emerge when the update rule is modified to allow skip connections, enabling adaptive depth.

This process depends on three main factors: the initial symmetry of the network, the geometric structure of the data, and the balance of regularization terms. As a result, classical architectures emerge from gradient-based optimization, rather than manual design.

2. Emergent Feature Learning and Representation Dynamics

A defining emergent capability in neural networks is their ability to learn new features tailored to data distributions, surpassing models built on fixed feature maps (Shi et al., 2022). Theoretical analyses show that, for supervised tasks where labels are determined by latent patterns (such as parities or pattern combinations), gradient descent rapidly aligns neuron weights along the most informative data directions:

After one or two steps, neurons cluster in weight space toward sums of “relevant” dictionary atoms; higher-order alignment is improved in subsequent updates.
Later training phases focus on tuning higher layers (e.g., readout weights), while feature detectors in hidden layers become relatively fixed.
In contrast, linear models with data-independent features (Neural Tangent Kernel, random features) require exponentially many feature dimensions to achieve comparable performance.
If the data structure is removed or hidden, even powerful statistical-query learning algorithms fail, highlighting the intrinsic data-driven nature of emergent feature discovery.

Empirical results demonstrate the formation of nearly optimal features and the importance of input structure in guiding this emergence. Fixed-feature baselines are systematically inferior, both in controlled and real-data scenarios.

3. Collective Dynamics, Phase Transitions, and Percolation Models

Emergent phenomena in neural networks often involve transitions between qualitative behaviors as a function of scale, dataset size, or training progress (Gokhale, 2023, Clauw et al., 16 Aug 2024, Lubana et al., 22 Aug 2024). Core findings include:

Grokking: Neural networks may abruptly transition from memorization to generalization after a period of stagnation, interpreted as a phase transition driven by collective, synergistic interactions among groups of neurons. This is measurable by higher-order mutual information (O-Information), which quantifies the interplay between synergy (joint coding exceeding the sum of parts) and redundancy (overlapping representations). Synergy peaks signal the onset of generalization.
Percolation Models: The acquisition of general structural knowledge (such as grammar or type constraints in LLMs) can be modeled as percolation on a bipartite graph constructed from the network’s exposure to data structures (Lubana et al., 22 Aug 2024). When the density of connections in the data exceeds a critical threshold, a “giant component” emerges, allowing sudden, simultaneous improvements in multiple downstream tasks—a formal analogy to phase transitions in physics.
Semantic Landscape: The training process can be viewed as a random walk or activated hopping between nodes (heuristic models) on a “semantic graph.” Emergent algorithms (internal solutions) occupy these graph nodes, and scaling laws or sudden capability jumps are unified under percolation and random walk statistics.

These frameworks account for the nonlinear, sometimes unpredictable appearance of new abilities and explain scaling behavior, grokking, and the impact of network or data size in a rigorous manner.

4. Symbol-like Representation and Emergence of Compositionality

Symbolic computation and compositional reasoning are classic hallmarks of human cognition. Several works demonstrate that neural networks can spontaneously develop symbol-like variables, compositional structures, or interpretable internal algorithms:

In recurrent and transformer-based models trained on sequence-based numeric tasks, “number variables” emerge as latent, mutable subspaces within the model’s activations (Grant et al., 10 Jan 2025). Causal intervention and subspace alignment methods (such as Distributed Alignment Search) reveal that these variables align with simplified symbolic algorithms, such as counters, even in the absence of explicit symbolic assignments.
The formation of internal symbols is not strictly discrete; representations are often graded, exhibiting a continuum between symbolic and distributed codes. Transformers, unless heavily parameterized, tend toward recomputing contextually rather than maintaining persistent states.
In communication games, neural networks generalize compositional functions, even when standard compositionality metrics or human judgments see the underlying mappings as non-compositional (Perkins, 2021). This highlights a disconnect between human-centric metrics and machine generalization.
Hybrid connectionist-symbolic systems (e.g., symbol-emergence artificial networks) create a mapping from continuous, trainable symbols to network configuration vectors, facilitating task switching, communication, and compositional generalization (Chen et al., 2023).

These phenomena collectively demonstrate how symbol-like and compositional properties are not hard-coded but arise from learned network dynamics and loss geometry.

5. Self-Organization, Weight Morphologies, and Universal Learning Mechanisms

Beyond local feature learning or modularity, deep neural networks can exhibit global self-organization of weights and connectivity:

During training, channel-like morphologies arise in weight matrices even when the architecture is initially homogeneous (Jong et al., 9 Jan 2025). Instabilities in the update dynamics, analyzed via connectivity ratios $r_n^{(l)}$ , lead to the emergence of dominant pathways (channels) carrying most of the information flow. Adjacent layers exhibit oscillatory modulations in channel width via lateral inhibition-like mechanisms.
This emergent wiring is mathematically analogous to pattern formation in condensed matter physics and has functional consequences for embedding dimensionality, generalization performance, and robustness. Notably, these morphologies arise independently of the data distribution, suggesting that certain forms of organization are intrinsic to gradient-based optimization.
On a higher level, large-scale self-organized phenomena (coherent coupling of critical neurons) produce Hebbian-like neural correlation graphs with power-law degree and clique-size distributions (Liu et al., 28 Aug 2025). As learning progresses, the network undergoes second-order and first-order phase transitions (connectivity and convergence transitions), culminating in scale-free task-specific sub-networks (“critical computational graphs”) and increased concentration of loss measures around robust optima.
These phenomena parallel avalanche-like dynamics observed in biological networks and point to universality in the self-organization of computation across artificial and natural systems.

6. Beyond Neural Scaling: Dynamical Systems, Ontology of Emergence, and Implications

Scaling up parameters, data, or compute improves model performance in predictable ways, but genuine emergent capabilities—such as reasoning, pattern abstraction, or symbolic manipulation—arise from changes in the internal dynamical structure of the system (Havlík, 6 Aug 2025). Key aspects include:

Neural networks, viewed as complex, nonlinear dynamical systems, display sensitivity to microscopic parameter changes and training randomness—a property akin to deterministic chaos.
Emergent behaviors are neither predicted by scaling laws alone nor reducible to the sum of neuron-wise operations; they are systemic effects driven by cooperative interaction and nonlinear transformation.
Empirical performance curves (e.g., power-law fits, grokking events) are “macroscopic” signatures that can be correlated to underlying structural transformations, but the mechanisms remain only partly understood.
Epistemologically, AI development is marked by “creation without understanding”: models acquire abilities whose micro-level origins are neither interpretable nor analytically derivable.
This realization motivates a shift from phenomenological definitions (e.g., “emergence = sudden performance jump”) to mechanistic investigation of how internal, system-level reorganizations instantiate new computational abilities.

A formal approach, inspired by statistical mechanics and complex systems, is necessary for describing and managing emergent capabilities in AI, especially as unpredictability grows with model scale.

7. Quantification, Metrics, and Measurement of Emergence

Quantitative frameworks have been proposed to measure the degree of emergence and its effect on network performance (AlShinaifi et al., 3 Sep 2024):

Emergence can be defined in terms of connectivity between “active” and “inactive” nodes, with formulas computing the number of paths linking these across layers. In feedforward and convolutional networks, higher emergence correlates with trainability, convergence speed, and adaptability.
Pruning (removal of redundant parameters) reduces absolute emergence but can increase relative emergence (effective information density per parameter), often yielding faster training and improved efficiency, though sometimes at a small cost in peak accuracy.
The concentration of local minima and ruggedness of the loss landscape, as revealed by emergence metrics, further illuminate network complexity and facilitate the design of optimized architectures and training schedules.

By quantifying these internal structural properties, practitioners can better anticipate when further training is beneficial or when network expressivity is sufficient for a given task.

Emergent capabilities in neural networks thus encompass a family of phenomena—structural reorganization, collective dynamics, symbolic abstraction, phase transitions, and self-organization—arising from the interplay of architecture, data structure, learning dynamics, and stochastic optimization. These findings have profound implications for both theoretical understanding and practical design of neural systems capable of continual learning, flexible adaptation, robust generalization, and complex autonomous behavior.