Directed Skill Graph in AI Systems

• Directed skill graphs are a formalism that represents discrete competencies as nodes in a DAG, supporting modular training and staged learning.
• The approach enforces acyclicity and sequential freezing of upstream skills to mitigate catastrophic forgetting and facilitate robust transfer.
• Graph representation learning and stochastic traversal strategies optimize inference, enabling adaptive, interpretable, and scalable AI system designs.

A directed skill graph is a formalism in which discrete capabilities ("skills") and the directed dependencies among them are represented as nodes and edges in a directed acyclic graph (DAG). This structure is used to factor complex systems—most notably, lifelong agents, automated vehicles, and LLMs performing inference—into modular, interdependent competencies that support staged training, interpretability, effective transfer, and tractable evaluation. Directed skill graphs have emerged as a unifying abstraction underlying hierarchical reinforcement learning, automated capability monitoring, skill-based curricula, and stochastic reasoning strategies in modern AI systems (Najar, 25 Jan 2026, Jatzkowski et al., 2021, Ellis-Mohr et al., 10 Jun 2025).

1. Formal Definition and Structural Principles

A directed skill graph is a tuple $G=(V, E)$ comprising a finite set of skills $V$ and a set of directed edges $E\subseteq V \times V$. Each edge $(u \to v) \in E$ typically indicates one of two interpretations:

• In most learning/control settings: "Skill $v$ is trained only after skill $u$ has been learned and frozen" (Najar, 25 Jan 2026).
• In cyber-physical and monitoring domains: "$u$ is a prerequisite or dependency for $v$" (Jatzkowski et al., 2021).

The core properties are:

• Acyclicity: $G$ is a DAG, preventing cyclic dependencies.
• Partial Ordering: There is an upstream-to-downstream (or abstract-to-concrete) order along the graph.
• Skill Categorization (in automated systems): Nodes may be annotated with one of several categories, e.g., System, Behavioral, Planning, Perception, Data Acquisition, Action, Actuation, supporting multiple layers of abstraction (Jatzkowski et al., 2021).
• Operational Design Domain (ODD) Conditioning: In safety-critical domains, $G$ may be parametrized by a set of environment or task features ("scene elements") $S$; an ontology maps $S$ to subsets of required skills and their dependencies (Jatzkowski et al., 2021).

2. Hierarchical Curricula and Sequential Training

The directed skill graph induces a hierarchical curriculum via topological ordering. In staged learning:

• Upstream skills (those with no prerequisites) are trained first and parameters are frozen after reaching competence.
• Downstream skills are then trained, using fixed upstream outputs as stable inputs.
• Per-skill specialization: Each skill $\pi_k$ is designed to map its narrow observation $o_t^k = \Omega_k(s_t)$ to a low-dimensional action $a_t^k$; the composed action vector is merged as $a_t = C(a_t^C, a_t^L, a_t^M, a_t^D, a_t^H)$ (Najar, 25 Jan 2026).
• Skill-specific reward shaping: Each skill is assigned a bespoke reward $r^k$ tailored to its functional scope (e.g., alignment for camera, lock-on indicator for targeting, positional error for movement, composite terms for dodging and decision policies).

Freezing upstream parameters reduces the exploration burden on later skills and enforces original specializations, thereby mitigating catastrophic forgetting. The result is a highly modular system: each phase only updates $\theta_k$ for the currently trained skill, with all $\theta_j$ ($j\prec k$) held constant (Najar, 25 Jan 2026).

3. Selective Adaptation and Lifelong Transfer

Directed skill graphs enable selective adaptation when the task or environment ("domain") shifts:

• Partition: Parameters are split into "upstream" $\Theta_{\mathrm{up}}$ (phase-invariant) and "downstream" $\Theta_{\mathrm{down}}$ (phase-sensitive).
• Zero-shot transfer: The entire pretrained stack is evaluated in the new domain without modification.
• Selective fine-tuning: Only those skills whose performance degrades under domain shift (typically downstream skills) are fine-tuned, freezing upstream modules. Empirically, fine-tuning only the two most phase-sensitive skills under budgeted interactions is sufficient to restore or surpass original performance, demonstrating strong modular transfer (Najar, 25 Jan 2026).
• Formal adaptation objective: Optimization is constrained to a small parameter subset with a bounded drift from prior weights, and subject to minimal data requirements.

This protocol sharply localizes retraining cost and memory requirements, as only the affected skills (and their immediate descendents) are updated, preserving broader competency and avoiding wholesale retraining.

4. Automated Construction and Ontological Encoding

In domains such as automated vehicle capability monitoring, skill graphs are constructed algorithmically using formal ontologies (Jatzkowski et al., 2021). The process is summarized as:

1. Base Graph Retrieval: Extract a core behavioral skill graph from ontology (T-Box).
2. ODD-driven Skill Insertion: Augment the graph with all skills required by the current operational design domain (ODD), utilizing mappings $\delta: S \rightarrow \mathcal{P}(V)$.
3. Dependency Closure: Iteratively include all prerequisite skills by following "dependsOn" edges until closure is reached.
4. Graph Extraction: Output the induced $G=(V, E)$.
5. Acyclic Consistency: Ensure all dependency insertions preserve acyclicity and ontology consistency.

Such knowledge-base driven pipelines ensure that skill graphs remain acyclic, consistent under ODD changes, and that changes to base competencies or new scene features propagate automatically.

5. Graph Representation Learning on Directed Skill Graphs

The asymmetry inherent in skill dependencies demands directed graph embedding methodologies:

• Dual-role embeddings (Tan et al., 2021, Khosla et al., 2018): Each skill $v$ is assigned both an incoming ("receiver") and outgoing ("sender") embedding, $h_v^{\mathrm{in}}, h_v^{\mathrm{out}}$, optimized to model the transition probability or link likelihood along directed edges.
• Alternating random walks (Khosla et al., 2018): Sampling is performed by alternating between following out-edges (prerequisite-to-dependent) and in-edges (dependent-to-prerequisite), exposing both “hub” and “authority” roles per skill.
• Edge likelihood regularization: Compatibility between $h_u^{\mathrm{out}}$ and $h_v^{\mathrm{in}}$ for edges $(u\to v)$ is directly optimized, preventing over-smoothing and maintaining local role discrimination in deep GNNs.
• Evaluation: Embeddings are validated via directed link prediction, node classification, and graph reconstruction. Negative sampling in both edge and directionality is crucial for accurate evaluation.

These graph neural network strategies enable robust, scalable representation and inference within large directed skill graphs, supporting automated curriculum design, link imputation, and transfer diagnostics (Tan et al., 2021, Khosla et al., 2018).

6. Stochastic Skill Search and Reasoning Strategies

A recent theoretical extension considers inference—particularly in LLMs—as stochastic traversal over a latent skill graph (DS3: Directed Stochastic Skill Search) (Ellis-Mohr et al., 10 Jun 2025):

• Trace as random walk: Each inference trace is a path in a directed graph $\mathcal{G}^{(i)}$, with transitions governed by a model-dependent kernel and emitting potential outcomes at each node.
• Control nodes: Special nodes ("branch", "stop") enable modeling of chain-of-thought (CoT), tree-of-thought (ToT), best-of-$N$ (BoN), and majority-vote reasoning.
• Closed-form analysis: The success probability and compute cost under different reasoning policies are derived in terms of the skill graph's directionality, required skill sequence length, and capability coefficient.
• Strategy selection: Empirical and theoretical results indicate that CoT is optimal for high-capability, low-difficulty regimes, while ToT, BoN, or MV become preferred under low-capability or high-difficulty, due to exponential increases in success via exploration amplification.
• Unified abstraction: DS3 subsumes various inference paradigms and enables principled selection or adaptivity among search/evaluation strategies, as a function of resource constraints and task structure.

This probabilistic view connects reasoning efficiency, compositionality, and transfer in LLMs directly back to the structure and learned traversal biases over a directed skill graph (Ellis-Mohr et al., 10 Jun 2025).

7. Applicability and General Guidelines

Directed skill graphs offer a principled template for building hierarchical, continually adapting agents and systems. General guidelines for their deployment include (Najar, 25 Jan 2026):

1. Identifying Core Competencies: Decompose the global task into discrete skill nodes.
2. Organizing as DAG: Layer skills such that all preconditions (prerequisites) are satisfied upstream.
3. Specialized Inputs/Rewards: Map observations and define rewards to respect skill granularity.
4. Sequential Freezing: Train each skill in graph order, freezing ancestors to protect established behaviors.
5. Localizing Adaptation: Upon environmental change, adapt only the most directly affected skill(s), freezing phase-agnostic components.

By adhering to these principles, directed skill graphs furnish a scalable, interpretable, and sample-efficient pathway toward robust, transferable, and modular lifelong agents across simulation, robotics, autonomous driving, and complex inference architectures (Najar, 25 Jan 2026, Jatzkowski et al., 2021, Ellis-Mohr et al., 10 Jun 2025).