Course Graph Structure Governs How Embedding Models Map Prerequisite Dependencies

Computer scientists evaluating geometric embedding models on structured educational curricula found that the structural topology of a course dictates how effectively machine learning architectures capture prerequisite dependencies and sequential study pathways. Investigating how well established knowledge graph embedding frameworks prioritize correct target learning resources within partially ordered graphs, the researchers observed that distance-based translational techniques excel when mapping linear curricula, whereas rotational complex-space models achieve superior precision in modular, interconnected course environments. The findings demonstrate that knowledge graph embeddings can encode directional educational constraints, establishing an empirical baseline for automated curriculum sequencing while revealing clear boundaries in how geometric algorithms represent sequential pedagogical structures.

Digital education platforms generate expansive collections of learning resources, yet learners navigating these repositories frequently encounter structural bottlenecks when automated systems fail to account for curricular sequencing. While traditional recommender systems rely predominantly on collaborative filtering and direct user-item interaction histories, they routinely neglect the strict directional requirements that govern educational progression. In formal learning, resources do not simply relate through topical affinity or associative proximity; rather, concepts require prerequisite mastery, structured remedial loops following diagnostic assessment, and deliberate sequencing. To address these operational constraints, computer scientists modeled educational curricula as partially ordered graphs, formalizing learning trajectories and remediation pathways under explicit mathematical properties of reflexivity, antisymmetry, and transitivity.

The experimental framework formalized educational course content through two core relational dimensions: an sequential progression relation designated as is previous of, which models the standard forward flow of instructional material, and a remedial dependency designated as if fail, revisit, which directs learners back to foundational resources upon failing an assessment. Because the forward progression relationship appears far more frequently than remediation links across academic curricula, the resulting graphs present sparse, directed, and asymmetric topologies. The authors sought to determine whether four prominent knowledge graph embedding models—TransE, TransH, TransR, and RotatE—could accurately project these asymmetric structures into continuous vector spaces and resolve downstream link prediction queries without compromising pedagogical logic.

Translational distance-based models formulate relationships as continuous geometric displacements. Under the baseline TransE formulation, entities and relations reside within a shared vector space where a source resource vector added to a relation vector is optimized to approximate the target resource vector. TransH extends this formulation by projecting entity vectors onto relation-specific hyperplanes prior to translation, allowing individual learning concepts to adopt distinct contextual representations depending on whether they act as progressive steps or remedial anchors. TransR further generalizes this projection mechanism by mapping entity embeddings into entirely distinct relational vector spaces through projection matrices. In contrast, RotatE abandons vector translation in favor of element-wise rotations in complex vector space using the Hadamard product, a geometric formulation designed to inherently capture symmetry, antisymmetry, inversion, and composition.

To evaluate these embedding geometries under realistic educational constraints, the researchers tested the models across three distinct curriculum topologies. The F-course represented an authored curriculum characterized by a predominantly linear progression with brief horizontal divergences that quickly converge back into the primary instructional sequence. The E-course and V-course represented authentic instructional datasets provided by university educators. Among these, the V-course embodied an interleaved modular structure comprising independent theoretical and practical tracks that students pursue in alternating order without strict mutual synchronization. Across all three datasets, the primary computational objective centered on triple completion, measuring a model’s ability to rank and score the most pedagogically appropriate successor or remedial resource given a source concept and a specific relational query.

Model training utilized contrastive learning governed by a margin ranking loss objective, pulling valid curriculum triples closer in the embedding space while separating invalid configurations. The researchers engineered negative curriculum samples using two structural heuristics: minimum distance constraints, which penalize direct recommendations between concepts separated by excessive graph distance, and alternating order inversions, which explicitly mark triples that reverse the prerequisite sequence as negative examples. Hyperparameter tuning optimized the learning rate, embedding dimensionality, contrastive margin, and batch size across ten random initializations per combination, training models for up to 600 epochs. Across evaluations, performance improved consistently as the contrastive margin and embedding dimensionality increased, with optimal separation emerging at margins between 1.5 and 2.0 and latent representations spanning 70 to 80 dimensions.

Because educational environments frequently present multiple pedagogically valid subsequent steps, the authors evaluated query completion using Mean Average Precision alongside a modified Mean Reciprocal Rank metric termed MMRR. Unlike traditional reciprocal rank formulations that evaluate solely the first correct prediction in a ranked list, MMRR sums the reciprocal ranks of all valid target resources and normalizes this sum against an ideal ranking. In the linear F-course, TransH and TransE achieved the highest performance, both registering MMRR and MAP@7 scores in the order of 0.8. A Kruskal–Wallis test confirmed statistically significant performance differences among the four evaluated models with a p-value in the order of 1e-5, though pairwise comparisons between TransE and TransH revealed no statistically significant divergence between the two translational formulations.

A parallel pattern emerged in the E-course evaluation, where TransE and TransH again led the benchmarking pool, each attaining MMRR scores near 0.65 and MAP@7 values around 0.62. In contrast, TransR consistently underperformed in both the F-course and E-course, generating substantially lower precision metrics than the simpler translational baselines. Kruskal–Wallis testing on the E-course confirmed significant differences across the broader model cohort at p-values near 1e-5, while demonstrating that TransE and TransH remained statistically indistinguishable from one another. The authors noted that the overall performance ceiling across all architectures was lower on the E-course than on the other datasets, suggesting that unmodeled topological features or structural misclassifications may have restricted representational accuracy.

The comparative hierarchy shifted decisively within the modular topology of the V-course. In this multi-track environment, RotatE emerged as the dominant architecture, surpassing scores of 0.7 across MMRR, MAP@5, and MAP@7. TransR placed second, while the standard translational models demonstrated reduced capacity to capture the branching pathways. A Kruskal–Wallis evaluation confirmed that the four models diverged significantly with a p-value in the order of 1e-3, and direct pairwise testing verified that RotatE maintained a statistically significant performance advantage over the second-place TransR model. The rotational geometry of RotatE proved substantially more capable of managing the horizontal breadth and concurrent learning tracks characteristic of modular curricula.

These experimental outcomes highlight that curriculum topology directly dictates the geometric suitability of knowledge graph embeddings. Translational frameworks such as TransE and TransH map directly to linear curricula because sequential course progression behaves mathematically as directional vector displacement along a constrained instructional trajectory. Conversely, complex rotational embeddings like RotatE accommodate modular environments where parallel instructional tracks require compositional and relational expressiveness without rigid coordinate displacement. While the evaluated embedding algorithms demonstrated clear potential to preserve order-constrained educational dependencies, the researchers emphasized that lower results on certain course topologies underscore the necessity of developing specialized, structure-aware embedding models designed specifically for hybrid educational recommendation systems.

In standard knowledge graph applications, embeddings typically target massive networks containing thousands or millions of entities. Educational course graphs, by contrast, frequently operate on a compact scale, sometimes comprising fewer than fifty learning resources per curriculum. Within these constrained environments, the primary computational priority shifts away from navigating vast relational networks toward ensuring exacting precision and pedagogical fidelity for branching decisions. The researchers selected triple completion over direct embedding-to-embedding comparison because measuring distance in latent space does not guarantee that a model reliably encodes directional instructional logic. Formulating recommendations as query-driven triple completion—such as predicting successor concepts after completing variables or identifying remedial review targets when failing a functions module—directly tests whether scoring functions accurately prioritize valid next steps.

The training mechanics further reflected these structural demands through architectural adaptations and calibrated negative sampling. While TransE, TransR, and RotatE relied on two embedding layers dedicated separately to entities and relations, TransH incorporated an additional third layer specifically to model relation-specific hyperplane normal vectors. Across hyperparameter optimization sweeps, the most effective learning rates clustered at 1e-2 and 1e-3 across training runs extending up to 600 epochs. To teach models the asymmetry of instructional flow, negative sample generation explicitly penalized distant jumps, such as pairing introductory material directly with advanced optimization, alongside temporal reversals, such as placing practical exercises before foundational introductory units. This contrastive framing trained the scoring functions to separate valid curriculum paths from structural inversions that violate prerequisite ordering.

These empirical observations directly address the broader research questions regarding how course characteristics govern embedding effectiveness. The diminished performance observed on the E-course highlighted that existing embedding formulations encounter clear boundaries when course graphs do not cleanly fit linear or modular typologies, suggesting that unmodeled topological factors or structural classification issues strongly affect representational success. To build on these foundational findings, the authors outlined future research directions that extend beyond standalone knowledge graph embeddings. Rather than relying solely on knowledge-based filtering over partially ordered graphs, subsequent work aims to integrate these geometric representations into hybrid recommendation architectures that combine structural curriculum constraints with collaborative filtering techniques, ultimately seeking to enhance personalized and adaptive pacing in online learning environments.

Subject of Research: Evaluation of knowledge graph embeddings on educational partially ordered graphs

Article Title: A first step toward structured educational recommendations: knowledge graph embeddings in partially ordered graphs

Article References: Martínez-Martínez, A., Montoliu, R., & Remolar, I. (2026). A first step toward structured educational recommendations: knowledge graph embeddings in partially ordered graphs. Applied Intelligence, 56(16), Article 541. https://doi.org/10.1007/s10489-026-07576-5

Image Credits: AI Generated

DOI: 10.1007/s10489-026-07576-5

Keywords: knowledge graph embeddings, partially ordered graphs, educational recommender systems, TransE, TransH, TransR, RotatE, prerequisite relationships, curriculum sequencing, contrastive learning, margin ranking loss, triple completion, mean average precision, link prediction, pedagogical dependencies, linear course structures, modular course topologies