JUSTIFICATION OF A HYPERGRAPH MODEL FOR ADAPTING A DIGITAL EDUCATIONAL SYSTEM TO DIVERSE LEARNING CONTEXTS

ОБОСНОВАНИЕ ПРИМЕНЕНИЯ ГИПЕРГРАФОВОЙ МОДЕЛИ ДЛЯ АДАПТАЦИИ ЦИФРОВОЙ ОБРАЗОВАТЕЛЬНОЙ СИСТЕМЫ К РАЗЛИЧНЫМ КОНТЕКСТАМ ОБУЧЕНИЯ

Икласова Кайнижамал

PhD, ассоциированный профессор Факультет инженерии и цифровых технологий Северо-Казахстанский университет имени Манаша Козыбаева,

Казахстан, г. Петропавловск

ABSTRACT

This paper substantiates the application of hypergraph theory as a universal mathematical framework for representing complex multi-relational interactions within educational environments. The aim of the study is to provide a formal justification of the advantages of the hypergraph model over traditional graph-based representations and to demonstrate its applicability across various educational scenarios. A hypergraph model of a digital educational system is proposed, comprising five vertex classes: learners, educational content, competencies, learning groups, and instructors. The study demonstrates that hypergraphs preserve the semantic integrity of educational events, naturally represent multidimensional relationships among entities, and support adaptation to different learning contexts without altering the formal structure of the model. Examples of implementation in higher education, corporate training, and massive open online courses are presented. The results confirm the potential of hypergraph-based structures for the development of adaptive digital educational systems and personalized learning pathways.

АННОТАЦИЯ

В статье обосновывается применение аппарата теории гиперграфов в качестве универсальной математической модели для представления сложных многосвязных отношений, возникающих в образовательной среде. Цель исследования заключается в формальном обосновании преимуществ гиперграфовой модели по сравнению с традиционными графовыми представлениями и демонстрации её применимости для различных сценариев обучения. Предложена гиперграфовая модель цифровой образовательной системы, включающая пять классов вершин: обучающиеся, образовательный контент, компетенции, учебные группы и преподаватели. Показано, что гиперграфы позволяют сохранять семантическую целостность образовательных событий, естественным образом описывать многомерные связи между объектами и обеспечивать адаптацию системы к различным образовательным контекстам без изменения формальной структуры модели. Рассмотрены примеры применения модели в высшем образовании, корпоративном обучении и массовых онлайн-курсах. Результаты исследования подтверждают перспективность использования гиперграфовых структур при разработке адаптивных цифровых образовательных систем и персонализированных образовательных траекторий.

Keywords: hypergraph, digital educational system, adaptive learning, learning analytics, personalized learning, competency model, recommender systems, graph theory.

Ключевые слова: гиперграф, цифровая образовательная система, адаптивное обучение, образовательная аналитика, персонализация обучения, компетентностная модель, рекомендательные системы, теория графов.

Designing modern digital educational environments requires flexible architectural solutions capable of dynamically adapting to changing learning contexts. The traditional approach to developing adaptive learning systems often relies on rigidly fixed data models oriented toward a specific type of educational process — whether massive open online courses (MOOCs) with content recommendation, intelligent trainers for skill practice, or offline and blended project-based learning systems. When migrating a system to a different context, developers face the need for a complete redesign of relationships.

As a universal mathematical invariant capable of solving the problem of cross-context adaptation, the apparatus of hypergraph theory is proposed. The scientific novelty of this approach lies in the creation of a unified multi-context data model, in which the dynamic change of the semantic role of vertices and hyperedges allows switching between different educational environments and scenarios without altering the physical structure of the database.

Modern digital educational systems (DES) are characterised by complex many-sided interaction structures: a single piece of learning content may simultaneously develop several competencies, a student may belong to multiple learning groups, and an instructor may teach across several programmes. Traditional graph models, in which an edge connects exactly two vertices, cannot adequately represent such many-sided relationships without introducing artificial auxiliary nodes.

A hypergraph, in which each hyperedge may be incident to an arbitrary number of vertices, is a natural tool for modelling such structures. The aim of this paper is to formally justify the advantages of the hypergraph model and to demonstrate its applicability across different educational contexts [1-4].

Consider a typical educational act: student s₁ in group g₁ studies module c₁, acquiring competencies k₁ and k₂ under the guidance of instructor t₁. In an ordinary graph

,                                                  (1)

 where E⊆ V×V, this event decomposes into five binary edges: (s₁, c₁), (c₁, k₁), (c₁, k₂), (s₁, g₁), (c₁, t₁). The semantic integrity of the single educational act is lost: it becomes impossible to determine whether competency k₁ was acquired specifically while studying module c₁ in group g₁, rather than in some other context.

Table 1.

Comparison of graph-based and hypergraph-based representations

Note that a multipartite graph, often proposed as an alternative, removes only some of the limitations: it allows links between different classes of entities, but remains restricted to binary edges. A hyperedge is fundamentally different in that it captures a single contextual act that cannot be decomposed into pairs without loss of meaning.

We define the educational hypergraph model as:

,                                            (2)

where V is a finite set of vertices, is a family of hyperedges, and w: E → ℝ⁺ is a weight function reflecting the importance of each relationship.

The vertex set is partitioned into five disjoint classes:

,                             (3)

where S denotes students, C denotes learning content (modules, assignments), K denotes competencies, G denotes learning groups, and T denotes instructors.

Each hyperedge e ∈ E contains vertices from at least two distinct classes and captures a specific educational act. Examples of hyperedges:

-      e₁ = {s₁, c₁, k₁, k₂, g₁, t₁} - student s₁ in group g₁ studies module c₁ and develops competencies k₁, k₂ under instructor t₁;

-      e₂ = {s₁, s₂, s₃, c₂, k₃} - a group project by three students using content c₂ to develop competency k₃;

-      e₃ = {c₁, c₃, k₁} - two modules jointly provide a single competency.

The incidence matrix B of size |V| × |E| enables efficient computation of competency coverage, instructor workload, and the connectivity of learning trajectories.

One of the key properties of the proposed model is its contextual adaptability: by varying the dominant vertex classes and the structure of hyperedges, the same formal framework describes fundamentally different learning environments.

All five vertex classes are active. Hyperedges are dense (5–7 vertices): groups are clearly defined and the competency matrix is set by the educational standard. The key analytical task is balancing instructor workload and diagnosing ‘bottleneck’ competencies covered by the fewest hyperedges.

Class G is replaced by roles or job positions, and class T is replaced by domain experts or mentors. Hyperedges emphasise links of the form {employee, task, competency, mentor}. The competency graph becomes directed: the sequence in which skills are acquired to meet the requirements of a specific position becomes important.

Classes G and T may be empty or minimal. Hyperedges are sparse: {student, content, competency}. The model scales easily to millions of vertices of type S owing to the sparsity of hyperedges, and admits efficient implementation using sparse incidence matrices.

Consider a small example: 3 students {s₁, s₂, s₃}, 2 modules {c₁, c₂}, 3 competencies {k₁, k₂, k₃}, 1 group {g₁}, 1 instructor {t₁}. We define the following hyperedges:

Table 2.

Hyperedges of the illustrative example

From the incidence matrix it is clear that competency k₁ is covered by two hyperedges (e₁ and e₂), indicating high reliability. Competency k₃ is covered only by hyperedge e₃ - a potential risk point that requires additional content. Such a conclusion cannot be drawn from an ordinary graph without an additional semantic layer.

This paper has shown that the hypergraph model surpasses the graph-based representation in three key respects: the semantic integrity of educational acts, the natural description of many-sided relationships, and contextual adaptability without any change to the formal structure. The proposed model (2) with five vertex classes is equally applicable in university, corporate, and distance learning settings.

Future work will focus on developing a personalisation algorithm for learning trajectories based on analysis of the hypergraph incidence matrix, as well as on empirical validation of the model using real-world academic process data.

This research is funded by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP23488869).

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