1 Introduction

Recommendation systems are integral to virtually every digital service platform, employing diverse criteria to rank items and provide tailored suggestions to users. This reciprocal interaction is advantageous for both service providers and users. A proficient recommendation system simplifies user decision-making by presenting a curated subset of items aligned with their probable interests (Isinkaye et al. 2015). For instance, in a course recommendation system, learners receive course suggestions based on their past course history, incorporating factors like user ratings, watch time, course descriptions, reviews, and overall course ratings. Additional considerations may involve metadata associated with courses and collaborative insights gathered from users who have undertaken similar courses. Top-N recommenders generate a curated list of ‘N’ courses anticipated to be attractive to the user.

Diverse recommendation systems exist in the literature, employing various methodologies. The predominant categories are content-based and collaborative recommendation systems (Pazzani 1999). Content-based systems rely on a user’s interaction history and course metadata to formulate recommendations, limiting exposure to new item types. On the other hand, collaborative recommendation systems aggregate collaborative inputs from all users to inform suggestions, often favoring popular items and facing challenges such as cold-start (Lika et al. 2014). To address these limitations, hybrid recommendation systems (Burke 2002) combine elements from both content-based and collaborative filtering approaches, leveraging their strengths while mitigating weaknesses such as cold-start or popularity bias. However, hybrid methods come with their own set of limitations, including increased complexity in model design and implementation, potential overfitting when merging diverse recommendation signals, and the need for extensive data to train and balance multiple recommendation algorithms effectively (Kulkarni et al. 2022). The intricate landscape of higher-order relationships among users and items requires careful processing to overcome these challenges and provide accurate recommendations.

In a noteworthy contribution to the literature, we propose a recommendation system harnessing the capabilities of a partially ordered neutrosophic hypergraph (Akram et al. 2018; Luqman et al. 2019; Ward 1954) to model higher-order relationships between users and items effectively. The potency of the morphological operation known as dilation is harnessed for recommendation generation, while morphological erosion on the dual of the embedded hypergraph efficiently produces meaningful explanations. While hypergraphs are commonly used in recommendation systems, introducing partially ordered neutrosophic hypergraphs enhances representational power by incorporating non-determinism and partial ordering. This proves instrumental in addressing scenarios where users with similar interests may exhibit different feedback on certain items, reflecting the inhomogeneity of user-item interactions.

Detailed discussions on partially ordered neutrosophic hypergraphs and the processes of morphological dilation and erosion follow in the subsequent sections. The ensuing list outlines our primary contributions.

  • Pioneering the literature, we employ partially ordered neutrosophic hypergraphs for the first time to model higher-order relationships between users and items effectively. The presence of the indeterminacy parameter in neutrosophic hypergraphs significantly enhances the efficiency in addressing the inhomogeneity of user-item interactions. At the same time, partial ordering contributes to the generation of practical top-N recommendations.

  • Recommendation generation is facilitated by applying the morphological operation known as dilation.

  • The morphological operation of erosion, applied to the dual of the hypergraph, is utilized to furnish explanations for recommendations.

  • Diverging from prevalent assumptions in the literature, our approach does not presuppose that similar users invariably share identical interests across all items, acknowledging the inhomogeneity of user-item interactions.

  • Demonstrating superiority over comparable methods under analysis, our approach excels in precision, recall, NDCG, and hit ratio, achieving minimal root mean square error.

2 Research hypothesis and objectives

Research Hypothesis: Our primary hypothesis is that employing partially ordered neutrosophic hypergraphs can effectively model the higher-order relationships between users and items, thereby improving the precision and accuracy of recommendations in environments characterized by user-item interaction inhomogeneity.

2.1 Research objectives

  • Explore the Influence of Partially Ordered Neutrosophic Hypergraphs: Investigate how this innovative structure addresses the complexities of user-item interactions, particularly focusing on inhomogeneity.

  • Verify precision Improvements: Assess how addressing inhomogeneity through this method improves the precision of recommendations compared to traditional approaches.

  • Investigate Morphological Dilation Efficacy: Analyze the effectiveness of morphological dilation in generating Top-N recommendations with high precision and Recall metrics.

  • Validate Explanation Mechanisms: Evaluate how morphological erosion contributes to generating insightful explanations of the recommendations provided.

2.2 Justification for using neutrosophic hypergraphs

While many recommendation systems employ content-based or collaborative filtering methods, there are notable limitations, even in hybrid approaches. Hybrid methods attempt to mitigate some of these issues by combining multiple techniques, but challenges persist, including:

  • Cold Start Problem: Despite advancements, hybrid systems may still struggle to make accurate recommendations for new users or items due to limited interaction data.

  • Oversimplification: Content-based systems might restrict exposure to novel items, while collaborative systems tend to favor popular items. Even hybrid methods, although combining both, can sometimes reinforce these biases.

  • Inhomogeneity of Interactions: Many existing models, including hybrids, often assume uniformity in user preferences, failing to account for the diversity and evolving nature of interests across users.

Partially ordered neutrosophic hypergraphs address these challenges by:

  • Incorporating Non-Determinism: This allows for a more nuanced representation of user preferences, capturing the variability in feedback for similar items across different users.

  • Modeling Higher-Order Relationships: By effectively representing complex relationships and interactions, our approach reflects the real-world dynamics of user-item interactions, which are rarely linear or uniform.

  • Leveraging Partial Ordering: This enables the model to consider the sequence and dependencies between user interactions, enhancing the recommendation process by providing context.

2.3 Challenges

In educational and e-commerce environments, user-item interactions are inherently complex and multifaceted. Users may exhibit varying degrees of interest in different items based on prior experiences, social influences, and contextual factors. Traditional methods often struggle with these complexities, leading to less personalized and less effective recommendations. The incorporation of partially ordered neutrosophic hypergraphs provides a robust framework for addressing these challenges by:

  • Recognizing Diverse User Interests: This approach acknowledges that users may have unique preferences for certain items, which is critical in environments with diverse offerings.

  • Enhancing Representational Power: The model captures intricate dependencies and relationships, which traditional models may overlook, ultimately leading to more accurate and contextually relevant recommendations.

3 Related works

Recommendation systems play a pivotal role across various digital service platforms and continue to gain widespread popularity. This section delves into the contemporary landscape of course recommendation systems. Our study centers specifically on course recommendations to facilitate a comprehensive comparative analysis and enhance our grasp of the advancements in this domain.

Alatrash et al. (2024) use attention mechanism for weight assignment for the pair of items which are directly connected in the Knowledge Graph(KG). This mechanism combines symmetric square root normalization from LightGCN (He et al. 2017) with a semantic similarity function based on SBERT. Determining path costs to identify the highest weighted paths involves visual and textual explanations, with cosine similarity employed for calculating path costs. Frej et al. (2023) propose an Unrestricted Policy-Guided Path Reasoning (UPGPR) that leverages recent enhancements in Reinforcement Learning(RL) for reasoning based on Knowledge Graphs in recommendation systems. This approach not only ensures high predictive performance but also maintains inherent interpretability. The UPGPR model employs an RL agent to traverse the KG by considering the relationships between entities. It starts with a learner and concludes at the recommended course, offering a clear and understandable rationale. Premalatha et al. (2022) accurately map students and courses using LSTM and GRU models, while Xiao et al. (2018) integrate collaborative filtering, content-based filtering, and association rule mining for an efficient course recommendation system. Bagunaid et al. (2022) use DBSCAN for student categorization and RNN for scoring, ensuring tailored recommendations. Hazar et al. (2022) extract insights from comments on educational videos using a convolutional neural network. Ibrahim et al. (2018) integrate ontology mapping with content-based and collaborative filtering for enhanced recommendation efficiency. Garanayak et al. (2020) predict educational institute rankings through time series analysis.

Meddeb et al. (2021) recommend Arabic learning materials using various techniques, Khalid et al. (2021) group learners into clusters using hyperspheres, and Lang and Wang (2020) construct a knowledge graph from students’ learning records for personalized knowledge point recommendations. Lin et al. (2022) introduce HRRL, a context-aware reinforcement learning approach that integrates an attention-based recommendation model with a profile reviser utilizing Recurrent Reinforcement Learning (RRL) to exploit temporal context. Sebbaq and El Faddouli (2022) incorporate transfer learning using word2vec and a bidirectional GRU, enhancing the model’s classification capabilities. Li et al. (2023) introduce an Explainable Self-supervised Model (ESM) for predicting MOOC dropout, employing self-supervised training based on neural networks. Lin et al. (2023) suggest a multi-level representation learning framework for enhancing KG’s semantic representation and relation, incorporating course-level and concept-level representation. Liang et al. (2023) introduce a technique for combining a knowledge graph by simplifying the Graph Convolution Network model and integrating multi-path with knowledge graph embedding. Afzaal et al. (2023) utilize an algorithm for generating data-driven recommendations to implement targeted interventions, fostering self-regulation among university students.

Cui et al. (2022) present RSL-GRU, a novel architecture tailored for sequential learning with gated recurrent units. This model prioritizes the analysis of users’ sequential behaviors alongside the associated Knowledge Graphs (KGs), organized in chronological order. Klasnja-Milicevic and Milicevic (2023) unveil NCO-A, a Neural Co-Attention Model tailored specifically for providing Top-N Knowledge Concept Recommendations in MOOCs by harnessing heterogeneous data. Ai et al. (2018) present a framework for learning representations of a knowledge base for recommendation purposes, utilizing soft matching for generating personalized explanations. He et al. (2017) establish a comprehensive NCF framework that combines neural networks with collaborative filtering, improving the modeling of latent features associated with users and items. Xian et al. (2019) introduce a reinforcement learning-based method that incorporates the utilization of a soft reward strategy, applies user-conditional action pruning, and employs a multi-hop scoring strategy to efficiently sample diverse reasoning paths in recommendation systems.

La Gatta et al. (2022) propose a music recommendation system that uses hypergraph embeddings. From the embedded hypergraphs, they generate random walks and vertex embedding and provide top-N recommendations for each user. Yangyang et al. (2022) propose a PoI recommendation system using hypergraph embedding and logical matrix factorization. They use hypergraph embedding technology to identify users with similar points of interest to make recommendations. Zheng et al. (2018) suggest a novel social network hybrid recommender system based on hypergraph topological structure for which they use Hybrid Matrix Factorization(HMF) and analyze social relations using hypergraph topology. Gharahighehi et al. (2021) introduce a fair multi-stakeholder news recommender system that employs hypergraph ranking. This method improves the coverage of under-represented authors by modifying the query vectors dynamically by changing their weights, considering author’s popularity among users.

Xia et al. (2021) proposed a social recommendation system with a hypergraph attention network as its backbone. It extracts user and item feature vectors using a simple graph convolution neural network from a user-item interaction graph. Wang et al. (2022) use a hyperedge-based graph neural network for MOOC recommendation. They also suggest a hyperedge-based graph attention network. Short-term patterns are inferred from learned representations using a GRU-based sequence model.

Wang et al. (2020) use sequential hypergraphs for next-item recommendations. It represents the short-term item correlations using hypergraphs, and multi-order connections are retrieved from the hypergraph using multiple convolutional layers. Yu et al. (2021) suggest using a self-supervised multi-channel hypergraph convolutional network for social recommendation. Each channel in the network encodes the high-order user relation patterns using hypergraph convolution. These embeddings are collectively analyzed to get proper recommendations.

Zhu et al. (2021) argue that social relations influence different products differently. Users may have differing interests in certain products. So, they follow a user-item-user interaction modeling using hypergraphs to represent users with similar interests in a particular item. Chen et al. (2020c) employ a signed hypergraph-based neural network to model user-item interactions where positive signs indicate a positive inclination of the user towards an item and negative signs indicate the user’s disapproval of the item. Potential paths are extracted from the graph to facilitate item recommendations. Xu et al. (2023) use the hypergraph attention model to embed complex user-item interaction information. It follows information maximization, taking into account the social consistency of the interactions. Karantaidis et al. (2021) employ a completely dynamic algorithm based on hypergraphs that use hyperedge weight updation using gradient descent and optimize the hypergraph structure to incorporate changing user preferences.

Karantaidis et al. (2021) propose an adaptive method for image and tag recommendation, optimizing the structure, weights, and ranking vectors of a hypergraph for efficient recommendations. Their work thoroughly investigates different hyperparameters to maximize the precision of these recommendations. Ouyang et al. (2024) present a method in the educational domain that combines fusion graphs and hypergraphs to capture both higher-order and lower-order information. Additionally, they implement a layer dropout method to prevent overfitting of the model. Yang et al. (2024) employ an attention mechanism on hypergraphs to directly extract interaction relationships in a group recommendation environment within the educational domain. Wang et al. (2023) utilize deep reinforcement learning on heterogeneous graphs with graph neural networks and an attention mechanism. Their approach dynamically extracts the interests of learners to provide an interactive course recommendation system within the educational domain.

Eleftherakis et al. (2024) highlight demographic bias in neighborhood-learning models for collaborative filtering, where discriminatory patterns can emerge despite strong ranking performance. They introduce two novel algorithms: an enhanced Balanced Neighborhood Sparse Linear Method (BNSLIM) using Alternating Directions Method of Multipliers (ADMM) to optimize similarities and reduce training time, and Fairly Sparse Linear Regression (FSLR), which induces controlled sparsity to improve fairness.

The primary features of the latest advancements are outlined in Table 1, with particular emphasis placed on elucidating explainable educational recommenders.

Table 1 Overview of recent educational recommendation systems

4 Problem formulation

The sequencing of course recommendations in a course recommendation system is crucial due to the existence of prerequisite relationships among courses. Additionally, the assumption that similar users consistently have identical interests in items may need to be verified. For instance, some students might share standard core courses, yet their preferences for minor programs could vary significantly. Unfortunately, numerous works in the literature overlook these heterogeneous interests. Traditional pairwise relationships between users and items are inadequate to handle such diverse relations. Incorporating user-user-item relationships becomes essential to identify and comprehend these intricate and varied connections (Zhu et al. 2021).

4.1 Problem definition

Conventional recommendation systems tackle the recommendation challenge by utilizing binary representations of user-item interactions. Nevertheless, this method falls short in capturing the nuanced and non-uniform user preferences, which may or may not preserve a specific order. To address this limitation, higher-order data representations become essential to adeptly handle the intricate relationships between users and items. The problem at hand can be formulated as follows:

  • Input: User-item interaction data supplemented with relevant information such as reviews.

  • Output: A recommendation model tasked with predicting the probability of user ‘u’ interacting with item ‘i’ and also a globally consistent partial ordering of items.

Using the predicted ratings and partial ordering (Feng et al. 2018; Trotter Jr and Moore Jr 1976; Sidorenko 1994; Middendorf and Timkovsky 1999), our model can pick the top-N items the subject user will be interested in.

5 Proposed system

A neutrosophic hypergraph, denoted as \(H = (V, E, \mu _{+c}, \mu _{-c}, w_{cl})\), is employed to characterize user-item interactions. In this representation, H stands for the hypergraph, V is the set of hypernodes (\(V = \{v_1, v_2,..., v_{|V|}\}\)), E is the set of hyperedges (\(E = \{e_1, e_2,..., e_{|E|}\}\)), \(\mu _{+c}\) denotes the membership degree of a hypernode c in relation to the hyperedges it is part of, \(\mu _{-c}\) represents the non-membership degree of a hypernode c with respect to the hyperedges it belongs to, and \(w_{cl}\) signifies the indeterminacy associated with the hypernode c in relation to the hyperedge l to which it belongs. Additionally, it is ensured that \(e_i \ne \emptyset\) and \(\bigcup _{i=1}^{|E|} e_i = V\). A list of abbreviations used in the paper is given in Table 2.

Table 2 Symbols used and meaning

Hyperedges are employed to model users, while hypernodes represent items. The association of a hypernode with a hyperedge indicates that the user linked to the hyperedge has interacted with the item corresponding to that particular hypernode. Since a hyperedge can establish connections with varying numbers of other hyperedges and encompass diverse quantities of hypernodes, this representation efficiently captures higher-order relationships among users and items. Notably, the nodes within the hypergraph are partially ordered, reflecting the chronological order of users’ interactions with courses. It can be mathematically formulated as follows:

  • Establishing partial ordering with respect to chronological order of user-item interactions:

    $$\begin{aligned} & \forall \; h_1, h_2 \in H, \quad h_1 \le h_2 \quad \text {if } \quad t_1 \le t_2 \quad \text {and} \quad (h_1, h_2) \in R\\ & H = \{h_1, h_2, \ldots , h_n\} \end{aligned}$$
  • Compute values for each hypernode using the timestamp function:

    $$\begin{aligned} f: H \rightarrow \mathbb {R}, \quad f(h_i) = t_i \end{aligned}$$
  • Generate extended orders incorporating transitivity through the transitive closure:

    $$\begin{aligned} h_i \le h_j \quad \text {if } \quad t_i \le t_j \quad \text {and} \quad (\exists k \in H: h_i \le h_k \wedge h_k \le h_j) \quad \text {with transitivity} \end{aligned}$$
  • Arrange hypernodes in order considering their chronological sequence and the transitive closure:

    $$\begin{aligned} h_1 \le h_2 \le \ldots \le h_n \quad \text {if } \quad t_1 \le t_2 \quad \ldots \le t_n \quad \text {and} \quad (h_1, h_2,\ldots ,h_n) \in R \quad \forall h_i \in H \end{aligned}$$

In a partially ordered set of nodes, some nodes will be ordered concerning the other nodes, but all the nodes need not have to have such a relation. That is, if \(h_1 \le h_2 \le h_5 \le h_0\) is a partial ordered relation, \(h_2\) and \(h_5\) may have a clear relation between them that \(h_2\) always precedes \(h_5\). However, the other nodes may not even have a particular relation. This helps us to encode course pre-requisite relations more efficiently.

A sub-hypergraph, created to represent user-item interactions in the COCO dataset (Dessì et al. 2018), is depicted in Fig. 1.

Fig. 1
figure 1

Sub-hypergraph representation for the COCO dataset: the diagram illustrates a subset of the hypergraph structure used to model course recommendations in the COCO dataset. In this subgraph, courses are represented as hypernodes, and learners are the hyperedges. The entire hypergraph comprises over 43,000 hypernodes, 2.4 million hyperedges, and approximately 4.5 million memberships and indeterminacies, capturing the complex relationships between learners and their course enrollments. This sub-hypergraph demonstrates how frequent course patterns and learner groups form tightly connected communities, facilitating more effective course recommendations

This illustration showcases the associations between learners and courses within the COCO dataset, where hyperedges represent learners and hypernodes denote the courses undertaken by the respective learners (associated with the hyperedges to which they belong). The workflow of the proposed method is given in Fig. 2. Our system uses Algorithm 1 to assign parametric values to the hypernodes.

Fig. 2
figure 2

Workflow of the proposed course recommendation method: this figure outlines the workflow of the proposed recommendation system utilizing a Partially Ordered Neutrosophic Fuzzy Hypergraph (PONFHG). It begins by constructing a hypergraph with users as hyperedges and items as hypernodes, capturing their interactions. The system estimates parameters to assess relationships between users and items, applies fuzzy spectral clustering for grouping similar users, employs morphological dilation to expand the recommendation pool and, erosion for explaining the recommendations. The process culminates in generating top N recommendations, complemented by user similarity heat-maps and recommendation paths that explain the rationale behind the suggested courses

Algorithm 1
figure a

Assigning Parametric Values to Hypernodes

Membership degrees of all nodes are initialized to zero, and non-membership degrees are initialized to one. As a new node is encountered, its membership degree and non-membership degrees are updated using a ‘membership degree changing factor’, \(\mu\)c. It is defined as given in Eq. (1).

$$\begin{aligned} \mu c = \frac{1}{\text {frequency of the most frequent item in the dataset+1} } \end{aligned}$$
(1)

The factor influencing the change in membership degree(\(\mu c\)) guarantees that membership and non-membership degrees will fall within the interval [0,1]. Neutrosophic hypergraphs introduce the flexibility for hypernodes to possess distinct membership degrees in various hyperedges. This non-deterministic feature is particularly significant in recommendation scenarios, where two users might express differing opinions on certain items despite sharing similar feedback on others. The indeterminacy of a hypernode is calculated according to Eq. (2). For a hypernode ‘c’ with a membership degree (\(\mu _{+{c}}\)) of 0.8, a non-membership degree (\(\mu _{-{c}}\)) of 0.2, and a user ‘u’ giving a rating of 4 out of 5, the indeterminacy of ‘c’ (\(w_{cu}\)) is calculated as ((0.8\(-\)0.2)+4)/(5+1) = 0.7667. Given that different users may rate the same course differently, the indeterminacy allows for varying degrees of uncertainty for the same course in different hyperedges. For example, if another user ‘k’ rates the same course ‘c’ as 2 out of 5, the indeterminacy for ‘c’ with respect to ‘k’ (\(w_{ck}\)) is computed as ((0.8\(-\)0.2)+2)/(5+1) = 0.433. This accommodates the heterogeneity in user-item interactions, allowing users with similar preferences on most items to differ significantly on other items.

$$\begin{aligned} w_{iu} = \frac{(\mu _{+i} - \mu _{-i}) + r_{u,i}}{maximum\_user\_rating+1} \end{aligned}$$
(2)

To ascertain the degree of association between a hypernode and a hyperedge, we consider the membership degree of the node, its non-membership degree, and the user rating for the corresponding item. Equation (2) ensures that the indeterminacy values for all nodes fall within the interval [0, 1], facilitating subsequent processing. Algorithm 1 generates an adjacency matrix where the indeterminacy of items (row entries) associated with respective users (column entries) serves as the cell values. This adjacency matrix becomes the input for the subsequent module. A partially ordered neutrosophic hypergraph is illustrated in Fig. 3.

Fig. 3
figure 3

Conceptual diagram of a partially ordered neutrosophic hypergraph

For instance, C4 is a course completed by students S2 and S3, with a membership degree of 0.8, a non-membership degree of 0.5, and an indeterminacy of 0.1. It’s important to note that the indeterminacy of a hypernode varies across the hyperedges to which it belongs, although a single value is provided in the figure for convenience. An explicit ordering exists among the nodes: \(c_5 \le c_6 \le c_4 \le c_2\). Another ordering present is \(c_1 \le c_2\). However, the node \(c_3\) does not participate in any ordering. The application of topological sorting generates the overall partial ordering: \(c_5 \le c_6 \le c_4 \le c_1 \le c_2 \le c_3\). Importantly, this partial ordering encompasses nodes from different hyperedges and considers the transitivity of node ordering. Algorithm 2 establishes a partial ordering for the hypernodes. In steps 1–8, the algorithm identifies frequent course orderings (course sets) in the dataset using minimum support. Steps 9–19 determine the transitive closure for these frequent patterns, leveraging the transitivity of partially ordered relations to discover more extended patterns. Finally, step 20 conducts a topological sort on the frequent patterns to obtain the ultimate partial ordering using the learner-action timestamp data as the reference. Figure a depicts the partial ordering of hypernodes for a subset of the COCO dataset. In contrast, Fig. 4b illustrates the refined partial ordering after considering transitive relations among hypernodes. Consequently, Fig. 4b incorporates additional edges, representing extra relationships between hypernodes that are absent in Fig. 4a.

Algorithm 2
figure b

Constructing Partially Ordered Neutrosophic Fuzzy Hypergraph

The partially ordered neutrosophic hypergraph returned from Algorithm 2 is subjected to spectral clustering (Yang et al. 2021) as per Algorithm 3. Step 1 of the algorithm performs KNN (K-Nearest Neighbours) (Zhang and Zhou 2007) on the neutrosophic graph to get the distance matrix (which contains the distances to the nearest ‘kn’ neighbors of each hypernode) and indices of nearest neighbors. Steps 2 to 4 process each distance value in this matrix and normalize them as per the Eq. (3), where dist_mean is the mean of distances.

$$\begin{aligned} distance = e^ {\text {-}{\frac{distance^2}{dist\_mean ^2}}} \end{aligned}$$
(3)

Step 6 creates a sparse matrix representation namely, dist_csr using the distance data. Steps 7 to 8 calculate the row and column-wise sum of the dist_csr matrix, respectively. Using these matrices, a matrix Z is created as per the Eq. (4).

$$\begin{aligned} Z = \frac{1}{\sqrt{nodelist }}\times dist\_csr \times \frac{1}{\sqrt{edgelist}} \times row\_matrix(1_{kn}) \end{aligned}$$
(4)

The nodelist and edgelist are sparse matrices representing row sum and column sum of dist_csr matrix. The graph Laplacian for the neutrosophic hypergraph is obtained by taking the dot product of Z with its transpose. ‘kc’ number of eigenvectors are calculated from the Laplacian, and these eigenvectors are subjected to the Fuzzy C Means algorithm to get fuzzy clusters.

Algorithm 3
figure c

Fuzzy Spectral Clustering

Fig. 4
figure 4

Global partial ordering of hypernodes

The values of the number of neighbors (kn) and clusters (kc) undergo fine-tuning in the algorithm’s training phase. In the testing phase, each user for whom recommendations are to be generated is added to the cluster with the lowest Minkowski distance (Roche-Newton and Rudnev 2015) from the user’s feature vector. The graph morphological operation dilation (Ayyub et al. 1998) is employed to generate the recommendation list, following the steps outlined in Algorithm 4. In the algorithm, Step 1 involves sorting the soft clusters generated by Algorithm 3 in descending order of Minkowski distance from the user_feature_vector. Step 2 initializes the hypercluster (the set of clusters to which a particular user belongs). Step 3 identifies the hyperedge corresponding to the subject user. Steps 4–8 add all the soft clusters the subject user belongs to, to the hypercluster. Step 9 initializes the recommendation list. Steps 10–12 apply the dilation operation recursively until a recommendation list of the required length is obtained. Figure 5 illustrates the process of morphological dilation applied in our method.

Algorithm 4
figure d

Generating Recommendations.

Fig. 5
figure 5

Dilation of structuring element on hypergraph clusters

The structuring element S (corresponding to the user for whom recommendations are to be generated) undergoes dilation, resulting in the set of users (and, consequently, the set of items interacted with) who share at least one common item with the subject user among users belonging to the same hypergraph cluster. Mathematically, let \(L^n\) be the hyperedge corresponding to the user (who has interacted with n items already) for whom recommendations are to be made, and let H be the neutrosophic hypergraph modeling user-item interactions. We define the dilation of \(L^n\) on H as,

$$\begin{aligned} \delta (H|L^n)= \bigcup _{e \in E} e \quad \text {if} \quad L^n \cap e \ne \emptyset \end{aligned}$$
(5)

Where E is the set of hyperedges in H. Dilation operation can be applied recursively by taking the result of one dilation operation as a structuring element to another. For example, the secondary-level dilation can be explained as,

$$\begin{aligned} \delta (H|\delta (H|L^n))= \bigcup _{e \in E} e \quad \text {if} \quad \delta (H|L^n) \cap e \ne \emptyset \end{aligned}$$
(6)

Steps 13–16 apply polynomial regression to predict ratings of the items in the recommendation list. Step 17 sorts the recommendation list obtained in step 11 using the ratings obtained in step 14 as sorting criteria. The algorithm can generate recommendations in different fields like educational recommendations, e-commerce, entertainment, etc. The results are discussed in detail in the results section.

5.1 Generating explanations

We generate explanations by employing the morphological erosion process on the hypergraph’s dual. In the original hypergraph, users are depicted as hyperedges, and items are represented as hypernodes. As the recommendation process concludes with users being recommended a set of items, it becomes imperative to utilize the features of these recommended items for generating feasible explanations. To achieve this, we turn to the dual of the hypergraph. Interchanging hypernodes and hyperedges derive the dual. Consequently, users are now represented as hypernodes, and items are depicted as hyperedges. A hypernode signifies users interacting with items represented by the corresponding hyperedges. The dual of a neutrosophic hypergraph described as \(H = (V, E, \mu _{+c}, \mu _{-c}, w_{cl})\) can be mathematically expressed as follows:

Let \(H^* = (\hat{V}, \hat{E}, \mu _{+e}, \mu _{-e}, w_{lc})\) represent the dual neutrosophic hypergraph, where:

\(\hat{V}\) is the set of hyperedges in the dual hypergraph, i.e., \(\hat{V} = {e_1, e_2,..., e_{|E|}}\). \(\hat{E}\) is the set of hypernodes in the dual hypergraph, i.e., \(\hat{E} = {v_1, v_2,..., v_{|V|}}\). \(\mu _{+e}\) denotes the membership degree of a hyperedge e in relation to the hypernodes it is part of. \(\mu _{-e}\) represents the non-membership degree of a hyperedge e with respect to the hypernodes it belongs to. \(w_{ve}\) signifies the indeterminacy associated with the hyperedge e in relation to the hypernode v to which it belongs.

Additionally, the conditions \(\bigcup _{i=1}^{|E|} e_i = V\) and \(e_i \ne \emptyset\) ensure that each hypernode in the original hypergraph is now a hyperedge in the dual hypergraph. The union of all hyperedges in the dual hypergraph covers the set of hypernodes in the original hypergraph. The dual of the sub hypergraph shown in Fig. 1 is given in Fig. 6.

Fig. 6
figure 6

Dual hypergraph representation for a subset of the COCO dataset: this diagram illustrates the dual hypergraph created from a small subset of the COCO dataset. In this representation, hypernodes represent learners, and hyperedges correspond to courses. The interconnected regions represent groups of learners who are frequently taking the same sets of courses. This dual perspective helps analyze patterns in learner behavior and how specific groups of learners engage with particular sets of courses

The process of morphological erosion is explained by Eq. (7) and is illustrated in Fig. 7.

Fig. 7
figure 7

Erosion of structuring element on dual of the sub-hypergraph

We are applying erosion constrained by a threshold \(\Theta\).

$$\begin{aligned} \varepsilon (G|S,\theta )= \bigcup _{E_i \in E^n} \quad \bigcup _{s \in E_i} s \quad \text {if} \quad |s \cap E_{i}^{n} | \quad \ge \Theta \end{aligned}$$
(7)

If the structuring element intersects with more than a threshold \(\theta\) number of points in the hypergraph cluster, those points are returned as the result of erosion. In our context, it gives us information regarding users who have done at least \(\theta\) number of similar courses, which is a good measure of user similarity.

Once this information regarding user similarity is obtained, the explanations for recommendations can be given to user in different visual representations like recommendation paths, user-similarity heatmaps etc. The recommendation paths connecting the similar users can be extracted from the partially ordered neutrosophic hypergraphs. User similarity heatmap as illustrated in Fig. 8, can be generated to elucidate the recommendations.

Fig. 8
figure 8

User similarity HeatMap

This heatmap, which is created for user-37, reveals that user-37 exhibits the highest similarity to user 3, followed by users 21 and 35. The heatmap effectively communicates to user-37, for whom a specific recommendation was generated, by prioritizing the partial ordering of courses completed by users 3, 21, and 35 over other users. This gives the user valuable insights into the rationale behind the specific recommendation.

6 Experimental setup

The datasets used, the performance measures adopted, algorithms against which the method is compared and parameter settings are discussed in this session.

6.1 Datasets

We have used the educational dataset COCO (Dessì et al. 2018), amazon review datasetsFootnote 1 namely Beauty, Health, Sports, Office and Software, Goodreads book review dataset,Footnote 2 Movielens 1 Million datasetFootnote 3 and the Netflix prize datasetFootnote 4 for conducting experiments. The results are presented in Table 5.

6.2 Performance measures

The following metrics are used to assess the performance of the proposed algorithm.

  • Precision: Total number of items recommended that are relevant/Total number of recommended items (Ting 2010).

    $$\begin{aligned} precision @ k= \frac{|test\_set \cap top\_n|}{|top\_n|} \end{aligned}$$
    (8)

    Test_set is the set of items user has interacted with as per the dataset and top_n is the set of items recommended by the algorithm.

  • Recall: Total number of items recommended that are relevant/Total number of relevant items in the database (Ting 2010).

    $$\begin{aligned} recall @ k= \frac{|test\_set \cap top\_n|}{|test\_set|} \end{aligned}$$
    (9)
  • NDCG: It measures the quality of ranking (Järvelin and Kekäläinen 2002). It compares the position of the recommended items in the recommendation list against the actual order in which the user interacted with the items.

    $$\begin{aligned} NDCG @ k= \frac{DCG @ k}{iDCG @ k}, \end{aligned}$$
    (10)

    where

    $$\begin{aligned} DCG @ k= \sum _{i=1}^{n}\frac{2^{relevance_i - 1 }}{log_2(i+1)} \end{aligned}$$
    (11)

    Here iDCG is the DCG of the ideal order and \(relevance_i\) is the relevance score for the \(i \text{th}\) recommendation.

  • Hit Ratio: The fraction of users for which the correct answer is included in the recommendation list of length N.

    $$\begin{aligned} HR= \frac{U_{hit}^N}{U_{all}} \end{aligned}$$
    (12)

    Where \(U_{hit}^N\) is the number of users for whom correct recommendations were made in the recommendation list and \(U_{all}\) is the total number of users in the test set.

  • RMSE:

    $$\begin{aligned} RMSE = \sqrt{\frac{1}{N}\Sigma_{i=1}^{N}(R_{u,i} -\hat{R_{u,i}})^2} \end{aligned}$$
    (13)

    Here \(R_{u,i}\) is the rating given by the user ‘u’ to the product ‘i’ and \(\hat{R_{u,i}}\) is the rating the system has predicted for user ‘u’ on item ‘i’.

For precision, recall, NDCG, and hit ratio, higher values indicate better recommendations. For a good recommendation system, RMSE should be as low as possible.

6.3 Competitive algorithms for comparison

The proposed system has been compared with the state-of-the-art methods listed below:

  • Pop: This approach suggests the most popular item to the user.

  • NeuMF: (He et al. 2017) proposes a method in which neural collaborative filtering and generalized matrix factorization are fused to create a framework called Neural Matrix Factorization(NeuMF), which makes use of a multi-layer perceptron network to find the inner product of user and item latent features. The authors have yet to attempt to explain the recommendations generated, but this work still has good results.

  • CFKG: (Ai et al. 2018) constructs a user-item knowledge graph integrating user behaviors and item knowledge, learns KBE with heterogeneous relations, and uses user and item embeddings for personalized recommendations. It introduces a soft matching algorithm to explain recommendations by finding explanation paths in the latent knowledge base embedding space. It demonstrates good performance and flexibility with multiple relation types in real-world datasets.

  • PGPR: (Xian et al. 2019) apply reinforcement learning on knowledge graphs to perform accurate recommendations and proposes a Policy Guided Path Reasoning (PGPR) method to interpret the recommendations.

  • UPGPR: (Frej et al. 2023) propose a framework named Unrestricted Policy Guided Path Reasoning(UPGPR) which improves the PGPR method suggested by (Xian et al. 2019) by incorporating paths of any types and length.

  • EHCF: (Chen et al. 2020a) use a non-sampling transfer learning approach to model user-item relations across heterogeneous data with improved efficiency and performance.

  • LRGCN: (Chen et al. 2020b) remove non-linearities and introduce a residual network structure tailored to user-item interaction modeling.

  • LightGCN: (He et al. 2020) is a simplified graph convolutional network (GCN) model focused on collaborative filtering. The method aggregates neighborhood information by linearly propagating user and item embeddings on the interaction graph, using a weighted sum of embeddings across layers for final predictions.

  • SelfCF: (Zhou et al. 2023) introduce a self-supervised learning (SSL) approach to collaborative filtering (CF) that eliminates the need for negative sampling, simplifying the recommendation process. By augmenting the latent embeddings generated by various CF backbone networks, including graph-based models, SelfCF enhances user-item representation learning without relying on explicit negative instances.

  • FairMF: (Yao and Huang 2017) propose a framework named Fair Matrix Factorization(FairMF) which ensures fairness in recommendation process by penalizing overestimation and underestimation.

  • FSLR: (Eleftherakis et al. 2024) suggest Fairly Sparse Linear Regression (FSLR), which induces controlled sparsity to reveal inter-group correlations and improve fairness.

  • BNSLIM-ADMM (Eleftherakis et al. 2024) propose a framework named Balanced Neighborhood Sparse Linear Method leveraging the Alternating Direction Method of Multipliers (BNSLIM-ADMM) which improves the BNSLIM framework by updating all the similarities in parallel thereby greatly reducing the time complexity of training process.

We have tested the methods on the COCO dataset. The results of comparisons are shown in Fig. 17, and the superiority of the proposed method, PONFHG (Partially Ordered Neutrosophic Fuzzy Hyper Graphs), can be understood from the results.

6.4 Parameter settings

We have used polynomial regression for predicting item ratings as per algorithm 4. The degree of the polynomial to use is a hyper-parameter. We have done hyperparameter tuning by plotting degree versus RMSE, and the lowest RMSE was found at degree=2 as given in Fig. 9a. The number of clusters required for the fuzzy_C_Means algorithm also affects the goodness of clustering. Hence, it was fine-tuned by plotting the Fuzzy Partition Coefficient(FPC) (Skfuzzy) against different cluster numbers as given in Fig 9b. Clustering was performed with 71 clusters using the Minkowski distance metric, and a constrained erosion threshold of 3 was applied. For frequent pattern mining, the support threshold was set to 5, with a confidence level of 60%. Additionally, only learners who interacted with at least 10 courses were considered in the analysis, ensuring the robustness of pattern discovery. Values used for various parameters are given in Table 3. The code is publicly available.Footnote 5

Table 3 Parameter values
Fig. 9
figure 9

Fine-tuning hyper parameters

7 Results

Fig. 10
figure 10

An illustrative example of recommendations produced by the proposed approach: this figure presents the results of the proposed recommendation system, showcasing two clusters, four users, and the courses they have completed. Notably, it highlights two course recommendations generated for the learner with ID 35306780. A mapping of Course IDs to course names, along with the global partial ordering produced by the algorithm are given to enhance the understanding of the recommendations and their context within the hypergraph structure

Figure 10 provides an illustrative example of course recommendations generated by the proposed method. In this example, the recommendations are tailored for Learner ID: 35306780. Since the algorithm uses fuzzy spectral clustering, the learner is a member of multiple clusters with varying degrees of membership. Within Cluster 1, learners share common courses such as Course ID: 544860 (Learn Social Psychology Fundamentals), Course ID: 822160 (Have Better Conversations Today), and Course ID: 509618 (Amazing Psychology Experiments). Based on the global partial ordering, Course ID: 32939 \(\rightarrow\) Course ID: 544860 \(\rightarrow\) Course ID: 822160 \(\rightarrow\) Course ID: 509618 \(\rightarrow\) Course ID: 341182, Course ID: 341182 (Master Your Brain – Neuroscience for Personal Development) is recommended to Learner ID: 35306780 from Cluster 1. This course shares characteristics with others the learner has previously completed, ensuring the recommendation aligns with their past choices and preferences.

On the other hand, the system also recommends a more diverse course from Cluster 2, where Learner ID: 35306780 is a member. In this case, Course ID: 32939 (Astronomy – State of the Art) is recommended. Cluster 2 represents a different group of learners with distinct preferences and behaviors. Therefore, recommending a course from this cluster allows the learner to explore new and diverse learning paths.

Our work addresses six important research questions.

  • RQ1: How important is the use of neutrosophic hypergraphs in the recommendation scenario? Does it improve the performance?

  • RQ2: How does the partial ordering of nodes help?

  • RQ3: How well is the proposed method addressing the inhomogeneity of user-item interactions?

  • RQ4: How is the algorithm’s performance in different real-life datasets?

  • RQ5: Does the proposed method work better than state-of-the-art algorithms?

  • RQ6: How apt are the morphological operations erosion and dilation in the process of recommendation and explainability?

The following sub-sections answer these research questions.

7.1 Addressing RQ1: Relevance of neutrosophic hypergraphs

Numerous research endeavors in the existing literature leverage hypergraphs for recommendation systems. Our work distinguishes itself as the pioneering attempt to employ neutrosophic hypergraphs in this domain. As expounded earlier in this study, the inherent indeterminacy associated with neutrosophic hypergraphs enhances their efficacy in efficiently modeling user-item interactions. To substantiate this claim, we conducted an experiment where we removed the indeterminacy component, transforming the neutrosophic hypergraph into a simple intuitionistic fuzzy hypergraph (Parvathi et al. 2009). The results, presented in Fig. 11a, showcase the recommendation accuracy in both frameworks, utilizing the metrics, precision, recall, NDCG, and HR. Clearly, the implementation utilizing a neutrosophic hypergraph outperforms its counterpart lacking indeterminacy, underscoring the significance of neutrosophic hypergraphs in generating recommendations.

Fig. 11
figure 11

Relevance of indeterminacy and fuzzy clustering

Furthermore, we performed fuzzy spectral clustering on the neutrosophic hypergraph to discern meaningful user groups. This process involves constructing a graph Laplacian from the adjacency matrix, followed by fuzzy c-means clustering of the eigenvectors, as outlined in Algorithm 3. We omitted the fuzzy clustering module to underscore the indispensability of fuzzy spectral clustering in producing high-quality soft clusters and employed the k-means algorithm instead. The comparative results in terms of precision, recall, NDCG, and HR, as depicted in Fig. 11b and Table 4, affirm that fuzzy spectral clustering outperforms the alternative approach.

Table 4 Results for fuzzy hypergraph (without indeterminacy) and dropping fuzzy clustering

7.2 Addressing RQ2: Relevance of partial ordering of nodes

Employing a partial ordering approach in a recommendation system yields numerous advantages. This methodology introduces flexibility into the recommendation generation process, deviating from strict adherence to the total order. When a user finds multiple items equally preferable, this approach accommodates such scenarios, resulting in more diverse and personalized recommendations. Addressing ties gracefully, especially when items have comparable relevance to a user, avoids arbitrary choices and respects the partial order by acknowledging the similarity between items.

The application of partial ordering considers the relationships between items, enabling the recommendation system to make more accurate predictions. This approach captures the underlying structure in user-item interactions, enhancing the precision of recommending items likely to pique the user’s interest. In instances with a long-tail distribution of items, where some are popular, and many are less so, partial ordering provides a nuanced understanding of user preferences across the entire spectrum. This subtle understanding aids in recommending niche or less popular items to interested users.

As user preferences evolve, the adaptability of a recommendation system becomes crucial. Partial ordering allows the system to adjust more effectively to changes in user preferences and the introduction of new items without necessitating drastic modifications. By incorporating partial ordering, the recommendation system inherently introduces diversity into its recommendations, catering to users with preferences spanning different categories or characteristics.

Additionally, partial ordering contributes to the system’s robustness in the face of noisy data or outliers in user-item interactions. This approach enables the system to focus on the overall preference structure rather than being unduly influenced by isolated and potentially misleading interactions. Finally, partial ordering facilitates the provision of explanations for recommendations. Users can comprehend why specific items are recommended based on their preferences and the relationships among items within the partial order. For instance, a path extracted from the hypergraph is given in Fig. 12. If the user has done the course numbered 93, by virtue of global partial ordering employed in the hypergraph, the probability is high that the next course the user will choose could be 3 or 199. This accounts for the high NDCG values for the proposed method. Figure 13 shows the results of evaluation metrics on a Neutrosophic Fuzzy Hypergraph (NFHG) without partial ordering and on a partially ordered neutrosophic hyper graph (PONFHG). PONFHG has higher values for all the parameters which clearly underlines the relevance of partial ordering. Theorem 1 mathematically verifies the same.

Fig. 12
figure 12

A partially ordered path extracted from the hypergraph

Fig. 13
figure 13

Significance of partial ordering of hypernodes

Theorem 1

Partial ordering improves NDCG values.

Proof

Let I: Set of all items, \(\le _t\): Partial ordering relation on I based on chronological order, R: Set of recommended items, \(rel\_i\): Relevance score of the i-th item in R, NDCG(R): NDCG score for a set of recommended items R.

\(\le _t\) is a partial ordering relation on I based on the chronological order of user-item interactions.

The NDCG is defined as: \(NDCG(R)=\frac{DCG(R)}{iDCG(R)}\)

Where DCG is calculated as: \(DCG(R)=\sum _{i=1}^{|R|}\frac{2^{relevance_i - 1 }}{log_2(i+1)}\)

Ideal Discounted Cumulative Gain (IDCG) is obtained by sorting items based on their relevance scores in descending order: \(IDCG(R)=\sum _{i=1}^{|R|}\frac{2^{relevance_i ideal - 1 }}{log_2(i+1)}\)

Let \(rel_i(t)\) represent the relevance score of the i-th item at time t. So, following the chronological order, \(i\le _t j \longrightarrow rel_i(t) \ge rel_j(t)\)

Expressing the NDCG as a function of time, \(NDCG_t(R)=\frac{DCG_t(R)}{iDCG_t(R)}\)

The DCG at time t is given by: \(DCG_t(R)=\sum _{i=1}^{|R|}\frac{2^{relevance_i - 1 }}{log_2(i+1)}\)

Formalizing IDCG at time t, \(IDCG_t(R)=\sum _{i=1}^{|R|}\frac{2^{relevance_i ideal - 1 }}{log_2(i+1)}\)

Now, let’s consider the NDCG scores at two different times, say \(t_1\) and \(t_2\), where \(t_1 < t_2\).

$$\begin{aligned} NDCG_{t_1}(R) = \frac{DCG_{t_1}(R)}{iDCG_{t_1}(R)} \\ NDCG_{t_2}(R) = \frac{DCG_{t_2}(R)}{iDCG_{t_2}(R)} \end{aligned}$$

Since the partial ordering \(\le _t\) is based on chronological order, items at time \(t_1\) are ordered chronologically earlier than items at time \(t_2\).

This chronological ordering implies that \(rel_i(t_1) \ge rel_j(t_2)\) for all i and j.

Now, let’s compare \(DCG_{t_1}(R)\) and \(DCG_{t_2}(R)\):

$$\begin{aligned} DCG_{t_1}(R)= & \sum _{i=1}^{|R|} \frac{2^{rel_i(t_1) - 1}}{\log _2(i+1)} \\ DCG_{t_2}(R)= & \sum _{i=1}^{|R|} \frac{2^{rel_i(t_2) - 1}}{\log _2(i+1)} \end{aligned}$$

Since \(rel_i(t_1) \ge rel_i(t_2)\) for all i, it follows that \(2^{rel_i(t_1) - 1} \ge 2^{rel_i(t_2) - 1}\).

Therefore, each term in the summation for \(DCG_{t_1}(R)\) is greater than or equal to the corresponding term in the summation for \(DCG_{t_2}(R)\).

This implies that \(DCG_{t_1}(R) \ge DCG_{t_2}(R)\).

Similarly, considering \(iDCG_{t_1}(R) \ge iDCG_{t_2}(R)\), we can conclude that \(NDCG_{t_1}(R) \ge NDCG_{t_2}(R)\).

Hence, the NDCG scores decrease over time, and this change is by partial ordering based on chronological order. The formalization of IDCG at different times ensures that the improvement is consistent with the relevance scores at those respective times. \(\square\)

7.3 Adressing RQ3: Inhomogeneity of user-item interactions

The concept of inhomogeneity in user-item interactions suggests that users may share similar interests only in a subset of items and might have conflicting preferences for other items. Unfortunately, many existing recommendation systems in the literature overlook this inhomogeneity, assuming that similar users always have congruent interests in similar items. This erroneous assumption can significantly diminish recommendation accuracy.

To address this limitation, we consider the inhomogeneity of user-item interactions by assigning varying weights to the same item in the hyperedges of different users. This is achieved by incorporating the indeterminacy component, as explained in the ‘Proposed System’ section. By considering and accommodating inhomogeneity, our approach yields improved performance. The Fig. 14 shows a histogram of keywords generated from descriptions of courses recommended to a user.

Fig. 14
figure 14

Histogram of keywords from recommended courses

The figure illustrates the top words extracted from course descriptions, where mathematical terms like “Statistics,” “Probability,” and “Mathematics” are among the most frequent. This indicates that many of the courses recommended to the user are related to mathematical subjects. However, there are also terms from other fields, such as “Psychology,” “Economics,” and “Electricity,” reflecting that the algorithm has captured the user’s diverse interests. This diversity arises because the user is a member of multiple fuzzy clusters, allowing for more comprehensive and well-rounded recommendations.

7.4 Addressing RQ4: Performance evaluation

To prove the efficiency of the proposed method, we have tested it using popular datasets in educational and e-commerce domains. In e-commerce datasets, information such as seller rating, user review, watch time, etc., can replace different parameters used in making educational recommendations.

Fig. 15
figure 15

Evaluation parameters at different values of N (dataset: COCO)

The results of evaluating the method on the famous educational dataset, COCO, for different sizes of the recommendation list(N) are given in Fig. 15. All the parameters achieve pretty good values. The proof that the same method works well in other domains is shown in Fig. 16 and Table 5.

The results indicate that the recommendation model shows strong performance in terms of NDCG@N, precision@, recall@N and HR@N, especially as the value of N increases. This suggests that the model successfully retrieves relevant items in the top N recommendations, with a high proportion of relevant items being identified as N grows. The global partial ordering of hypernodes plays a crucial role in improving the model’s NDCG values by ensuring that the most relevant items are ranked higher in the recommendation list.

Fig. 16
figure 16

Evaluation results for real-world datasets

Table 5 Evaluation parameters at different values of N

7.5 Addressing RQ5: Comparison with state of the arts

The system outperforms the state-of-the-art methods mentioned in the “6.3 Competitive Algorithms for Comparison” section. The comparison results are shown in Figs. 17,  18, Tables 6, and 7.

Fig. 17
figure 17

Comparison with state-of-the-arts on COCO dataset

Table 6 Comparisons with state-of-the-art methods on COCO dataset
Table 7 Comparisons with state-of-the-art methods on ML1M dataset
Fig. 18
figure 18

Comparison with state-of-the-arts on ML1M dataset

The results of the proposed approach, PONFHG, demonstrate competitive performance against state-of-the-art methods on the COCO dataset. PONFHG achieves the highest HR of 0.27, indicating its strong ability to recommend relevant courses that match users’ preferences. Additionally, it achieves a recall score of 0.1599, reflecting its effectiveness in retrieving relevant courses from the recommendation pool. While its NDCG (0.129) is slightly lower than the FSLR method (0.157), PONFHG offers a notable advantage by explaining its recommendations. In terms of recall, FSLR scores higher (0.197) compared to PONFHG (0.1599), but PONFHG strikes a balance between precision and recall, excelling in HR and precision, and maintaining transparency in top recommendations. In the e-commerce domain (ML1M dataset) also, PONFHG outperforms majority of the competing methods. The results highlight that PONFHG is well-suited for educational recommendations, primarily due to its innovative use of global partial ordering of nodes. This approach effectively captures and integrates prerequisite relationships between courses, allowing for more accurate and context-aware recommendations. These advantages position PONFHG as a good solution for educational recommendation systems.

7.6 Addressing RQ6: How apt are the morphological operations erosion and dilation in the process of recommendation and explainability?

In this section, we aim to provide a mathematical validation of the accuracy of the processes involved in morphological dilation (for generating recommendation lists) and erosion (for generating user-similarity estimates for explanations).

7.6.1 Mathematical verification of dilation

Theorem 2

\(\begin{aligned} Course\_Done&(S_i,C_m) \wedge Course\_Done(S_i,C_n) \wedge \\& Course\_Done(S_j,C_m) \Longrightarrow Course\_Done(S_j,C_n)\end{aligned}\) given that

  1. (i)

    \(Cluster(S_i,C) \wedge Cluster(S_j,C) \Longrightarrow Similar(S_i,S_j)\)

  2. (ii)

    \(Similar(S_i,S_j) \Longrightarrow \exists x Course\_Done(S_i,x) \wedge Course\_Done(S_j,x)\)

  3. (iii)

    \(Cluster(S_j,C)\) and

  4. (iv)

    \(Cluster(S_i,C)\)

Proof

Proof by resolution:

  1. 1.

    \(Similar(S_i, S_j) \Longrightarrow \exists x Course\_Done(S_i, x) \wedge Course\_Done(S_j, x)\) Applying Skolemization on (1):

  2. 2.

    \(Similar(S_i, S_j) \Longrightarrow Course\_Done(S_i, y) \wedge Course\_Done(S_j, y)\)

  3. 3.

    \(Cluster(S_i, C) \wedge Cluster(S_j, C) \Longrightarrow Similar(S_i, S_j)\)

  4. 4.

    \(Cluster(S_i, C) \wedge Cluster(S_j, C) \Longrightarrow Course\_Done(S_i, y) \wedge Course\_Done(S_j, y)\)

  5. 5.

    \(Cluster(S_i, C) \wedge Cluster(S_j, C) \Longrightarrow Course\_Done(S_i, C_m) \wedge Course\_Done(S_j, C_m)\)

  6. 6.

    \(Cluster(S_i, C) \Longrightarrow Course\_Done(S_i, C_m)\)

  7. 7.

    \(Cluster(S_j, C) \Longrightarrow Course\_Done(S_j, C_m)\)

  8. 8.

    \(Cluster(S_j, C) \Longrightarrow Course\_Done(S_j, C_n)\)

    Applying Resolution on (5), (7) and (8),

  9. 9.

    \(\begin{aligned} Course\_Done&(S_i, C_m) \wedge Course\_Done(S_i, C_n) \wedge \\& Course\_Done(S_j, C_m) \Longrightarrow Course\_Done(S_j, C_n)\end{aligned} \square\)

Here \(Course\_Done(S_j, x)\) indicates that the student \(S_j\) has done the course x, \(Cluster(S_j, C)\) indicates that the student \(S_j\) belongs to cluster C and \(Similar(S_i, S_j)\) denotes that both \(S_i\) and \(S_j\) are similar. Theorem 2 proves using resolution that if two users belong to the same clusters, it might imply that they have done similar courses in the past and will continue to do so in the future.

7.6.2 Mathematical verification of erosion

Theorem 3 gives the mathematical verification of erosion:

Theorem 3

\(\exists c Student\_Enrolls(S_i,c) \wedge Student\_Enrolls(S_j,c) \implies Similar(S_i,S_j)\) given that, \(\forall x \forall y, \exists c Student\_enrolls(x,c) \wedge Student\_enrolls(y,c) \implies Similar (x,y)\).

Proof

Given that,

  1. 1.

    \(\exists c Student\_Enrolls(S_i,c)\)

  2. 2.

    \(\exists c Student\_Enrolls(S_j,c)\)

  3. 3.

    \(\forall x \forall y Student\_enrolls(x,C) \wedge Student\_enrolls(y,C) \implies Similar (x,y)\) Dropping universal quantifiers,

  4. 4.

    \(Student\_enrolls(x,c) \wedge Student\_enrolls(y,c) \implies Similar (x,y)\) Applying existential instantiation,

  5. 5.

    \(Student\_enrolls(x, c_1) \wedge Student\_enrolls(y, c_1)\) following implication

  6. 6.

    Similar(xy)

\(\square\)

Here, the predicate \(Student\_enrolls(S_i, c)\) means that the student \(S_i\) has enrolled for the course c.

8 Conclusion

Unveiling higher-order relationships between users and items is pivotal in generating precise recommendations. Our approach introduces the utilization of Partially Ordered Neutrosophic Hypergraphs, a novel concept in the literature, to capture intricate relationships between users and items. This innovative modeling technique yields superior performance across key evaluation metrics such as precision (2.75% on average), recall (15% on average), NDCG (11.5% on average), hit ratio (26% on average), and RMSE (0.43 on average). The inherent non-determinism within neutrosophic hypergraphs proves advantageous in representing users’ diverse interests in various items, addressing the challenge of inhomogeneity and ultimately leading to personalized recommendations. Leveraging partial ordering of hypernodes enhances our ability to predict the sequential order in which users may interact with items, resulting in elevated NDCG values for our proposed model.

To generate top-N recommendations, we employ the morphological operation of dilation on fuzzy clusters of users obtained through fuzzy spectral clustering on neutrosophic hypergraphs. This operation effectively incorporates content-based and collaborative information, enriching our understanding of user-item interactions. Additionally, the morphological erosion operation facilitates the generation of insightful explanations for recommendations based on user similarity measures.

Our method surpasses state-of-the-art approaches across diverse domains, as evidenced by rigorous testing in course recommendation and e-commerce domains using publicly available datasets. The superior performance across various evaluation parameters underscores the efficiency of our approach in addressing the inhomogeneity of user-item interactions, leading to highly accurate recommendations.

While the inherent non-determinism of neutrosophic hypergraphs allows for a more nuanced representation of user preferences, managing this indeterminacy effectively poses a challenge. Developing methods to quantify and utilize this indeterminacy in a way that enhances recommendation accuracy is an area that requires further exploration. Our research will focus on exploring advanced statistical and mathematical techniques to better quantify the indeterminate relationships inherent in user-item interactions. By developing robust frameworks that accurately capture and represent the full spectrum of user preferences, we aim to enhance the model’s capability to provide more precise and personalized recommendations.