Dynamic granulovascular graph neural network working method for user and item recommendation

Abstract

The application provides a dynamic granular ball graph neural network working method for user and item recommendation, and belongs to the technical fields of recommendation systems and graph neural networks. The method comprises the following steps: S1, constructing user-side and item-side granular balls in a low-dimensional spectral space, updating the granular balls according to the node representation of each propagation layer, calculating the membership of nodes to the granular balls by using a radius-normalized distance, and defining the collaborative purity according to the interaction distribution of the granular balls on the opposite granular balls; S2, coupling fine-grained interaction modeling and coarse-grained collaborative abstraction, and performing message propagation along two complementary paths; S3, performing bilateral degree perception gating, and adaptively balancing the contributions of the two propagation paths for each node; and S4, aggregating the representations of each layer by weighted summation to obtain the final embedding of the user side and the item side, finally adopting a Bayesian personalized ranking loss to optimize the candidate item score, so as to play a good recommendation role in ranking the user and the item.

Description

Technical Field

[0001] This invention relates to the fields of recommendation systems and graph neural networks, specifically to a dynamic particle-sphere graph neural network working method for user and item recommendations. Background Technology

[0002] In existing technologies, recommender systems alleviate information overload by providing users with personalized content and are widely used in various internet services. In large-scale real-world applications, most recommender models rely primarily on implicit feedback such as clicks, browsing, and purchases. However, implicit signals are inherently noisy, sparse, and semantically uncertain, meaning that observed interactions cannot fully and reliably reflect users' true preferences. Therefore, how to learn robust representations from sparse and noisy implicit feedback remains a fundamental problem facing modern recommender systems.

[0003] Over the past decade, the modeling paradigm for recommender systems has continuously evolved. Early collaborative filtering methods, along with representation learning methods such as matrix factorization and deep neural networks, typically represented users and items in a shared latent space learned from observation interactions. In recent years, graph neural networks have gradually become an important paradigm for recommender systems: they explicitly model user-item interactions as graphs and propagate collaborative signals through multi-hop neighborhoods. Representative models such as NGCF (Neural Graph Collaborative Filtering) and LightGCN (Lightweight Graph Convolutional Recommendation Model) demonstrate that utilizing higher-order connectivity in the interaction graph can significantly improve recommendation performance, especially in scenarios with sparse interactions. To enhance representation robustness, graph contrastive learning and data augmentation methods further construct auxiliary views such as node discarding, edge perturbation, or subgraph sampling to provide self-supervised signals. Despite achieving good results, most of these methods still operate at the finest level of the recommender graph, i.e., using individual user, item, and observation interactions as message propagation units. Since observation edges are only incomplete and uncertain projections of users' true preferences, this paradigm remains vulnerable to sparse, noisy implicit feedback.

[0004] Fine-grained interaction propagation has two inherent limitations. First, implicit interactions are noisy and have ambiguous behavioral meanings. When unreliable edges participate in neighborhood aggregation, noise propagates repeatedly in the graph, causing shifts in user and item representations; this problem is more severe for low-degree users and long-tail items due to the scarcity and instability of available observations. As shown in Figure 1(a), on a training graph with multiple perturbations, the normalized Jensen-Shannon divergence of coarse-grained collaborative preference statistics is significantly lower than that of fine-grained user statistics, especially for low-degree users. Second, repeated propagation on fine-grained graphs amplifies unreliable signals and reduces representational diversity. As shown in Figure 1(b), as the propagation depth of LightGCN increases, the normalized effective rank and numerical rank of user representations continuously decrease, reflecting representational collapse and weakened personalization capabilities. These observations suggest that robust graph recommendation needs to simultaneously preserve fine-grained personalized interactions and utilize more stable coarse-grained collaborative patterns. Summary of the Invention

[0005] This invention aims to at least solve the technical problems existing in the prior art, and in particular, it innovatively proposes a working method of dynamic particle-sphere graph neural network for user and item recommendation.

[0006] To achieve the above-mentioned objectives of the present invention, the present invention provides a method for a dynamic particle-sphere graph neural network for user and item recommendations, comprising the following steps:

[0007] S1. Construct user-side and item-side spheres in the low-dimensional spectral space. Update the spheres according to the node representation of each propagation layer. Calculate the membership degree from the node to the sphere using the radius-normalized distance. Define the cooperative purity based on the interaction distribution of the spheres on the opposite spheres.

[0008] S2 couples fine-grained interaction modeling with coarse-grained collaborative abstraction, and propagates messages along two complementary paths;

[0009] S3 performs bilateral sensing gating, adaptively balancing the contributions of the two propagation paths for each node;

[0010] S4 aggregates the representations of each layer by weighted summation to obtain the final embeddings on the user side and the item side, and finally uses Bayesian personalized ranking loss to optimize the candidate item scores.

[0011] In a preferred embodiment of the above technical solution, constructing user-side and item-side spheres in the low-dimensional spectral space in step S1 includes:

[0012] S1-1, for the normalized interaction matrix Perform randomized singular value decomposition:

[0013] ;

[0014] in, , , , respectively denote the truncation of the left singular vector, the singular value matrix, and the right singular vector; transpose symbol This indicates that the rows and columns of the matrix are interchanged; For randomized singular value decomposition; Let be the set of real numbers; d is the embedding dimension; For the user feature matrix, It is a singular value matrix. The feature matrix of the items;

[0015] Based on randomized singular value decomposition, the initial user embedding and item embedding are constructed as follows:

[0016] ;

[0017] ;

[0018] in, The square root of the singular value matrix; embedding and As a trainable initial representation, it is updated through backpropagation during model optimization.

[0019] In a preferred embodiment of the above technical solution, step S1 further includes:

[0020] S1-2, in and Performing lightweight K-Means yields the initial user-side particle center. and the center of the sphere on the side of the item ,in For the first Each user's lateral granule center, For the first The center of the side particle of each item;

[0021] After initialization, the model further refines the spheres through radius-aware membership learning and collaborative purity evaluation;

[0022] For each initial sphere, initialize its radius with the average distance from its node to the initial center:

[0023] ;

[0024] ;

[0025] in, Let be the radius of the k-th initial user-side particle in layer 0. Let be the radius of the s-th initial item side particle in layer 0; The set of nodes assigned to the k-th initial user-side sphere. ϵ is the set of nodes assigned to the s-th initial item-side sphere; ϵ is a minimal positive constant used for numerical stability; k is the user-side sphere index, and s is the item-side sphere index; Embed the 0th layer input for user u. For item i, embed the 0th layer of the input; Let be the center vector of the k-th user granularity in layer 0. Let be the center vector of the s-th item at the 0th layer.

[0026] In a preferred embodiment of the above technical solution, updating the particle based on the node representation of each propagation layer in S1 includes:

[0027] S1-3, for user-side nodes With the User lateral granules Item side nodes With the Item side ball The normalized distance is defined as follows:

[0028] ;

[0029] ;

[0030] in, Let be the normalized distance from user u to the k-th user-side particle; Let be the normalized distance from item i to the s-th item's side particle sphere; The input embedding of user u before propagation at layer l-1, The input embedding for item i before propagation at layer (l-1);

[0031] Temporary membership degree is calculated as follows:

[0032] ;

[0033] ;

[0034] in, Let be the temporary membership degree of user u at layer l to the granular ball of user k. Let i be the temporary membership degree of item i in layer l to the side particle of item s; is the temperature coefficient; k is the user-side particle index, and s is the item-side particle index.

[0035] In a preferred embodiment of the above technical solution, step S1 further includes:

[0036] S1-4, using temporary membership, refresh the center of the particle on both the user side and the item side by weighted average:

[0037] ;

[0038] ;

[0039] in, For user side-sphere refresh center, Refresh the center of the item's side pellets;

[0040] To avoid excessively rapid prototype drift between different layers, an exponential moving average is used to update the particle center:

[0041] ;

[0042] ;

[0043] in, For user lateral sphere update center, Update the center for the side particles of the item; Control the smoothness of the update process;

[0044] The particle radius is also updated using a layer-adaptive approach. For the k-th user-side particle, the refresh radius is defined by the weighted root mean square distance.

[0045] ;

[0046] ;

[0047] in, Let the refresh radius of the k-th user-side particle in layer l be . Refresh the radius of the side particle sphere of the s-th item in the l-th layer;

[0048] To ensure stability, both the user-side particle radius and the item-side particle radius are updated using an exponential moving average:

[0049] ;

[0050] ;

[0051] in, Let be the radius of the k-th user-side particle in the l-th layer. Let be the radius of the s-th item's side particle in the l-th layer; This means truncating x to the interval. ; , where is the radius momentum; , These are the lower and upper limits of the radius to prevent particle degradation, respectively.

[0052] In the preferred embodiment of the above technical solution, the calculation of the membership degree from the node to the particle using the radius-normalized distance in step S1 includes:

[0053] S1-5, for user-side nodes With the User lateral granules Item side node With the Item side ball ,definition:

[0054] ;

[0055] ;

[0056] in, The normalized distance from user u to the k-th user side sphere is based on the updated user u. This is the normalized distance from the updated item i to the s-th item's side sphere;

[0057] Soft membership degree is calculated as follows:

[0058] ;

[0059] ;

[0060] in, Let be the soft membership degree of user u at layer l to the granular ball of user k. Let be the soft membership degree of item i in layer l to the side granular ball of item s. Temperature coefficient to control the sharpness of preference distribution; For the total number of side balls of the user, The total number of particles on the item side; the user-side membership matrix is ​​denoted as... The item-side membership matrix is ​​denoted as ;

[0061] To balance efficiency and interpretability, only the Top-T membership degrees of each node are retained and renormalized, where the user-side membership is taken as... Item side retrieval The T parameter is a sparsification hyperparameter, representing the membership degree to be retained after truncation and normalization. and Each row contains at most T non-zero elements.

[0062] In a preferred embodiment of the above technical solution, the definition of cooperative purity in S1 based on the interactive distribution of granules on the opposite granules includes:

[0063] S1-6, pp. The soft sphere-level interaction matrix of the layer is calculated as follows:

[0064] ;

[0065] in, This represents the aggregation interaction strength between the k-th user-side particle and the s-th item-side particle in layer l; based on For each particle, a normalized cooperative preference distribution is constructed, and its purity is measured using normalized entropy. The higher the purity, the more concentrated the cooperative preference of the particle, and the more reliable it is as a coarse-grained cooperative unit.

[0066] For user side particles Its cooperative preference distribution on the item side sphere is defined as:

[0067] ;

[0068] Its co-purity is measured using normalized entropy:

[0069] ;

[0070] Similarly, for the side particles of the object. ,definition:

[0071] ;

[0072] as well as,

[0073] ;

[0074] in, Let be the probability of collaborative preference between the k-th particle on the user side of layer l and the s-th particle on the item side. Let be the probability of collaborative preference between the s-th particle on the item side and the k-th particle on the user side; The interaction matrix of the l-th layer granular level The (k,s)th element, The interaction matrix of the l-th layer granular level The (s,k)th element; Let k be the co-purity of the k-th particle on the user side of layer l. Let be the cooperative purity of the s-th particle on the l-th layer of the item; The larger the value, the more concentrated the cooperative preference distribution of the corresponding particles, and the more reliable they are as coarse-grained cooperative units; Smaller values ​​indicate that the particles contain a mixture of multiple heterogeneous preference patterns.

[0075] In a preferred embodiment of the above technical solution, message propagation along two complementary paths in step S2 includes:

[0076] Fine-grained paths operate on the user-item interaction graph, while coarse-grained paths operate on a dynamically constructed bipartite granular graph. Fine-grained path propagation performs normalized neighborhood aggregation on the original user-item interaction graph to preserve local cooperative signals. Coarse-grained path propagation includes three stages: node-to-granularity abstraction, interaction between user-side granularities and item-side granularities, and refinement from granularities to nodes.

[0077] The normalized neighborhood aggregation is defined as follows:

[0078] ;

[0079] ;

[0080] in, This is the normalized adjacency matrix of the bipartite interaction graph; For the neighborhood aggregation embedding of user u in layer l, For the neighborhood aggregation embedding of item i in the l-th layer; For user u, the set of neighborhood items Let i be the set of neighboring users of item i;

[0081] The abstraction from node to particle is that node features are aggregated into the corresponding particle to form a prototype, and strict mean pooling normalization is performed using the hyper-edge degree matrix;

[0082] The hypermarginality matrix is ​​defined as follows:

[0083] ;

[0084] ;

[0085] in, Let l be the user-side hyperedge degree matrix of the l-th layer. Let be the hyperedge degree matrix of the l-th layer item side; diag(·) is the constructor for the diagonal matrix; the soft mass of each sphere is the sum of the membership degrees of the nodes assigned to that sphere:

[0086] ;

[0087] ;

[0088] in, Let k be the hyperedge degree corresponding to the k-th particle on the user side of layer l. Let be the hyperedge degree corresponding to the s-th particle on the side of the l-th layer of the item;

[0089] The mean pooling normalization is defined as follows:

[0090] ;

[0091] ;

[0092] in, and These are represented by spheres on the user side and the item side, respectively.

[0093] The interaction between the user-side and item-side spheres employs purity-aware cross-attention learning at the sphere level, denoted as follows: [Equation 1] [Equation 2] [Equation 3] [Equation 4] [Equation 5] [Equation 6] [Equation 7] [Equation 8] [Equation 9] [Equation 10] [Equation 11] and The purity-aware bias matrix is ​​defined as follows:

[0094] ;

[0095] as well as,

[0096] ;

[0097] The spherical attention matrix for purity perception is calculated as follows:

[0098] ;

[0099] as well as,

[0100] ;

[0101] in, For the user-side purity-perceived bias matrix, For the purity perception bias matrix of the item side; The spherical attention matrix is ​​from the user side to the item side. A spherical attention matrix from the item side to the user side; Let l be the user-side particle embedding matrix of the l-th layer. Embedding matrix for the side particles of the item; This indicates row-wise Softmax normalization. Control the purity bias strength; since the cooperative purity is within [0,1], the additive bias is bounded, which can avoid numerical instability caused by unbounded bias; when At that time, the module degenerates into a standard spherical cross-attention module without purity modulation;

[0102] The updated granular representation is as follows:

[0103] ;

[0104] ;

[0105] in, Let be the updated particle representation matrix on the user side of layer l. This is the updated particle representation matrix for the l-th layer item side; and These are particle-sphere representations enhanced with contralateral information;

[0106] The refinement from sphere to node is based on a soft-assignment distribution that projects the coarse-grained signal back to the node:

[0107] ;

[0108] ;

[0109] in, For the l-th layer user-side coarse-grained node embedding matrix, An embedding matrix is ​​used for the coarse-grained nodes on the l-th layer of the item side; through refinement from coarse to fine, each node receives collaborative information that has already been aggregated and filtered at the coarse-grained level.

[0110] In a preferred embodiment of the above technical solution, the bilateral degree sensing gating in S3 includes:

[0111] For user u, scalar gating is defined as:

[0112] ;

[0113] in, Scalar gating for user u at layer l; For the Sigmoid function; This is the transpose of the gate weight vector; For fine-grained neighborhood aggregation embedding of user u in layer l, For coarse-grained neighborhood aggregation embedding of user u in layer l; This indicates vector concatenation; The degree of user u; For degree perception weight coefficient; This is a gated bias term; The relative importance of the two paths is controlled; β is constrained to be non-negative, causing high-level nodes to tend to depend on fine-grained evidence.

[0114] The degree of user u is denoted as The merged user representation is as follows:

[0115] ;

[0116] The same gating mechanism is used on the item side, that is, for item i, the scalar gating is defined as:

[0117] ;

[0118] The fused item is represented as:

[0119] .

[0120] In the preferred embodiment of the above technical solution, the weighted summation aggregation layers in S4 are defined as follows:

[0121] ;

[0122] in, This is the final embedding for user u; according to LightGCN settings, the default is taken as... The final embedding of item i is also represented by multiple layers of all L layers. Aggregation yields:

[0123] ;

[0124] Leveraging user embedding and item embedding The preference score for user u interacting with item i is defined as follows:

[0125] ;

[0126] The Bayesian personalized ranking loss optimization candidate item score is defined as follows:

[0127] ;

[0128] in, The training triples consist of user u, their observed item i, and unobserved item z sampled for that user; For the Sigmoid function; Θ represents all trainable parameters; for Regularization coefficient; the optimization objective is to improve the robustness of the model under sparse and noisy implicit feedback by utilizing an adaptive coarse-grained collaborative structure while preserving fine-grained sorting signals.

[0129] In summary, due to the adoption of the above technical solution, the beneficial effects of the present invention are:

[0130] This invention no longer relies solely on deep propagation on fine-grained interaction graphs. Instead, it organizes users and items with similar behaviors into adaptively evolving coarse-grained spheres and uses them as stable cooperative units for robust representation learning. Specifically, this invention designs a radius-aware, layer-adaptive sphere construction mechanism to characterize overlapping cooperative patterns in the representation space. It further proposes a purity-aware sphere-level interaction module to conduct cooperative reasoning on a bipartite sphere graph and backpropagates coarse-grained evidence to nodes through cross-granularity refinement. Finally, it utilizes degree-aware fusion gating to adaptively combine fine-grained interaction signals with coarse-grained cooperative evidence, enabling sparse nodes to utilize more reliable granular-level information while preserving the personalized signals of high-frequency nodes. Experiments on three public datasets—Gowalla, Yelp, and Amazon—show that DGBRec (Dynamic Sphere Graph Neural Network Recommendation Model) outperforms the strongest baseline across all ranking metrics, with relative improvements ranging from 2.03% to 6.52%, and achieves more significant gains in sparse, long-tailed, and noisy scenarios.

[0131] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0132] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the description of the embodiments taken in conjunction with the following drawings, in which:

[0133] Figure 1(a) is a bar chart of the stability of prior art coarse-grained preference statistics;

[0134] Figure 1(b) is a rank-degenerate linear graph under prior art graph propagation;

[0135] Figure 2 This is the overall framework diagram of the present invention;

[0136] Figure 3 This is a linear graph comparing the robustness of NDCG@20 under different noise interaction ratios in Gowalla and Yelp according to the present invention;

[0137] Figure 4 This is a bar chart showing the relative decrease rate of NDCG@20 for different user groups of Gowalla and Yelp under 20% noise interaction according to the present invention;

[0138] Figure 5 This is a bar chart showing the relative decrease rate of NDCG@20 for different item degree groups in Gowalla and Yelp under 20% noise interaction according to the present invention;

[0139] Figure 6 This is a normalized effective rank linear graph representing users and items at different propagation depths according to the present invention;

[0140] Figure 7 This is a linear graph showing the effect of propagation depth on Recall@20 and NDCG@20 under different datasets in this invention;

[0141] Figure 8 This is a linear plot of the influence of membership temperature hyperparameters on different datasets in this invention;

[0142] Figure 9 This invention relates to the number of user-side spheres under different datasets. Sensitivity analysis linear plot;

[0143] Figure 10 This invention relates to the number of lateral spheres of items under different datasets. Sensitivity analysis linear plot;

[0144] Figure 11 This is a linear graph of the sensitivity analysis of the Top-T membership scale of this invention. Detailed Implementation

[0145] Embodiments of the present invention are described in detail below. Examples of these embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and should not be construed as limiting the present invention.

[0146] To achieve the above objectives, this invention discloses a dynamic particle-sphere graph neural network method for user and item recommendations, comprising the following: a Top-N recommendation process based on implicit feedback, where the user set and the item set are respectively... and The implicit feedback of observation is represented by the binary interaction matrix R: ;

[0147] When user u has interacted with item i ,otherwise :

[0148] .

[0149] Implicit interactions can be caused by factors such as exposure, curiosity, or short-term interest, so an observed edge is not necessarily a completely reliable indicator of a user's true preferences.

[0150] The fine-grained interaction graph execution process naturally defines the user-item bipartite graph through the interaction matrix:

[0151] ;

[0152] Where G is the fine-grained interaction graph; V is the set of user nodes and item nodes. E represents the set of edges representing user-item interactions. .

[0153] Each observation interaction corresponds to a fine-grained edge in graph G. Let be... The collection of items that user u has interacted with. Let i be the set of users who have interacted with item i. Define the user degree matrix. and item degree matrix :

[0154] ;

[0155] .

[0156] The normalized interaction matrix is ​​defined as:

[0157] ;

[0158] in Corresponding to the bipartite graph symmetric normalized adjacency matrix; this invention directly utilizes Perform spectral initialization and fine-grained propagation.

[0159] In the dual-granularity recommendation setting process, most GNN recommendation methods directly perform message passing on the fine-grained interaction graph G. This approach can preserve personalized evidence but is susceptible to sparse and noisy local interactions. Obtaining higher-order collaborative signals by repeatedly propagating on the same fine-grained graph can also amplify unreliable edges and cause oversmoothing. Therefore, this invention introduces a coarse-grained collaborative granularity beyond the original nodes, aggregating users and items with similar behaviors; users can be softly assigned to... Individual user-side particles, items can be soft-distributed. The membership matrix of an item's side particle is as follows:

[0160] ;

[0161] .

[0162] in For the user-side soft membership matrix, This is the soft membership matrix for the item side; The number of lateral granules for the user. This represents the number of item-side granularities. Each line describes the soft association between a node and multiple coarse-grained granularities. Based on membership, the coarse-grained interaction strength between user-side granularities and item-side granularities is:

[0163] ;

[0164] Matrix P summarizes coarser-grained collaborative interactions, providing a structural foundation for spherical collaborative reasoning. Given an interaction graph, the goal of Top-N recommendation is to learn low-dimensional user embeddings. and item embedding This allows preference scores for unobserved items to be accurately predicted:

[0165] ;

[0166] in, The preference score for user u interacting with item i; For the scoring function, the inner product is preferred. The embedding is learned by jointly utilizing fine-grained interaction evidence from G and coarse-grained collaborative evidence provided by the granular sphere, and the resulting scores are used to generate a sorted list of items for each user.

[0167] like Figure 2 As shown, this invention discloses a method for using Dynamic Granular-Ball Graph Neural Networks for Recommendation (DGBRec), which combines fine-grained interaction propagation with coarse-grained granular ball inference to alleviate the reliability-depth contradiction in graph recommendation. DGBRec includes: granular ball initialization and dynamic soft allocation, fine-grained propagation, coarse-grained propagation, and degree-aware fusion. DGBRec initializes user-side and item-side granular balls from the user-item interaction graph and simultaneously performs fine-grained graph propagation and coarse-grained granular ball propagation. The coarse-grained path sequentially completes node-to-ball abstraction, ball-level interaction, and ball-to-node refinement, finally fusing the two representations using degree-aware gating. The specific execution steps are as follows:

[0168] S1. Construct user-side and item-side spheres in the low-dimensional spectral space to obtain an initialization containing global topological priors. Then refine the spheres through radius-aware membership learning and collaborative purity evaluation to achieve adaptive evolution of the coarse-grained structure prototype.

[0169] S1-1, performing particle sphere initialization, in large-scale recommendation systems, directly constructing coarse-grained collaborative structures in the original sparse interaction space is computationally expensive and lacks stability. To obtain an initialization that includes global topological priors, this invention constructs user-side and item-side particle spheres in a low-dimensional spectral space. The normalized interaction matrix... Perform randomized singular value decomposition:

[0170] ;

[0171] in, , , , respectively denote the truncation of the left singular vector, the singular value matrix, and the right singular vector; transpose symbol This indicates that the rows and columns of the matrix are interchanged; For randomized singular value decomposition; is the set of real numbers; d is the embedding dimension, preferably 64 or 128 dimensions; Let U be the user's feature vector, with dimension |U|×d. for The singular value matrix, Let be the feature vector of the item, with a vector dimension of . Based on randomized singular value decomposition, the initial user embedding and item embedding are constructed as follows:

[0172] ;

[0173] .

[0174] in, The square root of the singular value matrix; embedding and As a trainable initial representation, it is updated through backpropagation during model optimization. Thus, subsequent particle initialization and layer-by-layer membership calculation are both completed in a representation space consistent with the trainable node embeddings.

[0175] S1-2, in and Performing lightweight K-Means yields the initial user-side particle center. and the center of the sphere on the side of the item ,in For the first Each user's lateral granule center, For the first The center of each object's side sphere. These centers serve as a warm-start structure prototype and are adaptively updated during subsequent propagation.

[0176] After initialization, the model further refines the spheres through radius-aware membership learning and collaborative purity evaluation. Therefore, these coarse-grained structural prototypes are adaptively evolving spheres, rather than fixed clusters.

[0177] For each initial sphere, initialize its radius with the average distance from its node to the initial center:

[0178] ;

[0179] ;

[0180] in, Let be the radius of the k-th initial user-side particle in layer 0. Let be the radius of the s-th initial item side particle in layer 0; The set of nodes assigned to the k-th initial user-side sphere. ϵ is the set of nodes assigned to the s-th initial item-side sphere; ϵ is a minimal positive constant used for numerical stability; k is the user-side sphere index, and s is the item-side sphere index; Embed the 0th layer input for user u. For item i, embed the 0th layer of the input; Let be the center vector of the k-th user particle in layer 0. Let be the center vector of the s-th item particle in the 0th layer. Calculate the L2 norm of a vector;

[0181] S1-3, DGBRec does not fix the particle sphere throughout the training process, but updates it based on the node representation of each propagation layer; in layer l, the input user embedding and item embedding are denoted as... and DGBRec updates the granular spheres based on the sphere center and radius inherited from layer l-1, and calculates the membership of the current layer: first, it calculates the temporary soft membership degree using the center and radius of the previous layer, and then refreshes the sphere center and radius with a weighted embedding according to the current node. To avoid drastic prototype drift, both the center and radius are updated using exponential moving average (EMA), so that the coarse-grained structure evolves smoothly with the representation space and remains consistent with the current cooperative geometry.

[0182] For user-side nodes With the User lateral granules Item side nodes With the Item side ball The normalized distance is defined as follows:

[0183] ;

[0184] ;

[0185] in, Let be the normalized distance from user u to the k-th user-side particle; Let be the normalized distance from item i to the s-th item's side particle sphere; The input embedding of user u before propagation at layer l-1, The input embedding is the item i before propagation at layer l-1.

[0186] Temporary membership degree is calculated as follows:

[0187] ;

[0188] ;

[0189] in, Let be the temporary membership degree of user u at layer l to the granular ball of user k. Let i be the temporary membership degree of item i in layer l to the side particle of item s; is the temperature coefficient; k is the user-side particle index, s is the item-side particle index, and j is the value sequence number.

[0190] S1-4, using temporary membership, refresh the center of the particle on both the user side and the item side by weighted average:

[0191] ;

[0192] ;

[0193] in, For user side-sphere refresh center, Refresh the center of the item's side-particle ball.

[0194] To prevent excessively rapid prototype drift between different layers, EMA is used to update the particle center:

[0195] ;

[0196] ;

[0197] in, For the center of the k-th user-side particle in the l-th layer, The center of the side particle of the s-th item in the l-th layer; Control the smoothness of updates.

[0198] The sphere radius is also updated using a layer-adaptive approach. For the k-th user-side sphere, the refresh radius is defined using the weighted root mean square distance:

[0199] ;

[0200] ;

[0201] in, Let the refresh radius of the k-th user-side particle in layer l be . The refresh radius of the side particle ball for the s-th item in the l-th layer.

[0202] To ensure stability, both the user-side and item-side particle radii are updated using the exponential moving average (EMA), meaning the particle radii will not fluctuate drastically with a single batch of data.

[0203] ;

[0204] ;

[0205] in, Let be the radius of the k-th user-side particle in the l-th layer. Let be the radius of the s-th item's side particle in the l-th layer; , where is the radius momentum.

[0206] S1-5, radius-aware soft membership allows each user or item to participate in multiple spheres, better reflecting the multifaceted nature of user preferences; the spheres are dynamically updated with the propagation layer, adapting to the evolving representation space rather than relying on a fixed partition. A key characteristic of sphere computation is that node membership should be determined by the relationship between the node and the bounded sphere region, rather than solely based on centroid similarity. Therefore, this invention utilizes radius-normalized distance to calculate the membership degree from a node to a sphere.

[0207] For user-side nodes With the User lateral granules Item side nodes With the Item side ball ,definition:

[0208] ; ;

[0209] in, The normalized distance from user u to the k-th user side sphere is based on the updated user u. This is the normalized distance from the updated item i to the s-th item's side sphere. The soft membership degree is calculated as follows:

[0210] ;

[0211] ;

[0212] in, Let be the soft membership degree of user u at layer l to the granular ball of user k. Let be the soft membership degree of item i in layer l to the side granular ball of item s. Temperature coefficient to control the sharpness of preference distribution; For the total number of side balls of the user, The total number of particles on the item side; the user-side membership matrix is ​​denoted as... The membership matrix of an item is denoted as... .

[0213] To balance efficiency and interpretability, only the Top-T memberships of each node are retained and renormalized, where the user-side membership is taken as... Item side retrieval T is a sparsity hyperparameter, representing the number of memberships retained after truncation and normalization. and Each row contains at most T non-zero elements. This design preserves the overlapping property of spheres while keeping the assignment matrix sparse.

[0214] S1-6, high-quality spheres should not only maintain consistency within the embedding space, but also exhibit relatively consistent cooperative behavior; purity-aware sphere-level interactions and cross-granularity refinement enhance node representations using reliable coarse-grained cooperative evidence. To measure the reliability of spheres as coarse-grained cooperative units, this invention defines cooperative purity based on their interaction distribution on opposite spheres.

[0215] No. The soft sphere-level interaction matrix of the layer is calculated as follows:

[0216] ;

[0217] in, This represents the aggregation interaction strength between the k-th user side sphere and the s-th item side sphere. Based on For each particle, a normalized cooperative preference distribution is constructed, and its purity is measured using normalized entropy. The higher the purity, the more concentrated the cooperative preference of the particle, and the more reliable it is as a coarse-grained cooperative unit.

[0218] For user side particles Its cooperative preference distribution on the item side sphere is defined as:

[0219] ;

[0220] Its co-purity is measured using normalized entropy:

[0221] .

[0222] Similarly, for the side particles of the object. ,definition:

[0223] ;

[0224] as well as,

[0225] .

[0226] in, Let be the probability of collaborative preference between the k-th particle on the user side and the s-th particle on the item side in the l-th layer. Let be the probability of collaborative preference between the s-th particle on the item side of the l-th layer and the k-th particle on the user side; The interaction matrix of the l-th layer granular level The (k,s)th element, The interaction matrix of the l-th layer granular level The (s,k)th element; Let k be the co-purity of the k-th particle on the user side of layer l. Let be the co-purity of the s-th particle on the l-th layer of the material. The larger the value, the more concentrated the cooperative preference distribution of the corresponding particles, and the more reliable they are as coarse-grained cooperative units; Smaller values ​​indicate that the spheres contain a mixture of multiple heterogeneous preference patterns. Compared to fixed coarse-grained grouping, this layer adaptive design allows the spheres to co-evolve with the node representations, providing a more reliable coarse-grained structure for subsequent layer propagation.

[0227] S2 couples fine-grained interaction modeling with coarse-grained collaborative abstraction, and propagates messages along both fine-grained and coarse-grained paths.

[0228] The core of DGBRec is the coupling of fine-grained interaction modeling and coarse-grained collaborative abstraction. Message passing proceeds along two complementary paths: the fine-grained path operates on the user-item interaction graph, while the coarse-grained path operates on a dynamically constructed bipartite granular graph. The former preserves local personalized collaborative evidence in observed interactions, while the latter models high-level dependencies between user-side and item-side granular graphs and feeds the resulting signals back to the nodes.

[0229] To formally describe coarse-grained aggregation, the soft mass of each sphere is defined as the sum of the node memberships assigned to that sphere:

[0230] ;

[0231] .

[0232] Therefore, the hyperedge degree matrices for the user side and the item side are as follows:

[0233] ;

[0234] .

[0235] in, Let the k-th hyperedge degree be the user u at layer l. Let the s-th hyperedge degree be the value of item i in the l-th layer. Let l be the user-side hyperedge degree matrix of the l-th layer. Let be the hyperedge degree matrix of the item side in the l-th layer; diag(·) is the constructor for the diagonal matrix.

[0236] Fine-grained path propagation: At a fine-grained level, DGBRec performs normalized neighborhood aggregation on the original user-item interaction graph to preserve local cooperative signals. For user u and item i, the l-th layer propagation is defined as:

[0237] ;

[0238] ;

[0239] in, This is the normalized adjacency matrix of the bipartite interaction graph; For the neighborhood aggregation embedding of user u in layer l, For the neighborhood aggregation embedding of item i in the l-th layer; For user u, the set of neighborhood items Let i be the set of neighboring users of item i.

[0240] Coarse-grained path propagation: This path propagation consists of three stages: the abstraction from node to particle, the interaction between user-side particle and item-side particle, and the refinement from particle to node.

[0241] The process of abstracting from nodes to granules involves aggregating node features into corresponding granules to form prototypes, and then using the hyper-edge degree matrix for rigorous mean pooling normalization.

[0242] ;

[0243] ;

[0244] in, and These are represented by granular spheres on the user side and the item side, respectively. This operation yields a coarse-grained prototype that summarizes the collaborative semantics of the granular spheres on both sides.

[0245] The interaction between user-side granular spheres and item-side granular spheres is modeled; to model coarse-grained collaborative dependencies, a bipartite granular graph is constructed between user-side granular spheres and item-side granular spheres. Unlike using fixed statistical connections, this invention employs purity-aware cross-attention learning for sphere-level interactions.

[0246] To incorporate particle-to-sphere reliability into sphere-level interactions, a bounded additive purity bias is designed; all user-side and item-side purity are denoted as... and The purity-aware bias matrix is ​​defined as follows:

[0247] ;

[0248] as well as,

[0249] .

[0250] The spherical attention matrix for purity perception is calculated as follows:

[0251] ;

[0252] as well as,

[0253] ;

[0254] in, For the user-side purity-perceived bias matrix, For the purity perception bias matrix of the item side; The spherical attention matrix is ​​from the user side to the item side. A spherical attention matrix from the item side to the user side; Let l be the user-side particle embedding matrix of the l-th layer. Embedding matrix for the side particles of the item; This indicates row-wise Softmax normalization. Control the purity bias strength. Since the cooperative purity lies within [0,1], the additive bias is bounded, avoiding numerical instability caused by unbounded bias. When At this point, the module degenerates into a standard sphere-level cross-attention module without purity modulation. The updated sphere representation is as follows:

[0255] ;

[0256] ;

[0257] in, Let be the updated particle representation matrix on the user side of layer l. This is the updated particle representation matrix for the l-th layer item side; and These are the particle-sphere representations enhanced with contralateral information.

[0258] The refinement from spheres to nodes is followed by the final projection of the coarse-grained signal back to the nodes based on the soft-distribution distribution.

[0259] ;

[0260] .

[0261] in, For the user-side coarse-grained node embedding matrix of layer l, This is the coarse-grained node embedding matrix for the l-th layer item side. Through coarse-to-fine refinement, each node receives collaborative information that has already been aggregated and filtered at the coarse-grained level. Therefore, DGBRec can capture a wider range of collaborative dependencies with shallower propagation, reducing reliance on deep node messaging and mitigating the risk of oversmoothing.

[0262] S3 performs bilateral sensing gating, adaptively balancing the contributions of the two propagation paths for each node.

[0263] The reliability of fine-grained and coarse-grained signals is inconsistent across different nodes. For highly active users and popular items, local interaction neighborhoods typically contain sufficient evidence of collaboration, making fine-grained representations more stable. For less active users and long-tailed items, local observations are insufficient, making coarse-grained collaborative abstractions more valuable. To accommodate this heterogeneity, this invention introduces degree-aware gating, adaptively balancing the contributions of the two propagation paths for each node.

[0264] For user u, scalar gating is defined as:

[0265] ;

[0266] in, Scalar gating for user u at layer l; For the Sigmoid function; This is the transpose of the gate weight vector; For fine-grained neighborhood aggregation embedding of user u in layer l, Coarse-grained neighborhood aggregation embedding for user u in layer l; This involves concatenating vectors. The degree of user u; For degree perception weight coefficient; The gating bias term is used; the gating weight vector, degree-aware weight coefficient, and bias term are all learnable parameters, without preset fixed values, and are obtained through backpropagation of the BPR loss function during training. The relative importance of the two paths is controlled; β is constrained to be non-negative, causing high-level nodes to tend to depend on fine-grained evidence. The degree of user u is denoted as... The merged user representation is as follows:

[0267] .

[0268] The same gating mechanism is used on the item side; that is, for item i, the scalar gating is defined as:

[0269] ;

[0270] in, Scalar gating for item i at level l; For the Sigmoid function; This is the transpose of the gate weight vector; Fine-grained neighborhood aggregation embedding for item i in layer l. For coarse-grained neighborhood aggregation embedding of item i in layer l; This involves concatenating vectors. Let i be the degree of item i; For degree perception weight coefficient; The gating bias term; the gating weight vector, degree-aware weight coefficient, and bias term are all learnable parameters, without preset fixed values, and are obtained through backpropagation optimization of the BPR loss function during training; The relative importance of the two paths is controlled; β is constrained to be non-negative, causing high-level nodes to tend to depend on fine-grained evidence. The degree of item i is denoted as... The fused item is represented as:

[0271] .

[0272] Therefore, degree-aware gating adaptively balances personalization and robustness, allowing high-degree nodes to rely more on fine-grained local evidence, while low-degree nodes can benefit from a more robust coarse-grained collaborative structure. Sparse users and long-tail items, on the other hand, benefit more from granular evidence. This adaptive trade-off makes DGBRec particularly suitable for sparse and long-tail recommendation scenarios.

[0273] S4 aggregates the representations of each layer by weighted summation to obtain the final embeddings on the user side and the item side, and finally uses Bayesian personalized ranking loss to optimize the candidate item scores.

[0274] For user u and item i, multi-layer embeddings are obtained respectively. and L represents the number of propagation layers. For trainable initial embeddings before propagation, For the first Fusion representation after layer-wise two-granularity propagation. Depict different levels of preference information.

[0275] This invention obtains the final embedding of user u by weighted summation and aggregation of the representations of each layer:

[0276] ;

[0277] According to LightGCN's settings, the default value is... .

[0278] The final embedding of item i is also represented by a multi-layered representation of all L layers. Aggregation yields:

[0279] ;

[0280] Leveraging user embedding and item embedding The preference score for user u interacting with item i is defined as follows:

[0281] ;

[0282] The model uses Bayesian Personalized Ranking (BPR) loss to optimize candidate item scores:

[0283] ;

[0284] in, The training triples consist of user u, their observed item i, and unobserved item z sampled for that user; Here, Θ represents the Sigmoid function; Θ represents all trainable parameters. for The regularization coefficient controls the contribution of the regularization term to the total loss, and its value ranges from [value missing]. The specific value is determined through a validation set or grid search.

[0285] The optimization objective is to improve the robustness of the model under sparse and noisy implicit feedback by retaining fine-grained sorting signals and utilizing an adaptive coarse-grained collaborative structure.

[0286] Complexity analysis: Let , and These are the number of users, the number of items, and the number of observed interactions, respectively. For the embedding dimension, and Here, T represents the number of spheres on the user side and the item side, and L represents the number of membership degrees retained. This represents the asymptotic upper bound of the algorithm's time complexity. Because... and Each row retains only the Top-T non-zero memberships. It can be calculated by traversing the observed edges, with a complexity of O(n). The complexity of randomized SVD and K-Means in a one-time initialization is approximately... and ; This represents the number of iterations in the K-Means clustering algorithm, with a fine-grained propagation complexity of O(n log n). The computational complexity of precise radius-aware membership is... The propagation complexity of sparse node-to-sphere and sphere-to-node is... The complexity of constructing spherical-level interaction statistics based on observation edges is... The purity of the perceptual sphere-level attention complexity is... .

[0287] Therefore, the overall complexity of each layer is . The main storage overhead is Because T is small, and , The overhead of DGBRec is much smaller than the number of users and items, and is moderate and manageable in practice.

[0288] This invention conducts experiments on three real datasets to verify the effectiveness of the dynamic particle-sphere diagram neural network model. The experimental setup and specific implementation examples are given below.

[0289] Datasets: The experiment used three real-world datasets: Gowalla, Yelp, and Amazon. Gowalla contains user location check-in records collected from January to June 2010; Yelp contains user location ratings from January to June 2018; and Amazon contains book ratings collected in 2013. These datasets are commonly used public benchmarks in recommendation research. During preprocessing, users and items with fewer than three interactions were filtered; each user was randomly divided into training, validation, and test sets in a 70:5:25 ratio; during evaluation, observed items from the training and validation sets were excluded from the candidate set; and all ratings were converted to binary implicit feedback. Dataset statistics are shown in Table 1.

[0290] Table 1. Dataset Statistics

[0291] Number of users 25557 42712 76469 Item count 19747 26822 83761 Number of interactions 294983 182357 966680 Average number of interactions 11.5 4.2 12.6 density 0.0585% 0.01159% 0.0151%

[0292] Comparison Methods: To verify the effectiveness of DGBRec, the following four representative methods were selected for comparison.

[0293] (1) Traditional recommendation model

[0294] BiasMF (Bias Matrix Factorization): Introduces user bias, item bias, and other terms into the matrix factorization framework to model user-item interactions using latent factors.

[0295] NCF (Neural Collaborative Filtering): Uses a multilayer perceptron to replace the inner product in traditional matrix factorization to characterize complex nonlinear user-item interactions.

[0296] AutoRec (Autoencoder Recommendation): Taking user or item rating vectors as input, it reconstructs ratings and predicts missing items through an autoencoder.

[0297] (2) Recommendation model based on graph neural network

[0298] PinSage: It uses random walk sampling to identify important neighbors and is a scalable graph convolutional embedding method suitable for large-scale graph recommendation.

[0299] NGCF (Neural Graph Collaborative Filtering): Embedding propagation is performed on the user-item bipartite graph to explicitly encode higher-order collaborative signals.

[0300] GCCF (Graph Convolutional Collaborative Filtering): Removes nonlinear transformations and introduces residual structures to alleviate oversmoothing in graph collaborative filtering.

[0301] LightGCN (Lightweight Graph Convolutional Network): Removes feature transformations and non-linear activations, retaining only neighborhood aggregation to efficiently perform collaborative filtering.

[0302] DGCF (Decoupled Graph Collaborative Filtering): Decouples user intent in the interaction graph and learns independent multi-preference representations.

[0303] (3) Recommendation model based on self-supervised learning

[0304] SLRec (Self-Supervised Recommendation): Introduces a self-supervised task to mine potential correlations in item features, reducing the reliance on sparse supervisory feedback.

[0305] SGL (Self-Supervised Graph Learning): Self-supervised graph learning is carried out by constructing comparative views through node dropping, edge dropping, and random walks.

[0306] SimGCL (Simple Graph Contrast Learning): Instead of relying on structure graph augmentation, it injects uniform noise into the embedding space to perform graph contrast learning.

[0307] HCCF (Hyperbolic Collaborative Graph Contrast Filtering): It uses hypergraph learning to characterize multi-order structures that go beyond pairwise relationships and employs cross-view contrast supervision to enhance robustness.

[0308] (4) Coarse-grained and multi-grained recommendation models

[0309] NCL (Neighborhood Enhanced Contrastive Learning): It integrates structural contrastive learning and semantic contrastive learning, and describes node correlation through graph neighbors and prototype semantic clusters.

[0310] UnKD (Unbiased Knowledge Distillation): Transfers knowledge between teacher and student models and mitigates data bias through unbiased knowledge distillation.

[0311] SimRec (Similarity-based Recommendation): Distills the comparative knowledge of graph neural networks into a multilayer perceptron to achieve efficient graph-free recommendation and alleviate oversmoothing.

[0312] ClusterGCF (cluster-based graph collaborative filtering): clusters nodes with similar preferences and performs graph convolution within subgraphs to improve representation learning and address oversmoothing issues.

[0313] LightGNN (Lightweight Graph Neural Network): Improves efficiency while maintaining performance by pruning redundant or noisy edges and embeddings through hierarchical knowledge distillation.

[0314] Evaluation metrics: Recall@ and NDCG@ Two commonly used Top- Ranking metrics. Recall@ Measure the retention of interactions in the test set to enter Top- The proportion of the recommended list; NDCG@ Considering sorting position, higher weight is given to hits at the beginning of the list. Following a general full-sorting evaluation protocol, the test positive samples are sorted for each user across all uninterrupted items. Experiment Report =20 and Recall at =40 With NDCG@ .

[0315] Parameter settings: DGBRec is implemented using PyTorch and the Deep Graph Library (DGL). To ensure fairness, all methods use the same data partitioning, negative sampling strategy, and full ranking evaluation protocol. The embedding dimension of all embedding methods is uniformly set to 64; the learning rate is from... Selected from, Regularization coefficient from Select from, batch size from The selection is made from the middle. During BPR optimization, one unobserved item is uniformly sampled as a negative sample for each observation interaction.

[0316] The key hyperparameters of DGBRec are tuned based on the validation set Recall@20. Consistent with the hyperparameter analysis in Section 5.6, the propagation depth... exist Search in the middle, the main experiment uses ; Belonging temperature exist Search in the middle; user-side and item-side particle count , from Select Top- Sparse membership size from Select from the options. Main Experiment Settings Stable performance can be achieved on different datasets while maintaining a moderate degree of overlap in the allocation.

[0317] Table 2 Performance comparison of different methods on three datasets

[0318] Amazon Gowalla Yelp

[0319] BiasMF 0.0211 0.0293 0.0324 0.0578 0.0579 0.0695 0.0867 0.1269 0.0090 0.0120 0.0198 0.0307 NCF 0.0234 0.0306 0.0367 0.0600 0.0674 0.0833 0.1019 0.1563 0.0143 0.0187 0.0304 0.0487 AutoRec 0.0318 0.0415 0.0525 0.0826 0.0690 0.0985 0.1477 0.2511 0.0222 0.0268 0.0491 0.0692 PinSage 0.0317 0.0402 0.0486 0.0773 0.0809 0.0994 0.0985 0.1882 0.0245 0.0315 0.0510 0.0743 NGCF 0.0353 0.0454 0.0551 0.0876 0.1135 0.1367 0.1757 0.2586 0.0336 0.0419 0.0681 0.1019 GCCF 0.0501 0.0625 0.0772 0.1175 0.1282 0.1532 0.2012 0.2903 0.0365 0.0466 0.0742 0.1151 LightGCN 0.0571 0.0697 0.0868 0.1285 0.1433 0.1670 0.2230 0.3181 0.0373 0.0474 0.0761 0.1175 DGCF 0.0372 0.0468 0.0617 0.0912 0.1312 0.1555 0.2055 0.2929 0.0347 0.0437 0.0700 0.1072 SLRec 0.0480 0.0598 0.0742 0.1123 0.1298 0.1540 0.2001 0.2863 0.0327 0.0418 0.0665 0.1032 NCL 0.0623 0.0764 0.0955 0.1409 0.1478 0.1745 0.2283 0.3232 0.0402 0.0505 0.0806 0.1230 SGL 0.0569 0.0704 0.0874 0.1312 0.1509 0.1780 0.2332 0.3251 0.0398 0.0502 0.0803 0.1226 HCCF 0.0578 0.0716 0.0885 0.1335 0.1482 0.1751 0.2293 0.3258 0.0391 0.0492 0.0789 0.1210 SimGCL 0.0605 0.0730 0.0921 0.1367 0.1506 0.1772 0.2328 0.3276 0.0395 0.0498 0.0788 0.1213 UnKD 0.0607 0.0745 0.0947 0.1376 0.1496 0.1766 0.2331 0.3301 0.0392 0.0493 0.0819 0.1202 SimRec 0.0734 0.0879 0.1067 0.1535 0.1592 0.1865 0.2434 0.3399 0.0414 0.0519 0.0823 0.1251 ClusterGCF 0.0843 0.0995 0.1224 0.1614 0.1671 0.1981 0.2607 0.3547 0.0417 0.0521 0.0838 0.1270 LightGNN 0.0820 0.0969 0.1189 0.1677 0.1684 0.1962 0.2610 0.3597 0.0443 0.0553 0.0879 0.1328 DGBRec 0.0898 0.1038 0.1273 0.1721 0.1750 0.2040 0.2663 0.3688 0.0463 0.0575 0.0911 0.1374 promote 6.52% 4.32% 4.00% 2.62% 3.92% 2.98% 2.03% 2.53% 4.51% 3.98% 3.64% 3.46%

[0320] Based on the above experimental setup, the technical solution of the present invention is verified through the following embodiments:

[0321] Example 1: Overall Performance Analysis

[0322] DGBRec is compared with traditional collaborative filtering, standard graph neural networks, self-supervised graph recommendation, and coarse-grained / multi-granularity collaborative models. Table 2 uses NDCG@20, NDCG@40, Recall@20, and Recall@40 to evaluate ranking quality and recall. DGBRec achieves the best results across all metrics on Amazon, Gowalla, and Yelp. Compared to the strongest baseline, DGBRec improves by 6.52%, 4.32%, 4.00%, and 2.62% on Amazon; by 3.92%, 2.98%, 2.03%, and 2.53% on Gowalla; and by 4.51%, 3.98%, 3.64%, and 3.46% on Yelp. The results demonstrate that dual-granularity modeling can simultaneously improve head ranking accuracy and overall hit coverage.

[0323] Further comparisons revealed that DGBRec achieved the most significant improvement on the highly sparse and long-tailed Amazon dataset. In this dataset, user-item interactions provide limited local evidence, and models relying primarily on fine-grained neighborhood propagation are susceptible to unreliable or insufficient signals. DGBRec adaptively aggregates users or items with similar behavioral patterns to form more stable collaborative priors, enabling sparse users and long-tailed items to benefit from group-level preference evidence rather than relying entirely on limited observed interactions.

[0324] (1) Traditional models and standard graph neural networks. BiasMF, NCF, and AutoRec mainly model pairwise interactions in isolation, making it difficult to extract high-order connectivity relationships, thus resulting in weak overall performance. Standard graph models such as NGCF, GCCF, and LightGCN significantly improve performance through multi-hop neighborhood aggregation, but their upper limit is still constrained by implicit feedback noise and sparsity. Since these models typically use all observed edges indiscriminately during message passing, noise amplification and oversmoothing are prone to occur as the network deepens.

[0325] (2) Self-supervised recommendation models. Methods such as SGL, SimGCL, and HCCF improve representation robustness and alleviate sparsity through auxiliary self-supervised tasks and data augmentation strategies such as node dropping and noise injection, and are generally superior to standard graph models. However, DGBRec still leads significantly, indicating that explicit discovery and modeling of the inherent coarse-grained structure in the graph can provide more reliable and targeted guidance than random perturbation or uniform contrast noise.

[0326] (3) Coarse-grained and multi-granularity recommendation models. NCL, ClusterGCF, and LightGNN utilize semantic clustering, subgraph clustering, or graph pruning to obtain high-level structural information, making them the most competitive baselines. ClusterGCF recognizes the value of node grouping for stable cooperative signals, but its clustering mechanism is relatively static, heuristic, and rigid. DGBRec achieves flexible and noise-resistant data granularization through soft allocation, avoiding the forced inclusion of nodes into mutually exclusive groups, and amplifies true preferences and suppresses anomalous interactions in a dual-granularity framework, thereby further improving representation robustness.

[0327] Table 3 Ablation Experiment Results

[0328] Amazon Gowalla Yelp

[0329] w / o Coarse 0.0695 0.0833 0.1026 0.1432 0.1545 0.1835 0.2358 0.3289 0.0386 0.0481 0.0779 0.1168 w / o Fine 0.0784 0.0965 0.1152 0.1579 0.1638 0.1927 0.2544 0.3572 0.0414 0.0515 0.0822 0.1235 w / o Gate 0.0827 0.0979 0.1204 0.1676 0.1705 0.1973 0.2618 0.3641 0.0428 0.0522 0.0863 0.1248 w / o Degree 0.0844 0.1000 0.1223 0.1694 0.1711 0.1987 0.2627 0.3645 0.0428 0.0526 0.0856 0.1256 w / o DynamicBall 0.0836 0.0990 0.1216 0.1682 0.1710 0.1994 0.2629 0.3640 0.0427 0.0526 0.0861 0.1265 w / o Radius 0.0815 0.0970 0.1210 0.1687 0.1700 0.1998 0.2611 0.3641 0.0430 0.0531 0.0884 0.1299 DGBRec 0.0898 0.1038 0.1273 0.1721 0.1750 0.2040 0.2663 0.3688 0.0463 0.0575 0.0911 0.1374

[0330] Example 2: Ablation Experiment

[0331] To analyze the contributions of each key module, the following variants were constructed: with / o Coarse, coarse-grained particle propagation was removed, retaining only fine-grained user-item propagation; with / o Fine, fine-grained paths were removed, using only coarse-grained particle inference; with / o Gate, fixed fusion replaced degree-aware adaptive fusion; with / o Degree, degree information was removed from the gating, but the learnable fusion function was retained; with / o Dynamic Ball, the initial particle was fixed, and layer-by-layer adaptive updates were disabled; with / o Radius, radius normalization was removed, and nodes were assigned only based on centroid similarity. The results are shown in Table 3.

[0332] The complete DGBRec module performs best across all datasets and evaluation metrics, indicating complementarity among its modules. The w / oCoarse module shows the most significant degradation, with an average decrease of approximately 15.43%, particularly noticeable on Amazon and Yelp, where interactions are sparser. For example, after removing coarse-grained paths on Amazon, NDCG@20 decreases from 0.0898 to 0.0695, and Recall@20 decreases from 0.1273 to 0.1026. These results demonstrate that granular inference can provide stable collaborative evidence beyond noisy individual interactions, which is crucial for sparse recommendation.

[0333] Fine-grained paths are equally essential. While w / o Fine can still utilize coarse-grained collaborative abstraction, the average performance decreases by approximately 8.16%. This indicates that granular paths cannot completely replace direct user-item interaction modeling because fine-grained propagation preserves personalized local preferences. The comparison between w / o Coarse and w / o Fine further demonstrates that optimal results can only be achieved by jointly modeling fine-grained personalization and coarse-grained robustness.

[0334] The fusion mechanism also plays a crucial role. The average performance of w / o Gate and w / o Degree decreased by approximately 5.14% and 4.44% respectively compared to the complete model, indicating that simple fixed combinations are insufficient to accommodate the reliability differences between the two paths. Introducing node degree allows gating to enable highly active users and popular items to rely more on fine-grained evidence, while sparse users and long-tail items absorb more coarse-grained collaborative signals.

[0335] The use of `w / o Dynamic Ball` and `w / o Radius` validates the necessity of adaptive sphere construction. Fixing the initial sphere weakens the model's ability to track the evolution of the representation space; removing radius-aware assignment reduces the flexibility and robustness of node-sphere membership. Both variants exhibit persistent degradation, demonstrating that layer-by-layer adaptive updates and radius-normalized soft assignments are crucial for constructing reliable overlapping spheres.

[0336] In summary, the ablation experiments validated the effectiveness of the dual-granularity framework: coarse-grained particle paths are the main source of robustness, fine-grained paths preserve personalized interaction signals, and degree-aware gating adaptively balances the two; dynamic particle updates and radius-aware membership relationships further improve the quality of coarse-grained collaborative abstraction.

[0337] Example 3: Robustness under Unreliable Interactions

[0338] Noise robustness analysis. To evaluate the robustness of DGBRec to implicit feedback noise, random false positive interactions of varying proportions were injected into the training set while keeping the validation and test sets unchanged. For each noise proportion... Unobserved user-item pairs were randomly sampled and added to the training interaction graph. All methods were retrained on the contaminated graph and evaluated using the same full-sorting protocol.

[0339] like Figure 3 As shown, NDCG@20 on Gowalla and Yelp is presented. The performance of all methods degrades with increasing noise levels, indicating that implicit noise feedback significantly impairs graph recommendation models. LightGCN, which propagates directly across all observation edges and lacks an explicit denoising mechanism, exhibits the most significant degradation. NCL improves robustness through prototype-enhanced contrastive learning, but its graph view is still constructed from noisy interactions. ClusterGCF further improves robustness through its cluster-level structure, while LightGNN achieves stronger noise resistance by pruning redundant or unreliable graph components.

[0340] DGBRec achieved the best results across all noise proportions on both datasets. When Gowalla injected 20% noise, DGBRec's NDCG@20 decreased by only 6.86%, while LightGNN, ClusterGCF, NCL, and LightGCN decreased by 10.21%, 11.61%, 14.21%, and 18.98%, respectively. On the sparser Yelp dataset, the degradation was more pronounced, but DGBRec still only decreased by 10.80%, while LightGCN decreased by 31.90%.

[0341] The results show that DGBRec is more robust to unreliable implicit feedback. This is because the dual-granularity design no longer simply propagates along noisy fine-grained edges, but instead abstracts users and items as adaptive spheres, conducting reliability-aware collaborative reasoning at the information granular level and refining node representations through cross-granularity feedback. Therefore, a single noisy interaction is less likely to dominate the final representation, making it particularly suitable for sparse recommendations.

[0342] Sparse User Analysis. To further examine the robustness of DGBRec to sparse users, under a 20% noise interaction condition, users were divided into three groups based on the interaction level of the training set: 1–10, 11–20, and greater than 20. The NDCG@20 descent rate for a relatively clean setup is reported; a smaller descent indicates stronger noise resistance. Results from Gowalla and Yelp can be found... Figure 4 .

[0343] like Figure 4 As shown, all methods degraded after noise injection, but the differences between different models were significant. LightGCN showed the largest degradation across all user degree groups, especially for low-degree users; NCL and ClusterGCF, which utilize prototype or cluster-enhanced modeling, were more robust than LightGCN; LightGNN further reduced degradation by simplifying the graph structure and suppressing redundant and noisy components.

[0344] DGBRec exhibits the smallest decrease across all user degree groups. In Gowalla, users with 1-10 interactions experience only an 8.18% decrease, while LightGNN, ClusterGCF, NCL, and LightGCN show decreases of 11.22%, 13.05%, 16.44%, and 21.86%, respectively. Low-degree users have limited fine-grained evidence, and noisy edges tend to dominate their sparse neighborhoods, making models relying on direct edge propagation more vulnerable. DGBRec effectively mitigates this problem by aggregating multiple uncertain observations into granular-level stable cooperative evidence.

[0345] The trends on Yelp are similar and even more pronounced. Due to Yelp's greater sparsity, all models exhibit greater degradation under noise. For example, LightGCN shows a decrease of 29.53%, 30.48%, and 30.82% in groups 1–10, 11–20, and greater than 20, respectively. DGBRec, on the other hand, shows a decrease of only 10.47%, 10.66%, and 9.97% in the three groups, indicating that it can significantly reduce the impact of unreliable fine-grained interactions even under severely sparsity conditions.

[0346] Long-tail item analysis. Noisy implicit feedback also significantly affects long-tail items. Since long-tail items are only relevant to a small number of users, their representation is particularly sensitive to false positive interactions. To verify the robustness of DGBRec to long-tail recommendations under noisy conditions, items were divided into three groups according to their interaction frequency in the clean training set: 1–10, 11–20, and greater than 20. Group 1–10 represents long-tail items with very few interaction records. The reported NDCG@20 decrease rate under a 20% noise setting is shown in the table below. Figure 5 .

[0347] Both datasets show that long-tailed items degrade more severely than popular items after noise injection. Long-tailed items have very few observation interactions, indicating they are more sensitive to false positive edges; when noisy edges are added to the training graph, their local neighborhoods are easily distorted, and fine-grained graph propagation thus learns the biased item representation.

[0348] LightGCN exhibits the greatest degradation across different item degree groupings because it treats all observed edges as reliable evidence of collaboration. False positive edges directly participate in message passing and contaminate item representations, particularly affecting long-tail items. NCL and ClusterGCF achieve better robustness through prototyping or cluster-enhanced modeling, while LightGNN further mitigates noise through graph simplification and redundant component pruning; however, these methods still primarily rely on fine-grained graph structures or relatively static coarse structures, remaining vulnerable when the neighborhood of long-tail items is severely contaminated.

[0349] DGBRec exhibited the smallest decrease across all item degree groups. In Gowalla, long-tail items with 1-10 interactions saw only a 9.33% decrease, while LightGNN, ClusterGCF, NCL, and LightGCN saw decreases of 14.52%, 16.39%, 21.37%, and 27.28%, respectively. On the sparser Yelp, DGBRec showed only a 12.79% decrease for long-tail items, while LightGNN and LightGCN saw decreases of 21.65% and 37.08%, respectively.

[0350] The above results further support the motivation for dual-granularity design. Long-tail items lack sufficient fine-grained evidence, and their representations are easily dominated by noisy edges. DGBRec constructs item side-granular spheres, enabling long-tail items to obtain stable cooperative evidence from behavior-related items. Purity-aware sphere-level interactions and cross-granularity refinement reduce the dependence of item representations on unreliable individual interactions. Therefore, DGBRec can improve long-tail item recommendations for both sparse users and noisy environments.

[0351] Example 4: Oversmoothing Analysis

[0352] To examine whether DGBRec can alleviate oversmoothing, the number of propagation layers was increased from 1 to 6, and compared with a model containing only fine-grained propagation. Recommendation performance and representation diversity were also reported. Normalized effective rank was used to measure representation diversity; a smaller value indicates more severe representation collapse.

[0353] like Figure 6 As shown, fine-grained propagation alone achieves the best NDCG@20 at 2 layers, and then gradually decreases with increasing depth. This indicates that simply stacking fine-grained propagation layers cannot continuously improve recommendation performance; instead, it introduces oversmoothing into the user-item graph and weakens personalization. DGBRec outperforms fine-grained propagation alone at all depths; although performance also decreases at excessive depths, the degradation is smaller.

[0354] Figure 6 The results representing diversity further support the above conclusions. When the number of propagation layers increases from 1 to 6, the normalized effective rank on the user side, which only propagates fine-grained data, decreases from 0.9194 to 0.5904, while DGBRec only decreases from 0.9362 to 0.7103; the same trend is observed on the item side, with DGBRec consistently maintaining a higher effective rank at the same depth.

[0355] DGBRec mitigates oversmoothing through coarse-grained granular inference and cross-granularity refinement. The model does not rely entirely on repetitive fine-grained message passing, but instead leverages adaptive granularities to acquire broader collaborative evidence, thus maintaining higher representational diversity at deeper propagation. However, excessive deepening still leads to performance degradation; therefore, the main experiment uses... As the default depth.

[0356] Example 5: Hyperparameter Analysis

[0357] Further analysis of the depth of dissemination Belonging temperature User-side and item-side particle count and And Top- Key hyperparameters such as membership size.

[0358] Overall, DGBRec performs stably within a reasonable range of hyperparameters. For example... Figure 7 As shown, both datasets are in The system achieves optimal performance at depth; performance degrades beyond two layers, with Yelp showing a more pronounced decline. This indicates that excessively deep, fine-grained propagation introduces noisy higher-order signals and exacerbates oversmoothing, while DGBRec, with its coarse-grained sphere inference, can capture effective cooperative dependencies using shallow propagation.

[0359] like Figure 8 As shown, the subordinate temperature Controls the smoothness of the distribution from control nodes to pellets. Gowalla at lower temperatures Best time, Yelp The optimal timeframe is consistent with the data sparsity: denser Gowalla granules can be more precisely assigned to identify reliable granules; sparser Yelp granules require softer assignments, allowing lower-density nodes to receive signals from multiple relevant granules. Temperature When the size is too large, both datasets degrade, indicating that excessive smoothing weakens the discriminative power of particles and spheres.

[0360] like Figure 9 and Figure 10 As shown, the number of spheres also affects the quality of coarse-grained collaborative abstraction. Gowalla in and The best time is when there are fewer balls, while Yelp is best when there are fewer balls. and Yelp performs best when the granularity is large enough. The optimal granularity depends on available interaction evidence: too few granules over-compress heterogeneous user preferences or item semantics, while too many granules fragment sparse interactions, making granular statistics unreliable. or The decrease in the value is more significant, further illustrating that sparse data requires a more conservative granularity setting.

[0361] like Figure 11 As shown, Top- Membership size controls the degree of overlap in particle assignment. Both datasets are in When it reaches its optimal state. At that time, the allocation is close to a hard partition, making it difficult to characterize overlapping behavioral patterns; Increasing it to a moderate value can improve performance, especially beneficial for sparse users in Yelp who can borrow signals from multiple relevant granularities; further increasing it... This introduces irrelevant particles and slightly degrades performance. Therefore, a balance needs to be struck between overlapping preference modeling and noise control.

[0362] In summary, the experiments validated the stability of DGBRec and the necessity of adaptive dual-granularity modeling. Shallow propagation, assignment smoothness matching the sparsity of the dataset, an appropriate number of spheres, and moderately overlapping membership relationships collectively support effective recommendations. As a sparser dataset, Yelp is more sensitive to granularity-related hyperparameters, while Gowalla is more stable under different settings.

[0363] Although embodiments of the invention have been shown and described, those skilled in the art will understand that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the claims and their equivalents.

Claims

1. A method for using a dynamic particle-sphere graph neural network for user and item recommendations, characterized in that, Includes the following steps: S1. Construct user-side and item-side particles in the low-dimensional spectral space. Update the particles according to the node representation of each propagation layer. Calculate the membership degree from the node to the particle using the radius-normalized distance. Define the cooperative purity based on the interaction distribution of the particles on the opposite particle. S2, which couples fine-grained interaction modeling with coarse-grained collaborative abstraction, propagates messages along two complementary paths; S3 performs bilateral sensing gating, adaptively balancing the contributions of the two propagation paths for each node; S4 aggregates the representations of each layer by weighted summation to obtain the final embeddings on the user side and the item side, and finally uses Bayesian personalized ranking loss to optimize the candidate item scores.

2. The method according to claim 1, characterized in that, The construction of user-side and item-side spheres in low-dimensional spectral space in S1 includes: S1-1, for the normalized interaction matrix Perform randomized singular value decomposition: ； in, , , , respectively denote the truncation of the left singular vector, the singular value matrix, and the right singular vector; transpose symbol This indicates that the rows and columns of the matrix are interchanged; For randomized singular value decomposition; Let be the set of real numbers; d is the embedding dimension; For the user feature matrix, It is a singular value matrix. The feature matrix of the items; Based on randomized singular value decomposition, the initial user embedding and item embedding are constructed as follows: ； ； in, The square root of the singular value matrix; embedding and As a trainable initial representation, it is updated through backpropagation during model optimization.

3. The method according to claim 2, characterized in that, S1 further includes: S1-2, in and Performing lightweight K-Means yields the initial user-side particle center. and the center of the sphere on the side of the item ,in For the first Each user's lateral granule center, For the first The center of the side particle of each item; After initialization, the model further refines the spheres through radius-aware membership learning and collaborative purity evaluation; For each initial sphere, initialize its radius with the average distance from its node to the initial center: ； ； in, Let be the radius of the k-th initial user-side particle in layer 0. Let be the radius of the s-th initial item side particle in layer 0; The set of nodes assigned to the k-th initial user-side sphere. ϵ is the set of nodes assigned to the s-th initial item-side sphere; ϵ is a minimal positive constant used for numerical stability; k is the user-side sphere index, and s is the item-side sphere index; Embed the 0th layer input for user u. For item i, embed the 0th layer of the input; Let be the center vector of the k-th user granularity in layer 0. Let be the center vector of the s-th item at the 0th layer.

4. The method according to claim 1, characterized in that, The step S1 of updating the particle based on the node representation of each propagation layer includes: S1-3, for user-side nodes With the User lateral granules Item side nodes With the Item side ball The normalized distance is defined as follows: ； ； in, Let be the normalized distance from user u to the k-th user-side particle; Let be the normalized distance from item i to the s-th item's side particle sphere; The input embedding of user u before propagation at layer l-1, The input embedding for item i before propagation at layer (l-1); Temporary membership degree is calculated as follows: ； ； in, Let be the temporary membership degree of user u at layer l to the granular ball of user k. Let i be the temporary membership degree of item i in layer l to the side particle of item s; is the temperature coefficient; k is the user-side particle index, and s is the item-side particle index.

5. The method according to claim 1, characterized in that, S1 further includes: S1-4, using temporary membership, refresh the center of the particle on both the user side and the item side by weighted average: ； ； in, For user side-sphere refresh center, Refresh the center of the item's side pellets; To avoid excessively rapid prototype drift between different layers, an exponential moving average is used to update the particle center: ； ； in, For user lateral sphere update center, Update the center for the side particles of the item; Control the smoothness of the update process; The particle radius is also updated using a layer-adaptive approach. For the k-th user-side particle, the refresh radius is defined by the weighted root mean square distance. ； ； in, Let the refresh radius of the k-th user-side particle in layer l be . Refresh radius of the side particle sphere of the s-th item in layer l; To ensure stability, both the user-side particle radius and the item-side particle radius are updated using an exponential moving average: ； ； in, Let be the radius of the k-th user-side particle in the l-th layer. Let be the radius of the s-th item's side particle in the l-th layer; This means truncating x to the interval. ; , where is the radius momentum; , These are the lower and upper limits of the radius to prevent particle degradation, respectively.

6. The method according to claim 1, characterized in that, The calculation of the membership degree from the node to the particle using the radius-normalized distance in S1 includes: S1-5, for user-side nodes With the User lateral granules Item side nodes With the Item side ball ,definition: ； ； in, The normalized distance from user u to the k-th user side sphere is based on the updated user u. This is the normalized distance from the updated item i to the s-th item's side sphere; Soft membership degree is calculated as follows: ； ； in, Let be the soft membership degree of user u at layer l to the granular ball of user k. Let be the soft membership degree of item i in layer l to the side granular ball of item s. Temperature coefficient to control the sharpness of preference distribution; For the total number of side balls of the user, The total number of particles on the item side; the user-side membership matrix is ​​denoted as... The item-side membership matrix is ​​denoted as ; To balance efficiency and interpretability, only the Top-T membership degrees of each node are retained and renormalized, where the user-side membership is taken as... Item side retrieval The T parameter is a sparsification hyperparameter, representing the membership degree to be retained after truncation and normalization. and Each row contains at most T non-zero elements.

7. The method according to claim 1, characterized in that, The definition of cooperative purity in S1 based on the interactive distribution of granules on opposite granules includes: S1-6, pp. The soft sphere-level interaction matrix of the layer is calculated as follows: ； in, This represents the aggregation interaction strength between the k-th user-side particle and the s-th item-side particle in layer l; based on For each particle, a normalized cooperative preference distribution is constructed, and its purity is measured using normalized entropy. The higher the purity, the more concentrated the cooperative preference of the particle, and the more reliable it is as a coarse-grained cooperative unit. For user side particles Its cooperative preference distribution on the item side sphere is defined as: ； Its co-purity is measured using normalized entropy: ； Similarly, for the side particles of the object. ,definition: ； as well as, ； in, Let be the probability of collaborative preference between the k-th particle on the user side of layer l and the s-th particle on the item side. Let be the probability of collaborative preference between the s-th particle on the item side and the k-th particle on the user side; The interaction matrix of the l-th layer granular level The (k,s)th element, The interaction matrix of the l-th layer granular level The (s,k)th element; Let k be the co-purity of the k-th particle on the user side of layer l. Let be the co-purity of the s-th particle on the l-th layer of the material; The larger the value, the more concentrated the cooperative preference distribution of the corresponding particles, and the more reliable they are as coarse-grained cooperative units; Smaller values ​​indicate that the particles contain a mixture of multiple heterogeneous preference patterns.

8. The method according to claim 1, characterized in that, The message propagation along two complementary paths in S2 includes: Fine-grained paths operate on the user-item interaction graph, while coarse-grained paths operate on a dynamically constructed bipartite granular graph. Fine-grained path propagation performs normalized neighborhood aggregation on the original user-item interaction graph to preserve local cooperative signals. Coarse-grained path propagation includes three stages: node-to-granularity abstraction, interaction between user-side granularities and item-side granularities, and refinement from granularities to nodes. The normalized neighborhood aggregation is defined as follows: ； ； in, This is the normalized adjacency matrix of the bipartite interaction graph; For the neighborhood aggregation embedding of user u in layer l, For the neighborhood aggregation embedding of item i in the l-th layer; For user u, the set of neighborhood items Let i be the set of neighboring users of item i; The abstraction from node to particle is that node features are aggregated into the corresponding particle to form a prototype, and strict mean pooling normalization is performed using the hyper-edge degree matrix. The hypermarginality matrix is ​​defined as follows: ； ； in, Let l be the user-side hyperedge degree matrix of the l-th layer. Let be the hyperedge degree matrix of the l-th layer item side; diag(·) is the constructor for the diagonal matrix; the soft mass of each sphere is the sum of the membership degrees of the nodes assigned to that sphere: ； ； in, Let k be the hyperedge degree corresponding to the k-th particle on the user side of layer l. Let be the hyperedge degree corresponding to the s-th particle on the side of the l-th layer of the item; The mean pooling normalization is defined as follows: ； ； in, and These are represented by spheres on the user side and the item side, respectively. The interaction between the user-side and item-side spheres employs purity-aware cross-attention learning at the sphere level, denoted as follows: [Equation 1] [Equation 2] [Equation 3] [Equation 4] [Equation 5] [Equation 6] [Equation 7] [Equation 8] [Equation 9] [Equation 10] [Equation 11] and The purity-aware bias matrix is ​​defined as follows: ； as well as, ； The spherical attention matrix for purity perception is calculated as follows: ； as well as, ； in, For the user-side purity perception bias matrix, For the purity perception bias matrix of the item side; The spherical attention matrix is ​​from the user side to the item side. A spherical attention matrix from the item side to the user side; Let l be the user-side particle embedding matrix of the l-th layer. Embedding matrix for the side particles of the item; This indicates row-wise Softmax normalization. Control the purity bias strength; since the cooperative purity is within [0,1], the additive bias is bounded, which can avoid numerical instability caused by unbounded bias; when At that time, the module degenerates into a standard spherical cross-attention module without purity modulation; The updated granular representation is as follows: ； ； in, Let be the updated particle representation matrix on the user side of layer l. This is the updated particle representation matrix for the l-th layer item side; and These are particle-sphere representations enhanced with contralateral information; The refinement from sphere to node is based on a soft-assignment distribution that projects the coarse-grained signal back to the node: ； ； in, For the l-th layer user-side coarse-grained node embedding matrix, An embedding matrix is ​​used for the coarse-grained nodes on the l-th layer of the item side; through refinement from coarse to fine, each node receives collaborative information that has already been aggregated and filtered at the coarse-grained level.

9. The method according to claim 1, characterized in that, The bilateral sensing gating in S3 includes: For user u, scalar gating is defined as: ； in, Scalar gating for user u at layer l; For the Sigmoid function; This is the transpose of the gate weight vector; For fine-grained neighborhood aggregation embedding of user u in layer l, For coarse-grained neighborhood aggregation embedding of user u in layer l; This indicates vector concatenation; The degree of user u; For degree perception weight coefficient; This is a gated bias term; The relative importance of the two paths is controlled; β is constrained to be non-negative, causing high-level nodes to tend to depend on fine-grained evidence. The degree of user u is denoted as The merged user representation is as follows: ； The same gating mechanism is used on the item side, that is, for item i, the scalar gating is defined as: ； The fused item is represented as: 。 10. The method according to claim 1, characterized in that, The weighted summation aggregation layers in S4 are defined as follows: ； in, This is the final embedding for user u; according to LightGCN settings, the default is taken as... The final embedding of item i is also represented by multiple layers of all L layers. Aggregation yields: ； Leveraging user embedding and item embedding The preference score for user u interacting with item i is defined as follows: ； The Bayesian personalized ranking loss optimization candidate item score is defined as follows: ； in, The training triples consist of user u, their observed item i, and unobserved item z sampled for that user; For the Sigmoid function; Θ represents all trainable parameters; for Regularization coefficient; the optimization objective is to improve the robustness of the model under sparse and noisy implicit feedback by utilizing an adaptive coarse-grained collaborative structure while preserving fine-grained sorting signals.

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