CN113918832A

CN113918832A - Graph convolution collaborative filtering recommendation system based on social relationship

Info

Publication number: CN113918832A
Application number: CN202111235549.5A
Authority: CN
Inventors: 刘小洋; 马敏; 赵正阳; 付晓杨
Original assignee: Chongqing University of Technology
Current assignee: Chongqing University of Technology
Priority date: 2021-10-22
Filing date: 2021-10-22
Publication date: 2022-01-11
Anticipated expiration: 2041-10-22
Also published as: CN113918832B

Abstract

The invention provides a graph convolution collaborative filtering recommendation system based on social relations, which comprises an initialization embedding module, a semantic aggregation module, a semantic fusion module, a recommendation module and an optimization module; the data output end of the initialization embedding module is connected with the data input end of the semantic aggregation module, the data output end of the semantic aggregation module is connected with the data input end of the semantic fusion module, the data output end of the semantic fusion module is connected with the data input end of the recommendation module, and the data output end of the recommendation module is connected with the data input end of the optimization module; the initialization embedding module is used for randomly initializing the embedding matrix of the node and inquiring to respectively obtain the initialization embedding of the user u and the article i; the semantic aggregation module is used for aggregating and updating the node embedding by using a semantic aggregation layer after the initial embedding of the node is obtained. The method can extract the social information of the user, has high expandability, and has rich mined semantic information and good recommendation effect.

Description

Graph convolution collaborative filtering recommendation system based on social relationship

Technical Field

The invention relates to a recommendation method, in particular to a graph convolution collaborative filtering recommendation system based on social relations.

Background

In the information explosion age, a recommendation system has become one of the most effective ways for helping users to find mass data which are interested in the users, and the core of the recommendation system is to estimate the possibility of the users receiving props according to historical interaction conditions of the users such as purchase and click. Generally, recommendation systems generally follow two steps: the vectorized representation (embedding) of the user and the item is learned and then the interaction between them is simulated (e.g., whether the user purchased the item or not). Collaborative Filtering (CF) is based on historical interactive learning node embedding on a user-item bipartite graph, and item recommendation is made by predicting user preferences based on parameters.

In general, there are two key components in the learnable CF model: 1) embedding, which converts users and items into vectorized representations; 2) interaction modeling, which reconstructs historical interactions based on embedding. For example, Matrix Factorization (MF) directly embeds user and item IDs as vectors and models user-item interactions using inner products; the cooperative deep learning expands the MF embedding function by integrating the deep representation of the rich side information of the article; the neural collaborative filtering model replaces MF interactive functions of inner products by a nonlinear neural network; the translation-based CF model uses euclidean distance metrics as interaction functions, etc.

While these methods are effective, they construct embedding functions using only descriptive features (such as ID and attributes) rather than considering user-item interaction information, which is only used to define the objective function of model training, whose embedding functions lack explicit coding of key synergy signals hidden in the user-item interaction data to produce insufficient embedding to generate satisfactory embedding for CF.

With the recent development of the graph neural network, the proposal of LightGCN makes the CF model implemented by the conventional method shift to the graph convolutional neural network. The method is a lightweight GCN network construction model, abandons the feature transformation and nonlinear activation of the traditional GCN, and verifies through experiments that the two operations are ineffective for collaborative filtering. LightGCN learns the embedding of users and items by linear propagation over the user-item interaction matrix, and finally takes the weighted sum of the embedding learned by all layers as the final embedding. Although the problem existing in the method is solved by the proposal of the LightGCN, the LightGCN is only limited to processing historical interaction data of a user-item, and cannot model the social interaction of the user so as to extract social characteristic information of the user, so that the expandability of the LightGCN is not high, and the mined semantic information is single, so that the recommendation effect is influenced.

Disclosure of Invention

The invention aims to at least solve the technical problems in the prior art, and particularly creatively provides a graph convolution collaborative filtering recommendation system based on social relations.

In order to achieve the purpose, the invention provides a graph convolution collaborative filtering recommendation system based on social relations, which comprises an initialization embedding module, a semantic aggregation module, a semantic fusion module, a recommendation module and an optimization module;

the data output end of the initialization embedding module is connected with the data input end of the semantic aggregation module, the data output end of the semantic aggregation module is connected with the data input end of the semantic fusion module, the data output end of the semantic fusion module is connected with the data input end of the recommendation module, and the data output end of the recommendation module is connected with the data input end of the optimization module;

the initialization embedding module is used for randomly initializing the embedding matrix of the node and inquiring to respectively obtain the initialization embedding of the user u and the article i;

the semantic aggregation module is used for aggregating and updating node embedding by using a semantic aggregation layer after the initial embedding of the nodes is obtained; firstly, introducing first-order semantic aggregation in a semantic aggregation layer, and then expanding the first-order semantic aggregation to each layer to realize high-order semantic aggregation;

the semantic fusion module is used for fusing the user embedded vectors of the social embedding propagation layer and the interactive embedding propagation layer after respectively obtaining the semantic aggregation embedded vector of the social embedding propagation layer and the semantic aggregation embedded vector of the interactive embedding propagation layer; then, weighting, summing and fusing each order of embedding obtained by each embedding propagation layer to obtain final user embedding and article embedding;

the fusion adopts an aggregation mode of firstly adding element by element, then activating a function and finally regularizing;

the recommendation module is used for recommending products for the user according to the embedding of the articles;

the optimization module is used for optimizing the products in the recommendation module.

Further, the first-order semantic aggregation in the semantic aggregation module comprises:

the interaction embedding propagation layer refines the embedding of the user by aggregating the embedding of the interaction articles, and refines the embedding of the articles by aggregating the embedding of the interaction users; the first-order semantic aggregation is respectively expressed by the formula (1) and the formula (2):

wherein e is_uRepresenting the embedding of user u obtained by semantic aggregation of interactive embedding propagation layers;

AGG (. cndot.) is the aggregation function;

H_ua first-order neighbor set representing the user u, namely an item set interacted with the user u;

e_irepresents the embedding of item i;

H_irepresenting a first-order neighbor set of an item i, namely a user set interacted with the item i;

the social embedding propagation layer refines the embedding of the user by aggregating friends, and records the embedding of the user performing semantic aggregation in the social embedding propagation layer as c, so that the first-order semantic aggregation process of the social embedding propagation layer is shown as formula (3):

wherein, c_uRepresenting the embedding of user u by semantic aggregation of the social embedding propagation layer;

c_vrepresenting the embedding of user v by semantic aggregation of social embedding propagation layers;

a user v is a first-order friend of a user u, and v is not equal to u;

AGG (. cndot.) is the aggregation function;

F_urepresenting a set of friends of user u.

Further, high-order semantic aggregation in the semantic aggregation module realizes high-order semantic aggregation by overlapping a plurality of first-order semantic aggregation layers; the high-order semantic aggregation comprises: semantic aggregation of social embedding propagation layer and semantic aggregation of interactive embedding propagation layer:

the semantic aggregation of the social embedding propagation layer comprises:

semantic aggregation of social embedding propagation layer capturing higher-order friend signals by overlapping a plurality of social embedding propagation layers to achieve the purpose of enhancing user embedding, the mathematical expression of the process is shown as formula (4) and formula (5):

wherein the content of the first and second substances,

an embedding vector representing a user u of a (k + 1) th layer obtained by semantic aggregation of the social embedding propagation layer;

F_ua set of friends representing user u;

F_va set of friends representing user v;

the embedding vector of the user v of the k layer is obtained through semantic aggregation of the social embedding propagation layer;

the embedding vector of the user v at the k +1 th layer is obtained through semantic aggregation of the social embedding propagation layer;

an embedding vector representing a user u of a k-th layer obtained by semantic aggregation of the social embedding propagation layer;

| DEG | represents the number of elements in the solution set;

the semantic aggregation of the interaction embedding propagation layer comprises the following steps:

semantic aggregation of interaction embedding propagation layers enhances user and article embedding by stacking multiple interaction embedding propagation layers to capture cooperative signals of interaction high-order connectivity, and the mathematical expression of the process is as shown in formula (6) and formula (7):

wherein the content of the first and second substances,

denotes the embedding of item i at layer k + 1;

H_ia first-order neighbor set representing item i;

H_ua first-order neighbor set representing user u;

represents the embedding of user u at layer k;

represents the embedding of user u at layer k + 1;

represents the embedding of item i of the k-th layer;

| · | represents the number of elements in the solution set.

Further, the process of fusion in the semantic fusion module includes:

wherein the content of the first and second substances,

representing fusion of user embedding vectors of a k level of a social embedding propagation layer and an interactive embedding propagation layer;

representing the embedding of the user u of the k layer obtained by semantic aggregation of interactive embedding propagation layers;

g (. cndot.) is the polymerization mode.

Further, the user embedding and the article embedding in the semantic fusion module comprise:

wherein the content of the first and second substances,

embedding a user u fusing a social embedding propagation layer and an interactive embedding propagation layer;

k represents the total number of layers;

α_kis the weight when the k-th layer aggregates the embedding of the user;

e_iis the embedding of item i;

β_kis the weight at which the kth layer aggregates the embedding of the item;

indicating the embedding of item i of the k-th layer.

Further, the aggregation mode that element-by-element addition is performed first, then an activation function is performed, and finally row regularization includes:

where norm (·) represents row regularization;

jh (-) is the activation function;

presentation pair

Adding element by element;

representing the embedding of the user u of the (k + 1) th layer obtained by semantic aggregation of interactive embedding propagation layers;

an embedding vector representing user u at level k +1 obtained by semantic aggregation of social embedding propagation layers.

The aggregation mode of firstly adding element by element and then regularizing can also be adopted:

where norm (·) represents row regularization;

presentation pair

Adding element by element;

The aggregation mode of firstly solving the Hadamard product and then regularizing the line can also be adopted;

an indication of a hadamard product;

the method can also adopt a mode of splicing firstly and then reducing the dimensionality to be the same as the original polymerization mode through the full connecting layer:

wherein f (-) is a fully connected layer;

w is a weight;

show that

And

splicing is carried out;

b is an offset.

Further, the recommendation module includes:

using the inner product of the user and the recommended item as a predictive score, as shown in equation (12):

a score representing the score of the prediction score,

representing the final embedding of the user u,

t represents the transpose of the image,

e_iindicating the embedding of item i.

Further, the graph convolution collaborative filtering recommendation system based on the social relationship may be implemented by using an SRRA, where the SRRA includes the following steps:

S-A, notation of user-item interaction matrix

Where M and N are the number of users and items, R, respectively_uiIs the value of the u row, i column of the R matrix, where R is the user u and item i if there is an interaction_uiNot all right 1, otherwise R _ui0; a adjacency matrix of the user-item interaction graph may then be obtained, as shown in equation (14):

wherein A is an adjacency matrix of a user and article interaction diagram;

r is an interaction matrix of the user and the article;

t represents transpose;

S-B, let the embedded matrix of layer 0 be E⁽⁰⁾The user or article embedding matrix for obtaining the (k + 1) th layer is shown as the formula (15):

wherein D is a degree matrix;

a is a adjacency matrix;

E^(k)is a user or item embedding matrix of the k-th layer;

S-C, user societyThe intersection matrix is recorded as

Where user u and user v are friends then S_uv1, otherwise S_uv＝0，S_uvIs the value of the u row and v column of the S matrix; a adjacency matrix of the user's social graph may be obtained, as shown in equation (16):

S-D, let the embedded matrix of layer 0 be

The user embedded matrix of the k +1 th layer is obtained as shown in the formula (17):

wherein, P is a degree matrix corresponding to the matrix B;

b is an adjacency matrix of the user social graph;

C^(k)embedding a matrix for the users of the k layer;

S-E, respectively intercepting matrix E^(k)And matrix C^(k)The parts of (2) related to user embedding are respectively marked as E_u ^(k)And C_u ^(k)，E_u ^(k)And C_u ^(k)All represent a user embedded matrix of the k-th layer, where E_u ^(k)Is derived from user-item interactions, and C_u ^(k)Is derived from social relationships;

then matrix E^(k)About the part of the article being embedded is denoted as E_i ^(k)Having E of^(k)＝concat(E_u ^(k),E_i ^(k)) Wherein concat (E)_u ^(k),E_i ^(k)) Denotes a reaction of E_u ^(k)And E_i ^(k)Splicing is carried out;

S-F, calculating a representation of the user according to equation (18):

wherein, sum (E)_u ^(k),C_u ^(k)) Represents a pair E_u ^(k)And C_u ^(k)Summing is carried out;

norm (·) represents a row regularization operation;

E_u ^(k)representing a user embedding matrix of a k layer obtained according to the user-article interaction relation;

C_u ^(k)representing a user embedding matrix of a k-th layer obtained by social relations;

S-G, obtaining final representations of the user and the item, respectively, by fusing the representations of the layers according to equation (19):

wherein the content of the first and second substances,

representing the final user embedding matrix;

k represents a k-th layer;

k represents the total number of layers;

α_kis the weight when the k-th layer aggregates the embedding of the user;

representing the final article embedding matrix;

β_kis the weight at which the kth layer aggregates the embedding of the item;

E_i ^(k)denotes the kthEmbedding the layered article into a matrix;

S-H, calculating a prediction score according to the formula (20):

wherein the content of the first and second substances,

represents a predicted score;

to represent

Transposing;

representing the final article embedding matrix;

S-I, calculating a loss function using BPR as shown in equation (21):

wherein L is_BPRRepresenting the BPR loss in matrix form;

m is the number of users;

u is the user;

i, j are both items;

ln σ (·) denotes the natural logarithm of σ (·);

σ (-) is a sigmoid function;

means that user u scores a prediction of item i；

The item j is predicted and scored by the user u;

λ represents control L₂The strength of the regularization is used to prevent overfitting;

E⁽⁰⁾an embedded matrix representing layer 0;

| | · | | represents a norm.

Further, in the optimization module, the optimization method is as follows:

wherein L represents BPR loss;

o represents paired training data;

u is the user;

i, j are both items;

ln σ (·) denotes the natural logarithm of σ (·);

σ (-) is a sigmoid function;

the item i is predicted and scored by a user u;

the item j is predicted and scored by the user u;

Θ represents all trainable model parameters;

is the square of the two norms.

Further, the intelligent mobile handheld terminal comprises a user searching and transmitting module, the data output end of the initialization embedding module is connected with the data input end of the semantic aggregation module, the data output end of the semantic aggregation module is connected with the data input end of the semantic fusion module, the data output end of the semantic fusion module is connected with the data input end of the recommending module, the data output end of the recommending module is connected with the data input end of the optimizing module, the data output end of the optimizing module is connected with the data input end of the user searching and transmitting module, and the user searching and transmitting module is used for transmitting the optimized recommended product to the corresponding intelligent mobile handheld terminal of the user.

In summary, due to the adoption of the technical scheme, the invention has the beneficial effects that:

(1) the social relationship is innovatively integrated into the training of the collaborative filtering recommendation method based on graph convolution, a graph convolution collaborative filtering recommendation model (SGCF) integrated with the social relationship is provided, and the embedding of the nodes is learned through the integration of social behaviors and high-order semantic information of interactive behaviors.

(2) An implementable recommendation algorithm (SRRA) is provided under a constructed SGCF model framework, high-order relations in user-item interaction data and social data are modeled respectively, and then the semantic information of the two high-order relations in each layer is fused to form final user and item expressions which are finally used for recommending tasks.

(3) And comparison experiments are carried out on a plurality of real data sets with social information and a baseline model, and the rationality, effectiveness and computing performance superiority of the SGCF model and the SRRA algorithm are verified.

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.

Drawings

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

FIG. 1 is a schematic diagram of a CF-based social recommendation.

Fig. 2 is a schematic diagram of the graph embedding principle.

Fig. 3 is a schematic diagram of a HIN recommendation system.

FIG. 4 is a schematic diagram of user social relationships.

FIG. 5 is a schematic view of a user-item interaction relationship.

Fig. 6 is a schematic diagram of a framework structure of the SGCF model proposed by the present invention.

FIG. 7 is a diagram showing the relationship between the performance improvement value of each evaluation index and S-sensitivity according to the present invention.

FIG. 8 is a schematic diagram of the SRRA and baseline model evaluation index training curves of the present invention.

Detailed Description

Reference will now be made in detail to embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein like or similar reference numerals refer to the same or similar elements or elements having the same or similar function throughout. The embodiments described below with reference to the accompanying drawings are illustrative only for the purpose of explaining the present invention, and are not to be construed as limiting the present invention.

1 research motivation

Based on the analysis, a graph convolution neural network collaborative filtering recommendation method fusing social relations is provided, so that the following basic problems are solved.

Heterogeneous data is difficult to exploit: the network containing both the user interaction information and the user social information in the heterogeneous graph is a more complex heterogeneous graph. How to deal with the complex structure information for recommendation is a problem which needs to be solved urgently.

High-order semantic information is difficult to extract: the retention of different long-term dependencies among high-order semantic information capturing nodes is the key for improving node embedding and relieving the cold start problem of a recommendation system. How to inject high-order semantics into node embedding is a fundamental issue of the recommendation system.

Various semantic information is difficult to fuse: in a data set needing to be processed, there are two types of semantic information, namely social information and interaction preference information, and how to fuse and inject the two types of semantic information into user embedding is a basic problem to be solved.

2 related work

2.1 traditional collaborative filtering recommendation Algorithm

Collaborative filtering algorithms have been widely used in the e-commerce industry, and many collaborative filtering algorithms have emerged in the academic and industrial communities over the past two decades. Roughly speaking, such algorithms can be divided into two categories: based on neighborhood collaborative filtering algorithm and model recommendation algorithm.

1) Neighborhood-based recommendation algorithm

The neighborhood-based algorithm principle is to rank the target users or target items according to their similarities to neighbors and predict according to the scores of the most similar top-k neighbors, which can find potential information from the users 'past behavior to directly predict the users' interests without any domain knowledge. Neighborhood-based collaborative filtering algorithms primarily use user-item interaction data or sample data to accomplish prediction, which can be further divided into user-based collaborative filtering algorithms and item-based collaborative filtering algorithms.

The principle of the user-based collaborative filtering algorithm is to predict the unknown rating of a user on an item by using the weighted average of all the ratings of similar users on the item, and the item-based collaborative filtering algorithm is to predict the rating of a user on an item based on the average rating of the user on similar items. The key problem with the neighborhood-based CF approach is to compute the similarity and how to weight the aggregated score.

2) Model-based recommendation algorithm

The main idea of the model-based recommendation algorithm is to embed both the user and the item into a common potential subspace and then predict through the inner product between the implicit factors of the user and the item.

Model-based methods apply data mining and machine learning techniques to find matching models from training data to predict unknown scores. Model-based CF is more comprehensive than neighborhood-based CF, and it can mine the underlying information of explicit scoring levels. Common model-based methods include random walk-based methods and factorization-based CF models. The CF method based on factorization is one of the most popular methods at present and is widely used to construct recommendation systems.

However, the conventional collaborative filtering recommendation method only uses the user-item interaction data, so the recommendation precision is limited.

2.2 socialized recommendation Algorithm

Most existing social recommendation systems today are based on CF technology. A CF-based social recommendation system, also referred to as a social recommendation system, is shown in fig. 1.

It can be seen in FIG. 1 that social recommendations have two inputs, namely user-item interaction information and social information. The generic CF-based social recommendation framework contains two parts: basic CF models and social information models.

According to different fusion mechanisms of user-item interaction data and social data, social recommendation systems can be divided into two main categories: regularization-based recommendation systems and feature-based shared social recommendation systems.

1) Socialized recommendation algorithm based on regularization

The regularized socialization-based recommendation algorithm is based on the assumption that: users trust friends in their social circle more than strangers, consistent with their preferences. The regularization-based recommendation is implemented by converting social data and scoring data into the same target space, and constraining limits to each other so that the social influence of the user can be considered before the user makes a decision. SocialMF and CUNE are two representative algorithms in this group.

The SocialMF is intended to constrain the user's preferences to approximate the average preferences of the user's social network. Socimf solves the transitivity of trust in a trust network because the potential feature vectors of a user depend on the potential feature vectors of direct neighbors, neighbor feature vectors can propagate in the network, and make the potential feature vectors of a user depend on potentially all users in the network.

Because explicit user-user relationships extracted directly from social information have many limitations, the CUNE proposes to extract implicit and reliable social information from user feedback, determine top-k semantic socialization for each user, and then add top-k semantic friend information to MF and BPR frames to solve the problems of score prediction and item ordering, respectively.

A model is established indirectly in a social network based on a regularized social recommendation algorithm, thereby helping the model reduce cold start problems and increase coverage of recommended items. However, since the social information is indirectly modeled, the contact degree and the association degree of the user-item interaction information and the social information are low, which results in that the recommendation algorithm cannot effectively integrate the social information and the scoring information.

2) Socialized recommendation algorithm based on feature sharing

The basic assumptions based on the feature sharing recommendation algorithm are: user feature vectors in the user-item interaction space and the user-user social space are shared. The principle of the method is that the user-article interaction information and the social information share the user feature vector, and the user-article interaction information and the social information can be converted into the same space to be subjected to joint learning so as to obtain the feature representation of the user. TnustSVD and SoRec are two representative recommendation systems for this approach.

TrustSVD not only models scoring data and user trust relationship data, but also considers implicit behavior data and social relationship data of users. Therefore, the method adds implicit social information on the basis of the SVD + + model to improve the recommendation precision.

The SoRec method is based on the assumption that there is diversity in the social preferences trusted by the user. The user low-dimensional feature vector is learned by decomposing the scoring matrix and the social relationship matrix at the same time, so that the learned user feature vector can take the scoring habits and social characteristics of the user into account.

The social recommendation prediction can be accurately realized when the social recommendation prediction task is completed based on the feature sharing recommendation algorithm. However, the algorithms currently proposed in the mainstream of society only use original social information, and therefore cannot fully utilize social data. At this point the graph embedding algorithm gradually walks into the people's field of view.

2.3 recommendation Algorithm based on graph embedding

Network embedding, also known as network representation learning and graph embedding, is one of the popular research directions in the field of graph data mining in recent years, and is a process for mapping graph data (generally a high-dimensional dense matrix) into a low-micro dense vector, so that the obtained vector form can have the capability of representation and reasoning in a vector space, and can be used as an input of a machine learning model, and further, the obtained vector representation can be applied to a recommendation task.

Network embedding may represent the representation of the graphical data in vector form. The vector form may preserve the structural information of the node in the graph, i.e. the more structurally similar in the graph, the closer its position in the vector space. The graph embedding principle is shown in fig. 2.

It can be seen from fig. 2 that

nodes

1 and 3 are similar in structure, so they remain symmetrically positioned in vector space;

nodes

4,5,6,7 are structurally equivalent so that their positions in vector space are the same.

The social recommendation system based on the isomorphic information network and the social recommendation system based on the heterogeneous information network can be divided according to the types of the networks. The principles and classification of these two types of algorithms will be described in detail below.

1) Recommendation algorithm based on isomorphic graph embedding

The homogeneous graph contains only one type of node and edge, which only needs to aggregate a single type of neighbor to update the node representation. Perozzi et al proposed a random walk (Deepwalk) algorithm suitable for homogeneous graphs, which uses a truncated random walk sequence to represent the neighbors of a node, and then combines the obtained sequences as sentences in natural language processing to obtain vector representation of the node.

However, the random walk strategy in Deepwalk is completely random, so node2vec was proposed. The node2vec further expands the Deepwalk algorithm by changing the generation mode of the random walk sequence, the mode of selecting the next node in the random walk sequence by the Deepwalk is uniformly and randomly distributed, and the node2vec introduces the width-first search and the depth-first search into the generation process of the random walk sequence by introducing two parameters p and q.

The problems of data sparsity and cold start of the recommendation system are well solved based on the isomorphic network algorithm. But the graph in the real world can be modeled, for the most part, naturally as a heterogeneous graph. Therefore, recommendation algorithms based on heterogeneous networks are receiving increasing attention.

2) Recommendation algorithm based on heterogeneous graph embedding

A Heterogeneous Information Network (HIN) is composed of various types of nodes and edges, and fig. 3 is an exemplary diagram of a HIN-based recommendation system.

As can be seen in fig. 3, a HIN includes two or more types of entities linked by a plurality of (two or more) relationships.

Under the heterogeneous network representation, the recommendation problem can be regarded as a similarity search task on the HIN. The basic idea of most existing HIN-based recommendation methods is to make recommendations on HIN using path-based semantic correlations between users and items, e.g. meta-path-based similarities. And several path-based similarity metrics are proposed to evaluate the similarity of objects in heterogeneous information networks. Wang et al propose to integrate social tag information as additional information into the HIN to overcome the problem of data sparsity. Most HIN-based approaches, however, rely on display meta-paths, which may not fully mine the potential features of users and items on the HIN for recommendations.

The advent of network embedding has demonstrated its ability to fully mine the underlying information of data, and researchers are increasingly focusing their attention on this. Deepwalk generates a sequence of nodes by random walk and then learns the node embedding representation by the Skip-Gram model. Furthermore, LNES and SDNE characterize the proximity of second order links and neighbor relationships.

Most graph embedding methods, however, focus on homogeneous networks, and therefore they cannot migrate and apply directly to heterogeneous networks. While the literature attempts to analyze heterogeneous networks through embedded methods, few have modeled the entire system as a heterogeneous network for social recommendations to capture the similarities of users that are implicit to each other on a social network, although these methods have made good improvements.

Definition of 3 problems

3.1 higher order connectivity

3.1.1 social high-level connectivity

The social relationship has a higher order connectivity, which is shown in FIG. 4 (c).

In fig. 4, the target node is marked with a double circle. In FIG. 4(c), path length, path and no direct links are shown, reflecting potential friends that may be. The closer the distance is to all the accessible paths, the more paths it occupies, and the greater the impact on it.

3.1.2 Interactive high-order connectivity

The interaction also has a high-order connectivity. Interactive high-order connectivity is shown in fig. 5 (c).

In FIG. 5(c), the user of recommended interest is marked with a double circle in the left sub-graph of the user-item interaction graph. The right subgraph shows the tree structure from the development. High-order connectivity means paths that are reached from any node with a path length greater than 1. This high-order connectivity contains rich semantic information with collaborative signals. For example, the path represents the similarity of behavior between sums, since both users interacted with; a longer path indicates a high probability of being taken because its similar users have interacted with before. Also, from the perspective of the path, items are more likely to be of interest than items, because there are two paths that are connected, and only one path is connected.

4 recommendation method

4.1 SGCF recommendation model

The basic idea of the SGCF is to learn node embedding of users and items by fusing the high-order semantics of social and interactive behaviors. The SGCF models the high-order relations in the user-article interaction data and the social data respectively to learn the embedding of the user and the article, finally fuses the semantic information of the two high-order relations in each layer to form a final user expression, and fuses the semantic information of the high-order interaction relations in each layer to form a final article expression for a final recommendation task. The overall frame structure of the model is shown in fig. 6.

As shown in fig. 6, the SGCF first adopts initialization embedding layer initialization node embedding, then performs semantic aggregation operation on the social embedding propagation layer and the interactive embedding propagation layer in the semantic aggregation layer to refine embedding of the user and the article, fuses the two user embedding parts in the semantic fusion layer, then weights and sums the embedding of the user and the article in each propagation layer to form a final embedded representation, and finally scores the embedding parts in the prediction layer for recommendation.

4.1.1 initialization embedding layer

Randomly initializing an embedding matrix of nodes and inquiring to obtain initialized embedding of a user u and an article i respectively

And

where g is the dimension of the node embedding.

To represent

Is the embedded vector of user u (one node); this vector is g-dimensional, and each component of the vector belongs to the real number domain;

to represent

Is the embedded vector of item i (a node); this vector is g-dimensional and each component of the vector belongs to the real number domain.

4.1.2 semantic aggregation layer

After obtaining the initialized embedding of the nodes, a semantic aggregation layer is proposed to aggregate and update the node embedding, so high-order semantic information can be well preserved. First-order semantic aggregation is introduced into a semantic aggregation layer and then the semantic aggregation is expanded to each layer, so that high-order semantic aggregation is realized.

1) First order semantic aggregation

The basic idea of the graph neural network GCN is to learn the representation of the nodes by smoothing the features on the graph. To achieve this, it iteratively convolves the graph, i.e. aggregates the features of the neighbors as a new representation of the target node. In the SGCF, the interaction embedding propagation layer refines the embedding of users by aggregating the embedding of interaction articles, and refines the embedding of articles by aggregating the embedding of interaction users. The first-order semantic aggregation is respectively shown as a formula (1) and a formula (2).

Wherein e_uIndicating the embedding of user u, e_iRepresenting the embedding of an item i, AGG (-) being a polymerization function, H_uA first-order neighbor set representing user u, i.e. a set of items with which user u has interacted, H_iA first-order neighbor set representing item i, i.e., a set of users who have interacted with item i. The above formula shows the embedding e of user u in the interaction_uBy aggregation of the embeddings of an item i, whose first-order neighbours are (directly interacting), while the embeddings e of an item i_iBy means of embedded aggregation of its first-order neighbors (directly interacted with) user u.

The social embedding propagation layer refines the embedding of the user by aggregating friends. In order to distinguish meanings well, a user embedding of semantic aggregation in the social embedding propagation layer is marked as c, and then a first-order semantic aggregation process of the social embedding propagation layer is shown as a formula (3)

Wherein c is_uAnd c_vAll are user embedding, user v is a first-order friend of user u, and v ≠ u; AGG (. cndot.) is the aggregation function, F_uRepresenting a set of friends of user u. The above formula shows that in social interaction, user u is embedded in e_uBy embedding e into its first-order neighbors (socializes)_vPolymerization occurs.

2) High-order semantic aggregation

The semantic aggregation layer realizes the aggregation of high-order semantics by overlapping a plurality of first-order semantic aggregation layers. It includes semantic aggregation of social embedding propagation layers and interaction embedding propagation layers.

Semantic aggregation of social embedding propagation layers

As known by social high-order connectivity, stacking k layers can aggregate information to k-order neighbors. Semantic aggregation of social embedding propagation layers captures higher-order friend signals by overlapping a plurality of social embedding propagation layers to achieve the purpose of enhancing user embedding, and mathematical expressions of the process are shown as formulas (4) and (5).

Wherein

An embedding vector representing user u at level k +1 obtained by semantic aggregation of the social embedding propagation layer,

an embedding vector, F, representing a user u of level k obtained by semantic aggregation of social embedding propagation layers_uSet of friends, F, representing user u_vA set of friends that represents user v,

refers to the embedding vector of the user v at the k +1 th layer obtained by semantic aggregation of the social embedding propagation layer,

refers to the embedding direction of the user v of the k layer obtained by semantic aggregation of the social embedding propagation layerAmount of the compound (A). It should be noted that

Embedded for initialization of user u. | · | represents the number of elements in the solution set.

Semantic aggregation of interaction embedding propagation layers

According to the interactive high-order connectivity, the overlapping even layer (namely, the path length from the user is even) can capture the similarity information of the user behavior, and the overlapping odd layer can capture the potential interactive information of the user on the article. Semantic aggregation of interaction embedding propagation layers user and item embedding is enhanced by overlaying multiple interaction embedding propagation layers to capture collaborative signals of high-order connectivity in the interaction, and mathematical expressions of the process are shown as formulas (6) and (7).

Wherein

And

respectively representing the embedding of user u and item i at layer k, H_iRepresenting a first-order neighbor set of item i, H_uRepresenting a first order neighbor set of user u.

4.1.3 semantic fusion layer

1) End-user embedding formation

Merging of embedding of users of the social part and embedding of users of the interactive part (say that the embedding of users of the social part has 3 layers, then correspondingly, the user embedding of the interactive partAnd 3 layers are also arranged, the social embedding of the users on the 1 st layer and the interactive embedding of the users on the 1 st layer are fused in a one-to-one correspondence manner during fusion, and the like. Wherein a layer means the order of captured information, a layer 1 represents capturing only 1 order information, a layer 2 represents capturing 2 order information, and so on), and the function of the part is to enable the final user to embed the information with social information and interactive information at the same time. Using the formula

Fusion of layers. The role of this part is to enable the end user to embed information that can capture the various orders. The formula used:

2) formation of the final article inlay

Different from final user embedding, the final user embedding uses social information and interactive information, and the final article embedding only uses the interactive information, so that the article embedding method only fuses article interactive embedding of all layers, namely only 2 in 1), and the second step.

The formula is used:

the weighting is used only when the layers are fused, and the formula is already embodied.

Specifically, the method comprises the following steps: the user embedding by fusing the social embedding propagation layer and the interactive embedding propagation layer can be enabled to carry certain social information, so that the quality of the user embedding is enhanced. After the semantic aggregation embedded vector of the social embedding propagation layer and the semantic aggregation embedded vector of the interactive embedding propagation layer are obtained respectively, the user embedded vectors of the two layers are fused, and the fusion process is shown as a formula (8).

Wherein the content of the first and second substances,

the user embedding vectors of the k level of the social embedding propagation layer and the interactive embedding propagation layer are fused, wherein g (-) can be in a multiple aggregation mode, and the formula (9) is adopted, and the user embedding vectors are added element by element and then are normalized.

Where norm (·) represents regularization,

presentation pair

The addition is carried out element by element,

indicating the embedding of user u at layer k +1,

and the embedding vector of the user u at the (k + 1) th layer obtained by semantic aggregation of the social embedding propagation layer is represented, namely the user social embedding at the (k + 1) th layer.

Furthermore, g (-) can add an activation function on the basis of the formula (9); or by first taking the Hadamard product and then line regularization, i.e.

Or can first

Splicing the two parts, wherein the dimension is changed to 2 times of the original dimension, and then reducing the dimension to be the same as the original dimension through a full connection layer f (·), namely

Then, carrying out weighted summation and fusion on each order of embedding obtained by embedding and propagation of each layer to obtain final user embedding

And article embedding e_iAs shown in formula (11).

Wherein

Representing the fusion of user embedded vectors of a K level of a social embedded propagation layer and an interactive embedded propagation layer, wherein K represents the K level, K represents the total number of layers, and alpha represents the total number of layers_kIs the weight, β, at which the k-th layer aggregates the embedding of the user_kThe weight of each layer may be the same or different, and if the weight of each layer is the same, the embedding of each layer contributes the same to the finally formed embedding, and the larger the weight is, the larger the contribution is.

4.1.4 prediction layers

The last part of the model recommends a product for the user based on the embedding of the item, where the inner product of the user and the recommended item is used as the prediction score, as shown in equation (12).

A score representing the score of the prediction score,

represents the final embedding of user u,. T represents transpose,. e_iRepresents the embedding of item i;

the BPR loss is then calculated and the model parameters are optimized according to the calculated BPR loss as shown in equation (13).

Where L represents BPR loss, σ (-) is a sigmoid function,

refers to the user u scoring the prediction of positive sample i,

the method refers to the prediction of the user u on the negative sample j and scoring; o { (u, i, j) | (u, i) ∈ R⁺,(u,j)∈R^-Represents paired training data, u is user, i, j are both items, i ≠ j, except that i is a positive sample, which appears in the interaction list of u, j is a negative sample, which does not appear in the interaction list of u. R⁺Representing observable interactions, R^-Representing an unobservable interaction. Θ represents all trainable model parameters, where the parameters of the model include only the initialized embedded vectors for user u and item i

And

λ represents control L₂The strength of the regularization is used to prevent overfitting. ln σ (-) represents the natural logarithm of σ (-),

is the square of the two norms.

4.2 recommendation algorithm SRRA

For ease of implementation, the SRRA Algorithm is proposed under the framework of the SGCF model (see Algorithm 1 for details).

Noting the user-item interaction matrix as

Where M and N are the number of users and items, respectively, whichUser u and item i if there is an interaction R_uiNot all right 1, otherwise R _ui0. A adjacency matrix of the user-item interaction graph may then be obtained, as shown in equation (14).

Where A is the adjacency matrix of the user-item interaction graph, R is the interaction matrix of the user-item, and T represents the transpose.

Let the embedded matrix of layer 0 be

Where G is the dimension of the embedding vector, the user or article embedding matrix at the k +1 th layer can be obtained as shown in equation (15).

Where D is a degree matrix, which is a diagonal matrix with dimensions (M + N) x (M + N), M and N being the number of users and items, respectively; the value of the ith row and ith column of the matrix D is denoted as D_ii，D_iiFor degree of node i, i.e. each element D_iiRepresenting the number of non-zero values located in the ith row vector of the adjacency matrix a.

Similarly, the social matrix of the user is recorded as

Where user u and user v are friends then S_uv1, otherwise S_uv＝0，S_uvIs the value of the u-th row, v-th column of the S-matrix. The adjacency matrix of the user social graph may be obtained as shown in equation (16).

Let the embedded matrix of layer 0 be

The user embedded matrix of the k +1 th layer can be obtained as shown in the formula (17).

Wherein, P is the degree matrix corresponding to the matrix B, and B is the adjacency matrix of the user social graph.

Then respectively intercepting the matrix E^(k)And matrix C^(k)With respect to the user-embedded part, i.e. the truncation matrix E^(k)And matrix C^(k)The first M rows of (1), respectively denoted as E_u ^(k)And C_u ^(k)，E_u ^(k)And C_u ^(k)All represent the user embedding matrices of the k-th layer, but they are distinct E_u ^(k)Is derived from user-item interactions, and C_u ^(k)Is derived from social relationships. Then matrix E^(k)About the part of the article being embedded is denoted as E_i ^(k)Having E of^(k)＝concat(E_u ^(k),E_i ^(k)) I.e. E^(k)Is actually composed of E_u ^(k),E_i ^(k)The two matrixes are spliced; wherein C is_u ^(k),

Finally, the user's representation is calculated according to equation (18).

Wherein sum (E)_u ^(k),C_u ^(k)) Represents a pair E_u ^(k)And C_u ^(k)The summation is carried out, norm (-) represents the row regularization operation, and the row regularization is normalized by each row unit of the matrix, namely, the elements of the row are summed first, and then each element of the row is usedThe sum is divided by the elements, respectively, and the resulting value replaces the line.

The final representations of the user and the item, respectively, are obtained by fusing the representations of the layers according to equation (19).

α_kIs the weight, β, at which the k-th layer aggregates the embedding of the user_kIs the weight at which the k-th layer aggregates the embedding of the item.

Calculating a prediction score according to equation (20):

the loss function is calculated using the BPR as shown in equation (21).

Wherein H_uA first-order neighbor set representing the user u, namely an item set interacted with the user u; e⁽⁰⁾Representing the embedded matrix of layer 0, M being the number of users, | | · | |, representing the norm.

Equation (21) essentially corresponds to equation (13) except that (21) is in matrix form and the model parameters Θ are only E⁽⁰⁾Nothing else is included.

5 results and analysis of the experiments

The experiment uses 6 real data sets, which all contain social data and user behavior data, and the statistical data of the data sets are shown in table 2. The proposed SRRA algorithm is compared with two leading-edge baseline algorithms DSCF and LightGCN to verify the reasonability and effectiveness of the proposed SRRA algorithm.

5.1 data set

1) Brightkite this data set includes user check-in data and user social network data, which can be used for location referrals.

In order to ensure the quality of the data set, the lower limit of interaction of the users is limited to 100, and the upper limit of interaction is limited to 500, that is, each user has 100 check-in places and at most 500 check-in places.

2) Gowalla this is a check-in dataset obtained from Gowalla where users share their location through check-in. Similarly, the lower limit of interaction of the users is limited to 100, and the upper limit of interaction is limited to 500, that is, each user has at most 500 check-in places and at most 100 check-in places.

3) LastFM is a data set published by the second recommended system information isomerism and convergence international seminar. The data set includes music artist data to which the user listens and social network data of the user. The user interaction is limited to a lower limit of 10, i.e. each user has up to 10 favorite artists.

4) FilmTrust is a small dataset that was crawled from the FilmTrust website in 2011 at 6 months. Including the scoring information of the movie by the user and the social information among the users. The user is limited to an interaction with a lower limit of 10, i.e. each user goes to movies with 10 scores.

5) The Delcious data set contains social networks, bookmarks and tag information among users from the Delcious social bookmarking system. The lower limit of interaction of the users is limited to 10, and the upper limit of interaction is 500, namely, each user has up to 10 social bookmarks.

6) Epins this data set contained scores for 139,738 items of 49,290 users, each item being scored at least once, and contained trust relationships between users, for a total of 487,181 user trust pairs. The lower limit of interaction for the users is limited to 10, i.e. each user has up to 10 interactive items.

TABLE 2 statistics of the data set

Dataset	User#	Item#	Interaction#	Connection#	R-Density	S-Density
							Brightkite	6,310	317,448	1,392,069	27,754	0.00069	0.00070
Gowalla	14,923	756,595	2,825,857	82,112	0.00025	0.00037
							Epinions	12,392	112,267	742,682	198,264	0.00053	0.00129
FilmTrust	58	657	1,530	590	0.04015	0.17539
							Delicious	479	23,341	103,649	6,180	0.00927	0.02694
LastFM	1,860	17,583	92,601	24,800	0.00283	0.00717

Note: interaction is the number of user-item interactions, Connection is the number of user social connections, R-sensitivity is the Density of the user-item matrix, and S-sensitivity is the Density of the social matrix

5.2 Experimental setup

To evaluate the experimental results, each data set was individually evaluated at 7: 3, dividing the proportion into a training set and a testing set, and taking Pre @10, Recall @10 and NDCG @10 as evaluation indexes of the model.

Referring to LightGCN, dimensions of the embedding vectors of all models are set to 64, and embedding parameters are initialized with Xavier method. SGCF was optimized using Adam. The default learning rate is set to 0.001 and the default mini-batch is set to 1024. The regularization factor is searched within range, L₂Is 2 norm regular. And (4) selecting an optimal value through experiments, and setting the sum of the polymerization factors of all the layers as the representative number of the layers. 1000 rounds of training were performed for all models and experiments were performed with values of 1 to 5 respectively, which showed that the best performance of the model was achieved when 4.

5.3 analysis of results

The provided algorithm SRRA is improved on the basis of LightGCN, so that performances of Pre @10, Recall @10 and NDCG @10 of the two models under the same convolution layer number are specially compared, the SRRA and the LightGCN are respectively trained for 1-5 layers, and specific experimental results are shown in Table 3.

Table 3 comparison of the Performance of different layers of LightGCN and SGCF

As can be seen from Table 3, the proposed SRRA algorithm is respectively improved by 8.14%, 10.47% and 15.79% on the three indexes of Pre @10, Recall @10 and NDCG @10 in comparison with the existing algorithm. Furthermore, the proposed SRRA algorithm has different degrees of improvement over LightGCN in all three criteria under the same number of layers trained, with greater performance improvement in the FilmTrust, derilicous and LastFM three datasets, and on average 11.00%, 10.79% and 11.14% improvement in the Pre @10, Recall @10 and NDCG @10 algorithms, respectively, while on average the improvement in the brightkit, Gowalla and epoinions three datasets is less, with average improvement of 7.54%, 7.61% and 8.60%, respectively. And as can be seen from table 3, the SRRA algorithm achieves the best effect when Layer is 4. The magnitude of the increase in the algorithm is related to what factor, and the relationship between it and the quality of social data in the dataset, i.e. the Density of the social matrix (S-sensitivity), is explored below.

FIG. 7 analyzes the relationship between the Density (S-sensitivity) of the social matrix corresponding to each data set and the algorithm performance improvement value under the three indexes of Pre @10, Recall @10 and NDCG @10,

it can be seen from fig. 7 that the performance improvement amplitude of the SRRA algorithm is positively correlated with the S-sensitivity of the data set, that is, the higher the Density of the social matrix, the better the performance of the algorithm, which explains why the algorithm improves the recommendation effect to a greater extent after adding the social data for the three data sets of FilmTrust, dericious and LastFM, and improves the recommendation effect to a lesser extent after adding the social data for the three data sets of brightkit, Gowalla and epins.

And controlling, namely setting the training layers of the proposed SRRA algorithm and the baseline algorithm as 4 layers, comparing the training layers on the evaluation indexes of Pre @10, Recall @10 and NDCG @10, and obtaining an experimental result shown in the table 4.

TABLE 4 SRRA vs baseline algorithm Performance comparison

As can be seen from table 4, the SGCF model generally achieves a better effect than the single index on the single data set.

To observe the difference in training and computational performance between the SRRA algorithm and the two baseline algorithms, all algorithms were trained for 1000 rounds in the experiment and the Pre @10, Recall @10 and NDCG @10 values for 3 algorithms were recorded every 20 epochs during the training of each dataset, all of which can be visualized as fig. 8. FIG. 8 shows the Pre @10, Recall @10 and NDCG @10 indices of the SGCF and baseline algorithms as a function of the number of training rounds on the 6 datasets BrightKite, Gowalla, Epinions, Filmtroust, Delicious, LastFM, respectively.

As can be seen from fig. 8, from the performance of the three evaluation indexes, the proposed SRRA algorithm generally has the best performance compared with the baseline algorithm in each training round; in terms of convergence rate, the SRRA algorithm performs well in most data sets compared to the baseline algorithm, i.e., it can converge to a good result at a relatively fast rate, which indicates that the SRRA algorithm has relatively good computational performance.

6 summary of the invention

The invention provides a graph convolution collaborative filtering recommendation system based on social relations. Firstly, a general collaborative filtering recommendation model SGCF is constructed, wherein the model comprises 4 parts, namely an initialization embedding layer, a semantic aggregation layer, a semantic fusion layer and a prediction layer, wherein the semantic aggregation layer and the semantic fusion layer are the core of the model SGCF and respectively play roles in extracting high-order semantic information and fusing various semantic information. Then, an implementable algorithm SRRA is provided on the basis of the model, the algorithm is improved on the basis of the LightGCN, social data are merged into the algorithm in addition to the user-item interaction data, and potential relations between the users and the items can be mined by using richer social information, so that the recommendation quality is improved. Experiments on 6 real data sets showed that: 1) compared with a baseline algorithm, the proposed SRRA algorithm generally has a better performance effect. 2) The quality (S-sensitivity) of the data set influences the performance improvement range of the proposed SRRA algorithm, and the larger the S-sensitivity value is, the better the performance of the SRRA algorithm is. 3) The proposed SRRA algorithm has superior computational performance compared to the baseline algorithm.

While embodiments of the invention have been shown and described, it will be understood by those of ordinary skill in the art that: various changes, modifications, substitutions and alterations can be made to the 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 graph convolution collaborative filtering recommendation system based on social relations is characterized by comprising an initialization embedding module, a semantic aggregation module, a semantic fusion module, a recommendation module and an optimization module;

2. The system of claim 1, wherein the first-order semantic aggregation in the semantic aggregation module comprises:

AGG (. cndot.) is the aggregation function;

e_irepresents the embedding of item i;

a user v is a first-order friend of a user u, and v is not equal to u;

AGG (. cndot.) is the aggregation function;

F_urepresenting a set of friends of user u.

3. The system for collaborative filtering recommendation of graph convolution based on social relationships as claimed in claim 1, wherein high-order semantic aggregation in the semantic aggregation module implements aggregation of high-order semantics by superimposing a plurality of first-order semantic aggregation layers; the high-order semantic aggregation comprises: semantic aggregation of social embedding propagation layer and semantic aggregation of interactive embedding propagation layer:

the semantic aggregation of the social embedding propagation layer comprises:

wherein the content of the first and second substances,

F_ua set of friends representing user u;

F_va set of friends representing user v;

| DEG | represents the number of elements in the solution set;

wherein the content of the first and second substances,

denotes the embedding of item i at layer k + 1;

H_ia first-order neighbor set representing item i;

H_ua first-order neighbor set representing user u;

represents the embedding of user u at layer k;

represents the embedding of user u at layer k + 1;

represents the embedding of item i of the k-th layer;

| · | represents the number of elements in the solution set.

4. The system according to claim 1, wherein the merging process in the semantic merging module comprises:

wherein the content of the first and second substances,

g (. cndot.) is the polymerization mode.

5. The system of claim 1, wherein the user embedding and the item embedding in the semantic fusion module comprise:

wherein the content of the first and second substances,

k represents the total number of layers;

α_kis the weight when the k-th layer aggregates the embedding of the user;

e_iis the embedding of item i;

β_kis the weight at which the kth layer aggregates the embedding of the item;

indicating the embedding of item i of the k-th layer.

6. The system of claim 1, wherein the aggregation mode that element-by-element addition is performed first, then an activation function is performed, and finally row regularization is performed comprises:

where norm (·) represents row regularization;

jh (-) is the activation function;

presentation pair

Adding element by element;

7. The system of claim 1, wherein the recommendation module comprises:

wherein the content of the first and second substances,

a score representing a predictive score;

represents the final embedding of user u;

representing a transpose;

e_iindicating the embedding of item i.

8. The system of claim 1, wherein the system is implemented using an SRRA, and the SRRA comprises:

S-A, notation of user-item interaction matrix

Where M and N are the number of users and items, R, respectively_uiIs the value of the u row, i column of the R matrix, where R is the user u and item i if there is an interaction_uiNot all right 1, otherwise R_ui0; a adjacency matrix of the user-item interaction graph may then be obtained, as shown in equation (14):

wherein A is an adjacency matrix of a user and article interaction diagram;

r is an interaction matrix of the user and the article;

representing a transpose;

wherein D is a degree matrix;

a is a adjacency matrix;

E^(k)is a user or item embedding matrix of the k-th layer;

S-C, recording the social matrix of the user as

Where user u and user v are friends then S_uv1, otherwise S_uv＝0，S_uvIs the u-th row of the S matrixThe value of column v; a adjacency matrix of the user's social graph may be obtained, as shown in equation (16):