Introduction

With the rapid development of online information, recommender systems play an indispensable role in helping users make sensible decisions from massive online information quickly and efficiently [12, 17, 33, 52, 54]. Based on historical user behaviors, recommender systems fit users’ potential interests by recommending items for them [37]. Many companies have applied the recommendation techniques and obtained excellent profits.

Collaborative filtering (CF) is the most representative technique for recommender systems. The assumption that users share similar item preferences with other users who have similar preferences or common rated items in the past is the basic motivation of CF [13]. Despite the popularity of CF-based methods in recent years, they have faced the problem of data sparsity, where users typically rate only a few items, making it difficult to precisely measure user similarities using rating information alone. This problem also makes it challenging for recommender systems to accurately predict the most relevant items based on users’ interests. To address this issue, many recommendation methods have been proposed to use social information to learn more robust representations[11, 14, 20, 38, 47, 53].

Social recommendation depends on the basic assumption that the preferences of users are affected by their social neighbors [24, 27, 30]. Most social recommendation methods have only been proposed for unsigned networks, which consist of only positive links. However, in the real world, some networks have both positive links(e.g., trust relations) and negative links (e.g., distrust relations). Typical matrix factorization methods for social recommendation are based on the hypothesis that users with trust/distrust relations have similar/dissimilar item preferences. Therefore, these methods restrain the user’s embedding closer to his/her friends than his/her foes [20, 29, 38, 48].

The past few years have witnessed the great developments of graph neural networks in graph representation learning [46]. Recent works have shown an advantage to boosting performance in social recommendations when utilizing GNN techniques to obtain robust user/item representations. Graph neural networks are capable of capturing the high-order connectivity from social graphs and user-item interaction graphs through iterative propagation [10, 43, 45, 46].

Although these previous works have demonstrated that social relations are helpful to improve recommendation performance, however, the social inconsistency problem [49] is a challenge for GNN-based signed social recommender systems. Specifically, the social inconsistency problem indicates that the evidence of item preferences provided by the social information and user-item interactions information are not necessarily consistent. For example, in Fig. 1, \(u_{1}\) and \(u_{2}\) have a trust link, however \(u_{1}\) and \(u_{2}\) have dissimilar item preference (i.e. \(u_{1}\) like \(i_{4}\), \(i_{5}\) and \(i_{6}\), however \(u_{2}\) like \(i_{1}\), \(i_{2}\) and \(i_{3}\)). Similarly, \(u_{2}\) distrust \(u_{3}\), however \(u_{2}\) and \(u_{3}\) have similar item preference (i.e. \(i_{2}\) and \(i_{3}\) are \(u_{2}\) and \(u_{3}\) both like). This situation indicates the social inconsistency problem. In other words, social information probably contains lots of noise. Aggregating the information containing noises from inconsistent social neighbors could negatively affect recommendation performance.

To alleviate the social inconsistency in signed social recommender systems, we present a novel GNN-based signed social recommendation method ESSRec. Specifically, ESSRec first performs embedding propagation on the user-item graph to capture collaborative information from user-item interactions. Then, it reconstructs the signed social graph by separating signed social relations into consistently positive relations, consistently negative relations, and inconsistent relations based on the item-space user embedding, which reveals the user’s item preference. After reconstructing the signed social graph, ESSRec propagates embedding on the reconstructed signed social graph to learn robust representations for users. Finally, the meaningful representations for user and item are adopted for rating prediction.

Our contributions are summarized as follows:

  • We provide a method to reconstruct the signed social graph by extracting consistent social relations and inconsistent social relations from signed social networks.

  • We present a novel GNN-based signed social recommendation framework ESSRec, which can model information from the reconstructed signed social graph and user-item graph through embedding propagation.

  • We conduct extensive experiments on real-world dataset. Experimental results reveal the effectiveness of ESSRec.

The remaining of this article is organized as follows. We review work related to our framework in Section 2. In Section 3, we introduce the proposed social recommendation model. We perform experiments on real-world dataset to evaluate the recommendation performance of ESSRec in Section 4. Finally, we conclude this work in Section 5.

1 Related work

Recommender systems aim to help users select items they may prefer from massive possible options. Collaborative filtering (CF) is the most successful recommendation technique, which is based on the hypothesis that similar item preferences are shared by users who have rated similar items in the past [12]. Memory-based CF and model-based CF are two types of CF, which depend on how to process the rating data [5].

Memory-based CF approaches recommend items for a target user based on similar users, which are selected by using different similarity measures[32, 33, 35]. However, the similarity values are unreliable when common items are few, since the calculation process is based on these items.

Fig. 1
figure 1

A example of social inconsistency problem in signed social recommendation

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Model-based CF approaches train a model by use rating data to predict missing rating, and have many different approaches [3, 4, 6, 18, 40]. Matrix Factorization (MF) is the representative technique in the model-based CF [22]. Generally, MF-based methods train prediction models by learning low-rank representations from a user-item matrix.

Despite CF-based methods having demonstrated their effectiveness, they still suffer data sparsity [19, 33] and cold start problem [34] because available rating data are not enough to learn a robust model for recommendations. However, many recommendation methods with social information have emerged to improve recommendation quality [25, 26, 39].

It is believed that a person usually consults others connected by social relations. It implies that social information might help to improve recommendation performance.

To achieve this goal, many methods introduce trust information into CF by making the user embedding close to the embedding of its trusted user. For example, Jamali et al. introduce trust information by introducing the distance between a user and its trusted user as regularization terms [20]. However, Yang et al. not only map user-item matrix into low-dimensional spaces but also map the trust matrix into low-dimensional spaces [48].

It is acknowledged that distrust relationships contain additional information along with the trust relationships to improve performance in many analytic tasks [23, 36, 41]. Recently, some attempts have been made to incorporate the signed social networks in recommendation in matrix factorization based method. For example, Tang et al. consider the gap between user and trusted users should be smaller than its distrusted users [38]. In [1], Bai et al. obtain two representations of users by mapping trust relations and distrust relations into identical low-dimensional space.

Although factorization-based methods have achieved great success, these methods are insufficient to learn robust representations since the implicitly higher-order connectivity is not to be captured.

Recently, GNNs have achieved tremendous success in many application domains. GNNs is capable of alleviating sparsity problem by performing iterative embedding propagation to capture the higher-order connectivity. Moreover, it efficiently integrates side information(e.g., social relationship or knowledge graph) in the network structure.

Since the data used in recommender systems can be considered as a graph, many GNN-based recommendation models have been proposed [2, 42, 50, 51]. For example, Wang et al. model user and item representation through embedding propagation on user-item interaction graph to capture collaborative signal [43]. Fan et al. integrate user embedding in item space from the user-item graph and user embedding in social space from the social graph by aggregating information from neighbors [10]. Wu et al. model user representation via historical behaviors and social relationships [45].

Although these previous works have demonstrated their effectiveness, few pay attention to the social inconsistency in the signed social recommendation. In this paper, we present a novel GNN-based signed social recommendation framework to fill this gap.

2 Proposed method

In this section, we will firstly introduce the definitions in this paper, then describe the proposed method ESSRec. The remainder of this section is organized as follows. Section 3.1 introduces the use of definitions and notations. Section 3.2 shows the overview of the proposed framework. Section 3.3 introduces the embedding propagation on user-item graph. Section 3.4 presents the method of reconstructing the signed social graph to alleviate the social inconsistency problem. Section 3.5 introduces the embedding propagation on the reconstructed signed social graph. In Sections 3.6 and 3.7, we introduces the rating prediction and model training, respectively.

2.1 Problem description

Suppose that there is a recommender system which include n users and m items, we use \(U=\left\{ u_{1},u_{2},...,u_{n} \right\} \) to denote the users set and \(V=\left\{ v_{1},v_{2},...,v_{m} \right\} \) to denote the items set Let \(R\in \mathbb {R}^{n \times m}\) denote the observed ratings in user-item matrix. The rating of item j rated by user i is denoted by \(R_{ij}\) ,and we set \(R_{ij} = 0\) if the rating is not observed. In addition, we use \(S\in \mathbb {R}^{n \times n}\) to denotes the signed social networks, where \(S_{ij} = 1\) if user i has a positive relation (i.e. trust) with user j ,and \(S_{ij} = -1\) if user i has a negative relation (i.e. distrust) with user j ,and zero otherwise. Given a signed social network S and observed rating R, the aim of signed social recommendation is to infer missing values in R.

Following mainstream recommender models [7, 16], we utilize two embedding vector to represent user i and item j, which are denoted as \(p_{i} \in \mathbb {R}^{d}\) and \(q_{j} \in \mathbb {R}^{d}\), where d is the embedding size. In Table 1, we summarize the used notations and definitions.

Table 1 Notation
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2.2 Model architecture

The architecture of ESSRec is demonstrated in Fig. 2. There are four components in ESSRec: collaborative modeling, reconstruct signed social graph, reconstructed signed social modeling and rating prediction.

Fig. 2
figure 2

The architecture of ESSRec

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Collaborative modeling is the first component to obtain item-space user embedding and user-space item embedding from the user-item graph.

We reconstruct the signed social graph in the second component, which is to reconstruct the signed social graph by extracting consistent social relations and inconsistent social relations from signed social networks via exploiting reliable user similarity based on item-space user embedding.

The third component is reconstructed signed social modeling, which propagates user embedding on the reconstructed signed social graph.

ESSRec integrates user and item embeddings to predict the missing ratings in R in the fourth component.

Next, we will introduce each model component in more depth.

2.3 Collaborative modeling

The target of collaborative modeling is to learn representations of users and items based on collaborative information from the user-item graph. We first obtain the initial user embedding \(p_{i}\) for user i as \(e_{u_{i}}^{\left( 0\right) }\) and the initial item embedding \(q_{j}\) for item j as \(e_{v_{j}}^{\left( 0\right) }\) via embedding layer. To learn robust embedding, we perform embedding propagation to exploit collaborative information in user-item graph.

There are relations (i.e., ratings) between users and items in the sparsity user-item graph, and there are no relations between users or between items. To alleviate the data sparsity problem, we exploit GNN to extract implicit user-user relations and item-item relations by considering 2-hop neighbors from the user-item graph. An embedding propagation layer exploits the direct interacted items/users to refine user/item representations by performing an aggregation function. By stacking an even number of embedding propagation layers, the information from high-order user-user/item-item neighbors is aggregated for the target user/item based on its direct neighbors. Therefore, the multiple embedding propagation layers can alleviate the data sparsity problem in the user-item graph. Following [15], we define aggregation function for user-item graph as:

$$\begin{aligned} e_{u_{i}}^{\left( k+1\right) } = \sum _{v_{j} \in N_{u}} \frac{1}{ |N_{u} |} e_{v_{j}}^{\left( k \right) } \end{aligned}$$
(1)
$$\begin{aligned} e_{v_{j}}^{\left( k+1\right) } = \sum _{u_{i} \in N_{i}} \frac{1}{ |N_{i} |} e_{u_{i}}^{\left( k \right) } \end{aligned}$$
(2)

where \(e_{u_{i}}^{\left( k \right) }\) represents user i’s embedding after k-th embedding propagation on user-item graph. Similarly, we get \(e_{v_{j}}^{\left( k+1\right) }\) represents item j’s embedding after k-th embedding propagation. \(N_{u}\) is the interactive item set for user i, and \(N_{i}\) is the interactive user set for item j. In addition, we adopt a mean operation that controls the scale of the node embedding and makes training more stable.

To combine user and item representations, we define item-space user embedding by aggregating item embedding from the rated item set for each user as:

$$\begin{aligned} h_{i}^{I} = W\left( \sum _{v_{j} \in N_{u}} \frac{1}{ |N_{u} |} e_{v_{j}}^{\left( k \right) }\oplus e_{u_{i}}^{\left( k \right) }\right) \end{aligned}$$
(3)

where \(h_{i}^{I}\) represents item-space user embedding that combines rated item information into user embedding.

In collaborative modeling, we utilize the simple definition of embedding propagation since we emphasize the solution of the social inconsistency problem and the embedding propagation on reconstructed signed social networks.

After k-th embedding propagation in collaborative modeling, we obtain the item embedding \(e_{v_{j}}^{\left( k\right) }\) and the user embedding in item space \(h_{i}^{I}\). Let \(z_{j}\) denote the final item embedding for item j by renaming \(e_{v_{j}}^{\left( k\right) }\).

figure a

Collaborative modeling in ESSRec.

2.4 Reconstruct signed social network

The fundamental motivation for most social recommendation methods is the assumption that users have similar/dissimilar item preferences with other users they trust/distrust. This hypothesis is likely to make mistakes since the social inconsistency problem. It is common that some friends may share similar preferences and some foes may have similar preferences in the real world. Hence, social recommendation methods based on the assumption mentioned above could degrade recommendation performance.

Similarly, traditional CF-based methods assume that users share similar item preferences with other users who have similar preferences or common rated items in the past. It is hard to measure the precise similarity between users since most similarity computation methods are based on items rated by users, and observed ratings are sparse.

Based on the incomplete accuracy of the above two assumptions, social recommendations have the social inconsistency problem. For example, there is a positive link between user A and user B (i.e., A trusts B), but the similarity between A and B based on rated items is low. In this case, it is difficult to choose what assumptions should the recommender system exploit.

To alleviate the social inconsistency problem, we refine signed social relations by extracting consistent and inconsistent social relations based on users’ preferences. Specifically, we define four situations by double-checking evidence from social information and rating information, that is,

  • Consistent positive relations: positive link with high similarity of preference indicates consistent positive relations since the two information sources of preference similarity offer the same similar evidence.

  • Inconsistent positive relations: positive link with low similarity of preference indicates inconsistent positive relations since a social inconsistent problem occurred to spoil the reliability of the assumption that a positive link represents a similar preference.

  • Inconsistent negative relations: negative link with high similarity of preference indicates inconsistent negative relations since social inconsistent problem.

  • Consistent negative relations: negative link with low similarity of preference indicates consistent negative relations, which offer more reliable evidence of dissimilar.

In this way, we reconstruct signed social relations as:

  • Similar relations: consistent positive relations indicate that social information and rating information offer the same evidence of similar.

  • Neutral relations: inconsistent positive and negative relations indicate that social information and rating information offer different evidence.

  • Dissimilar relations: consistent negative relations indicate that social information and rating information offer the same evidence of dissimilar.

Then, we will introduce how to compute the similarity between users. Most of the existing recommendation methods typically use Vector Space Similarity (VSS) and Pearson Correlation Coefficient (PCC) as the similarity computation methods. However, PCC and VSS are both calculated on the users’ intersection, which is empty when users share no items in common. Therefore rating information is not enough to precisely measure user similarities since data sparsity problem. To solve this problem, we use item-space user representations aggregated from the item set following the cosine function as,

$$\begin{aligned} sim\left( u_{i},u_{j}\right) = \frac{ h_{i}^{I} h_{j}^{I} }{|h_{i}^{I} ||h_{j}^{I} |} \end{aligned}$$
(4)

where \(h_{i}^{I}\) is the item-space user representation. It alleviates imprecise similarity arising from the data sparsity problem since \(h_{i}^{I}\) is a low dimensional dense vector. After computing similarity between users, we define high preference similarity and low preference similarity as follows:

  • high preference similarity, when \(sim(i,j)\geqslant \beta \)

  • low preference similarity, when \(sim(i,j)<\beta \)

where \(\beta \) is a threshold for similarity to control the preference similarity level between users.

figure b

Reconstruct signed social network in ESSRec.

2.5 Reconstructed signed social modeling

There are three classes of relations in the reconstructed signed social graph: similar, dissimilar, and neutral. To simulate different incentives obtained by different social relations, we define three embeddings aggregated from different relations for each user: the similar embedding \(h^{P}\), the dissimilar embedding \(h^{N}\), and the neutral embedding \(h^{O}\). Then, we discuss the embedding propagation on the reconstructed signed social graph.

Embedding propagation on the reconstructed signed social graph indicates that users refine their preference by social interaction. Specifically, users clear what they prefer by interaction with similar neighbors(i.e., to obtain the similar embedding \(h^{P}\)), clear what they tolerate by interaction with neutral neighbors(i.e., to obtain the neutral embedding \(h^{O}\)), and clear what they are disgusted by interaction with dissimilar neighbors(i.e., to obtain the dissimilar embedding \(h^{N}\)). Then we first introduce how to obtain these embeddings.

For initial similar embedding \(h^{P}\) for user i, we define the aggregation function as:

$$\begin{aligned} h_{i}^{P(0)} = W(\sum _{j \in N_{u}^{+}} \frac{h_{j}^{I}}{|N_{u}^{+} |}\oplus h_{i}^{I}) \end{aligned}$$
(5)

where \(h_{i}^{P(0)}\) denotes the similar embedding for user i after the aggregation layer, and \(W \in \mathbb {R}^{d \times 2d}\) is a linear transformation matrix. More specifically, we aggregate similar neighbors’ (i.e., positive link with high similarity) item-space representation for each user to clarify what they prefer.

Similarly, we define the aggregation functions for dissimilar embedding \(h^{N}\) and the neutral embedding \(h^{O}\) for a user i as:

$$\begin{aligned} h_{i}^{O(0)} = W(\sum _{j \in N_{u}^{o}} \frac{h_{j}^{I}}{|N_{u}^{o} |}\oplus h_{i}^{I}) \end{aligned}$$
(6)
$$\begin{aligned} h_{i}^{N(0)} = W(\sum _{j \in N_{u}^{-}} \frac{h_{j}^{I}}{|N_{u}^{-} |}\oplus h_{i}^{I}) \end{aligned}$$
(7)

where \(h_{i}^{O(0)}\) denotes the neutral embedding by aggregating item-space embedding from neutral neighbors(i.e., positive link with low similarity and negative link with high similarity), moreover, \(h_{i}^{N(0)}\) denotes the dissimilar embedding by aggregating item-space embedding from dissimilar neighbors(i.e., negative link with low similarity).

Then, we refine these embeddings by propagating user embedding on the reconstructed signed social graph based on balance theory [8].

In general, balance theory implies “the friend of my friend is my friend” and “the enemy of my enemy is my friend”. That means if A trusts B, B trusts C, then we consider A trusts C, and if A distrusts B, B distrusts C, then we consider A trusts C. The reconstructed social relations make the hypothesis more reliable since it directly indicates that users have similar or dissimilar preferences.

Considering the balance theory, we define the aggregation function following [9].

For similar embedding \(h^{P}\) as:

$$\begin{aligned} h_{i}^{P(l+1)} = mean( \sum _{j \in N_{u}^{+}} \frac{h_{j}^{P(l)}}{|N_{u}^{+} |}, \sum _{k \in N_{u}^{-}} \frac{h_{k}^{N(l)}}{|N_{u}^{-} |} ) \end{aligned}$$
(8)

where \(h_{i}^{P(l+1)}\) denotes the similar embedding for user i after (l+1)-th embedding propagation, \(N_{u}^{+}\) and \(N_{u}^{-}\) denote the sets of similar and dissimilar neighbors of user. This function aggregates \(h^{P}\) embedding from similar neighbors and \(h^{N}\) embedding from dissimilar neighbors. It indicates that users obtain a positive incentive of preference from neighbors by clear what similar neighbors like and what dissimilar neighbors dislike.

For dissimilar embedding \(h^{N}\) for a user, we define the aggregation function as:

$$\begin{aligned} h_{i}^{N(l+1)} = mean( \sum _{j \in N_{u}^{-}} \frac{h_{j}^{P(l)}}{|N_{u}^{-} |}, \sum _{k \in N_{u}^{+}} \frac{h_{k}^{N(l)}}{|N_{u}^{+} |} ) \end{aligned}$$
(9)

where \(h_{i}^{N(l+1)}\) represents dissimilar embedding \(h^{N}\) for user i after (l+1)-th embedding propagation. This function aggregates dissimilar embedding \(h^{N}\) from similar neighbors and similar embedding \(h^{P}\) from dissimilar neighbors. It indicates that users obtain a negative incentive of preference from neighbors by clear what similar neighbors dislike and what dissimilar neighbors like.

For neutral embedding \(h^{O}\) for a user, we define the aggregation function as:

$$\begin{aligned} h_{i}^{O(l+1)} = mean( \sum _{j \in N_{u}^{+}} \frac{h_{j}^{O(l)}}{|N_{u}^{+} |}, \sum _{k \in N_{u}^{-}} \frac{h_{k}^{O(l)}}{|N_{u}^{-} |}, \sum _{a \in N_{u}^{o}} \frac{h_{a}^{P(l)}}{|N_{u}^{o} |}, \sum _{b \in N_{u}^{o}} \frac{h_{b}^{N(l)}}{|N_{u}^{o} |}, \sum _{c \in N_{u}^{o}} \frac{h_{c}^{O(l)}}{|N_{u}^{o} |} ) \end{aligned}$$
(10)

where \(h_{i}^{O(l+1)}\) represents neutral embedding for user i after (l+1)-th embedding propagation. This function aggregates neutral embedding \(h^{O}\) from similar and dissimilar neighbors and all embeddings from neutral neighbors. It indicates that users obtain a neutral incentive of preference from neighbors by clear what neutral neighbors pay attention to and what similar neighbors and dissimilar neighbors are tolerable.

To learn robust user representations, similar user representations \(h^{P}\), dissimilar user representations \(h^{N}\), and neutral user representations \(h^{O}\) are needed to be considered together since these representations capture different incentives from neighbors(i.e., positive incentive, negative incentive, and neutral incentive). To distinguish the incentive between consistent neighbors and inconsistent neighbors, we first combine similar user representations \(h^{P}\) and dissimilar user representations \(h^{N}\) to obtain consistent user representations as:

$$\begin{aligned} h_{i}^{consis} = W( h_{i}^{P(l)} \oplus h_{i}^{N(l)}) \end{aligned}$$
(11)

Then we concatenate consistent user representations and inconsistent user representations(i.e., neutral user representations) to get the final user representations as:

$$\begin{aligned} h_{i} = W( h_{i}^{consis} \oplus h_{i}^{O(l)}) \end{aligned}$$
(12)
figure c

Reconstructed signed social modeling in ESSRec.

2.6 Rating prediction

We obtain final user and item representations after collaborative modeling and reconstructed signed social modeling. Finally, we predict the missing rating by the inner-product function as:

$$\begin{aligned} \hat{R_{ij}} = h_{i}^{\textrm{T}}z_{j} \end{aligned}$$
(13)

where \(h_{i}\) denotes the final user embedding, and \(z_{j}\) denotes the final item embedding.

2.7 Model training

We have introduced the proposed framework ESSRec to model information from the user-item interaction graph and reconstruct the signed social graph. The loss function is defined as:

$$\begin{aligned} Loss = \sum _{(i,j) \in \tau }(R_{ij} - \hat{R_{ij}})^{2} \end{aligned}$$
(14)

where \(\tau \) is the rating set, \(|\tau |\) is the size of \(\tau \).

In this paper, we utilize the ADAM [21] as the optimizer to optimize this objective function. The trainable model parameters include two initial embedding (i.e. \(p_{i}\) and \(q_{j}\)) and all linear transformation matrices.

Furthermore, following [15], we drop the non-linear activation function and unnecessary transformation matrices. We will discuss the effectiveness of this choice.

3 Experimental evaluation

3.1 Experimental settings

3.1.1 Dataset

To evaluate the effectiveness of our method, we use the Epinions datasetFootnote 1, the only signed social recommendation dataset publicly available [28]. Epinions contains 180202 users, 7555761 items, 13668320 ratings, and 841372 signed social links. It allows users to create positive (trust) and negative (distrust) links to other users, resulting in a signed network S. Ratings in Epinions with scores ranging from 1 to 5. We randomly select 80% of user-item interaction as the training set and 10% as the validation set and test set. The statistics of the Epinions dataset are demonstrated in Table 2.

Table 2 Statistics of epinions dataset
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3.1.2 Evaluation metrics

Following typical ways to evaluate the recommendation performance, we choose two popular metrics: MAE and RMSE, which are defined as follows:

$$\begin{aligned} MAE = \frac{1}{|\tau |}\sum _{(i,j) \in \tau }|R_{ij} - \hat{R_{ij}} |\end{aligned}$$
(15)
$$\begin{aligned} RMSE = \sqrt{\frac{1}{|\tau |}\sum _{(i,j) \in \tau }(R_{ij} - \hat{R_{ij}})^{2}} \end{aligned}$$
(16)

where \(\tau \) is the rating set, \(|\tau |\) is the size of \(\tau \). A smaller values of MAE and RMSE indicate better recommendation performance.

3.1.3 Baselines

To assess recommendation performance, we compared our model with existing recommender systems methods as follows:

  • SocialMF [20]: This method constrains latent user factors closed to his/her trust user by matrix factorization based on the assumption that users share similar preferences with friends.

  • TrustMF [48]: This method proposes a truster model and a trustee model by mapping users into truster space and trustee space and combining them to the TrustMF model.

  • TDRec [1]: This method makes use of trust information to learn user latent factors and uses distrust information to regularize user latent factors by matrix factorization.

  • RecSSN [38]: This method exploits both explicit trust and distrust relationships based on the user is similar to his/her trust neighbor than a distrust neighbor.

  • GraphRec [10]: This is a GNN-based social recommendation method, which integrates user embedding in item space from the user-item graph and user embedding in social space from the social graph by aggregating information from neighbors.

The parameters for each baseline method are shown in Table 3.

Table 3 Optimal parameters
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Table 4 The result of overall comparison
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3.2 Overall comparison

Table 4 shows the results of MAE and RMSE for all methods. From the experimental results, we have the following main findings:

  • TDRec and RecSSN perform better than SocialMF and TrustMF. SocialMF and TrustMF only use the trust information, while TDRec and RecSSN leverage both the distrust information and trust information. These findings support that distrust information is capable of improving the recommendation performance.

  • GraphRec achieves a second-best performance without distrust information. It demonstrates the power of GNNs in graph representation learning.

  • Our method Model performs better than all baseline methods. Compared to methods based on matrix factorization, our method leverages the ability of GNNs for representation learning on graph data. In addition, our method not only simply utilizes trust information and distrust information but also reconstruct a more reliable similarity relation from two information sources.

3.3 Impact of reconstruct similar relation

To investigate the impact of reconstructed similarity relation on the performance of ESSRec, we introduce variants of the model. These two variants are defined in the following:

  • ESSRec-A: The process of extracting consistent and inconsistent relations is removed. This variant only uses trust relation and distrust relation from the original signed social network to learn the user’s final embedding by performing embedding propagation according to formulas (8) and (9).

  • ESSRec-B: This variant removed neutral embedding \(h^{O}\) after the embedding propagation on the reconstructed signed social graph. ESSRec-B uses similar embedding \(h^{P}\) and dissimilar embedding \( h^{N}\) of the user to learn the final user embeddings.

Fig. 3
figure 3

Impact of reconstructed similarity relation on Epinions dataset

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Figure 3 demonstrates the recommendation performance of ESSRec and its variants on Epinions. We have the following findings:

  • Reconstruct signed social graph: The aim of reconstructing the signed social graph is to alleviate the social inconsistency problem by extracting consistent and inconsistent relations. The experimental result shows that ESSRec-A performs worse than ESSRec. It supports that reconstructing the signed social graph is capable of alleviating the social inconsistency problem and thus improving the recommendation performance.

  • Neutral embedding \(h^{O}\): Neutral embedding \(h^{O}\) can be regarded as noisy information since it aggregates information from inconsistent social neighbors. However, \(h^{O}\) can also be regarded as a neutral incentive of preference from neighbors, which is between a similar incentive and a dissimilar incentive. The performance of rating prediction has deteriorated without neutral embedding \(h^{O}\). It justifies that neutral embedding \(h^{O}\) can help to boost the performance of recommendation.

3.4 Impact of \(\beta \) on the results

We will now analyze the precise impact of \(\beta \) on the performance of ESSRec. The experiments in Fig. 5 were conducted with embedding size of 16 and \(\beta \in \left\{ -0.5,-0.3, -0.1, 0, 0.1, 0.3, 0.\right. \)\( \left. 5 \right\} \). The results of MAE and RMSE on Epinions dataset are demonstrated in Fig. 4(a) and (b) respectively. We make the following observations:

Fig. 4
figure 4

Impact of \(\beta \) on the performance of ESSRec

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With the increase in \(\beta \), more social relations are regarded as dissimilar item preferences. The results in Epinions(i.e., MAE and RMSE) are both obtain the best performance with \(\beta = 0\). Moreover,the result curve is approximate symmetrical and the model achieve worse performance with \(\beta = -0.1\) and \(\beta = 0.1\).

3.5 Impact of embedding size

Figure 5 presents the performance comparison of embedding size on Epinions dataset. The complexity of our model will significantly boost with the increase of embedding size. From the results, we have the following findings:

The MAE on Epinions obtains the best performance with an embedding size of 16 and gets the best RMSE value with an embedding size of 32. In addition, ESSRec achieves a second-best performance on RMSE with an embedding size of 16. To sum up, the embedding size of 16 is the best comprehensive performance to balance performance and complexity.

Fig. 5
figure 5

Impact of embedding size on the performance of ESSRec

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4 Conclusion

In this paper, we addressed the social inconsistency problem in signed social recommendation. We present a novel GNN-based signed social recommendation method ESSRec to alleviate this problem by checking signed social relations via user embedding in item space which combine collaborative information. This method exploits consistent positive relations, consistent negative relations, and inconsistent relations in the reconstructed signed social graph to model user preference more accurately. Experimental results on Epinions dataset show the effectiveness of ESSRec.

This work represents an initial attempt to reconstruct signed social graph by checking original links via rating information, since both rating information and signed social information indicate user similarity. The ablation experiment shows that embedding propagation on reconstructed graph can learn node representation better. In this work, we exploit rating information to extract consistent and inconsistent relations from original graph, however there are many other forms of additional information can be useful for extracting consistent and inconsistent relations,such as community information[31],content based information and context-aware data[44]. For example, by integrating content descriptions associated with users, we can compute the similarity between users and reconstruct original graph, which help filter noises from original links.

Some limitations should be pointed out. In collaborative modeling, we perform embedding propagation without considering different rating scores from user to item. Future works will integrate rating scores into similarity computation, which is now computed by the low dimensional dense vector.