Graph Bottlenecked Social Recommendation

Introduction

Learning informative user and item representations is the key to building modern recommender systems. Classic collaborative filtering paradigm factorizes user-item interaction matrix to learn user and item representations, which is widely researched but usually limited by sparse interactions. With the proliferation of social media, social recommendation has become an important technique to provide personalized suggestions (Tang et al., 2013). Both user-item interactions (Shao et al., 2022, 2024) and user-user social relations (Wu et al., 2019, 2020a) are available on social platforms, prompting the development of various social recommendation methods designed to exploit these behavior patterns (Ma et al., 2008; Konstas et al., 2009).

Following the social homophily (McPherson et al., 2001) and social influence theory (Marsden and Friedkin, 1993), many efforts are devoted to characterizing social relation effects on user preferences. Early works mainly focus on exploiting first-order social relations, i.e., social regularization that assumes socially connected users share similar preference (Jamali and Ester, 2010), and social enhancement that incorporates user-trusted friends’ feedback as auxiliary for the target user (Guo et al., 2015). Recently, witnessed the power of graph neural networks (GNNs) on machine learning (Kipf and Welling, 2017; Wu et al., 2020b, e; Cai et al., 2024b; Chen et al., 2023), graph-based recommendations have attracted more and more attention (He et al., 2020; Wu et al., 2020d; Yang et al., 2023a; Cai et al., 2024a). Graph-based social recommendations (Fan et al., 2019; Wu et al., 2019, 2022) achieve impressive progress in improving recommendation performances by formulating users’ high-order interest propagation and social influence diffusion with GNNs.

Despite the effectiveness, current graph-based social recommendations rarely notice the social noise problem, i.e., social graphs are inevitably noisy with redundant social relations. Those redundant relations are caused by unreliable social relations and low preference-affinity social relations (Sun, 2023; Quan et al., 2023). Consequently, directly using the observed social graph may hinder precise user preference characterization, leading to sub-optimal recommendation results. We conduct an empirical study to illustrate the social noise problem. As shown in Figure 1, we compare LightGCN with current SOTA graph-based recommendation methods, including SocialLGCN (Liao et al., 2022) and DiffNet++ (Wu et al., 2020a). To avoid the effect of the message-passing mechanism of different methods, we additionally implement the extension of LightGCN, called LightGCN-S which additionally performs social neighbor aggregation for user representation learning. We can find that compared with LightGCN, graph-based social recommendations do not present significant strength on both metrics, even worse on the Douban-Book dataset. This indicates that social networks are usually noisy, it’s necessary to filter redundant social relations to enhance the robustness of social recommendations. However, identifying and removing redundant social relations is non-trivial due to a lack of ground-truth labels. Besides, how can guarantee the recommendation accuracy while removing social relations?

In this paper, we focus on learning the denoised social graph structure to facilitate recommendation tasks from an information bottleneck perspective. Specifically, we propose a novel Graph Bottlenecked Social Recommendation (GBSR) framework to tackle the social noise problem. Let $\mathcal{G}^{S}=\{U,\mathbf{S}\}$ denote the user-user social graph and $\mathbf{R}$ denote the user-item interaction matrix, where $U$ is userset and $\mathbf{S}$ is social structure matrix. The optimal denoised social graph structure $\mathbf{S^{\prime}}$ should satisfy: the minimal from $\mathbf{S}$ yet efficient for infer $\mathbf{R}$. To achieve this goal, we first introduce user preference signals to guide the social graph denoising process, then optimize the learning process via the Information Bottleneck (IB) principle. Specifically, GBSR maximize the mutual information between the denoised social graph structure $\mathbf{S^{\prime}}$ and interaction matrix $\mathbf{R}$, meanwhile minimizing it between the denoised social graph structure $\mathbf{S^{\prime}}$ and the original $\mathbf{S}$. Therefore, the learning objective is formulated as: $Max:I(\mathbf{R};\mathbf{S^{\prime}})-\beta I(\mathbf{S^{\prime}};\mathbf{S})$.

Nevertheless, optimizing the objective of GBSR for social recommendation is still challenging due to the following two challenges. For the maximization of $I(\mathbf{R};\mathbf{S^{\prime}})$, social graph and sparse interaction matrix are two non-Euclidean data, which are hard to compare directly. For the minimization of $I(\mathbf{S^{\prime}};\mathbf{S})$, estimating the upper bound of MI is an intractable problem. Although some works (Alemi et al., 2017; Cheng et al., 2020) leverage variational techniques to estimate the upper bound, they heavily rely on the prior assumption. To address the above two challenges, GBSR is implemented as follows. First, regarding the hard-comparable issue of $I(\mathbf{R};\mathbf{S^{\prime}})$, we take all nodes into intermediary and derive the lower bound of $I(\mathbf{R};\mathbf{S^{\prime}})$ for maximization. Second, we introduce the Hilbert-Schmidt independence criterion (HSIC) (Ma et al., 2020) to replace the minimization of $I(\mathbf{S^{\prime}};\mathbf{S})$. HSIC (Gretton et al., 2005) is a statistic measure of variable dependency, minimizing HSIC approximate the minimization of mutual information. Our contributions are summarized as follows:

In this paper, we revisit the social denoising recommendation from an information theory perspective, and propose a novel Graph Bottlenecked Social Recommendation (GBSR) framework to tackle the noise issue.

Technically, we derive the lower bound of $I(\mathbf{R};\mathbf{S^{\prime}})$ for maximization, and introduce the Hilbert-Schmidt independence criterion (HSIC) to approximate the minimization of $I(\mathbf{S^{\prime}};\mathbf{S})$.

Empirical studies on three benchmarks clearly demonstrate the effectiveness and generality of the proposed GBSR , i.e., GBSR achives over $17.06\%$, $10\%$, and $11.27\%$ improvements of NDCG@20 compared with the strongest baseline.

Preliminaries

where $\mathcal{P}$ denote distribution of training data, and $\theta$ denote GNN parameters. However, user social networks are usually noisy with redundant relations (Quan et al., 2023), directly using $\mathbf{S}$ to infer interaction probability may decrease the recommendation accuracy. In this work, we focus on learning robust social structure $\mathbf{S}^{\prime}$ to facilitate recommendation performance:

where $\mathcal{F}_{\phi}$ denotes social denoising function with the parameters $\phi$. Consequently, the final optimization of graph-noised social recommendation is described as follows:

2. Information Bottleneck Principle

Information Bottleneck (IB) is a representation learning principle in machine learning, which seeks a trade-off between data fit and reducing irrelevant information (Tishby et al., 2000; Tishby and Zaslavsky, 2015). Given input data $X$, $Z$ is the hidden representation, and $Y$ is the downstream task label, which follows the Markov Chain $<X\rightarrow Z\rightarrow Y>$. IB principle describes that an optimal representation should maintain the minimal sufficient information for the downstream tasks (Tishby and Zaslavsky, 2015; Saxe et al., 2019):

where $I(Y;Z)$ denotes the mutual information between the hidden representation $Z$ and label $Y$, $I(X;Z)$ denotes the mutual information between the hidden representation $Z$ and input data $X$ two variables, $\beta$ is the coefficient to balance these two parts. IB principle has been widely applied in machine learning tasks, such as model robustness (Wu et al., 2020c; Wang et al., 2021b), fairness (Gronowski et al., 2023), and explainability (Bang et al., 2021). In this work, we introduce the IB principle to robust social denoising learning, which aims to seek the minimal yet sufficient social structure for recommendation tasks.

The Proposed GBSR Framework

In this section, we introduce our proposed Graph Bottlenecked Social Recommendation (GBSR) framework for social denoising based recommendation. Essentially, GBSR aims to learn the minimal yet efficient social structure to facilitate recommendation tasks, which is guaranteed by the information bottleneck principle. Next, we first give the overall optimization objective of GBSR , then introduce how to implement each component of GBSR in detail. Finally, we instantiate GBSR with LightGCN-S backbone.

As shown in Figure 2, we present the overall objective of our proposed GBSR framework for the social recommendation. Instead of directly using the original social structure $\mathbf{S}$, we aim to learn a denoised yet informative social structure $\mathbf{S}^{\prime}$ to enhance recommendation. Due to the lack of available prior for social denoising, we introduce user preference signals to guide social graph denoising. To guarantee the trade-off between social denoising and recommendation tasks, we optimize GBSR via graph information bottleneck principle. Thus, the goal of GBSR is: $Max:I(\mathbf{R};\mathbf{S^{\prime}})-\beta I(\mathbf{S^{\prime}};\mathbf{S})$. Due to the intractability of $I(\mathbf{R};\mathbf{S^{\prime}})$, we take all nodes into an intermediary for calculation. Thus, we obtain the final optimization objective of GBSR :

where the first term is encouraging that the denoised social graph preserves the essential information to facilitate recommendation tasks. The second term is the compression of the original social graph, aiming to filter redundant social relations.

2. Preference-guided Social Denoising

To achieve the above objective of GBSR , we first need to refine the denoised social graph. The challenge is that although the social graph has noisy relations, there are no available labels to guide the denoising process. Based on social homogeneity social-connected individuals have more similar behavior similarity, we inject user preference signals into the social denoising process, i.e., users with similar preferences are more likely to have social relations.

Formally, we formulate the social denoising process as a graph edge dropout problem. Given the original social graph structure $\mathbf{S}$, the denoised one is defined as:

where $\mathbf{e}_{a}$ and $\mathbf{e}_{b}$ denotes user $a$ and user $b$ preference representations, respectively. $g()$ is the fusion function, we employ MLPs to realize it. However, $\mathbf{S^{\prime}}$ is not differentiable with the parameter $\rho$ of Bernoulli distribution, so we use the popular concrete relaxation method (Jang et al., 2017) to replace:

Given the denoised social graph $\mathbf{S}^{\prime}$, we first present how to maximize the mutual information $I(\mathbf{R};U,V,\mathbf{S^{\prime}})$, which ensures the denoised social graph satisfy recommendation tasks. Specifically, we derivate the lower bound of $I(\mathbf{R};U,V,\mathbf{S^{\prime}})$ as follows:

where $\mathcal{G}(\cdot)$ is any graph-based social recommender as we mentioned in the preliminaries, $\sigma(\cdot)$is the sigmoid activation, $\mathcal{D}=\{(a,i,j)|r_{ai}=1\!\wedge\!r_{aj}=0\}$ is all training data. Next, we introduce each derivation step as follows: (a) is the definition of mutual information; (b) is the non-negative property of $H(\mathbf{R})$; (c) is that $p(r|a,i,\mathbf{S^{\prime}})\leq 1$, and we split all samples into observed interactions and non-observed interactions; (d) $\sigma(\mathcal{G}(a,i,\mathbf{S^{\prime}}))$ is the variational approximation of $p(r_{ai}=1|a,i,\mathbf{S^{\prime}})$; (e) is due to $log(\sigma(x))-log(\sigma(y))\geq log(\sigma(x-y))$.

According to the above derivation, we can find that the popular BPR ranking loss (Rendle et al., 2009) is the lower bound of mutual information $I(\mathbf{R};U,V,\mathbf{S^{\prime}})$. Therefore, we employ BPR loss as the objective of mutual information maximization.

Next, we introduce how to minimize $I(\mathbf{S^{\prime}},\mathbf{S})$, which aims to reduce the redundant social relations in the original graph. Estimating the upper bound of mutual information is an intractable problem. Although some works (Alemi et al., 2017; Cheng et al., 2020) leverage variational techniques to estimate the upper bound, but heavily rely on the prior assumption. Therefore, we introduce Hilbert-Schmidt Independence Criterion (HSIC (Gretton et al., 2005)) as the approximation of the minimization of $I(\mathbf{R};\mathbf{S^{\prime}})$.

HSIC brief. HSIC serves as a statistical measure of dependency (Gretton et al., 2005), which is formulated as the Hilbert-Schmidt norm, assessing the cross-covariance operator between distributions within the Reproducing Kernel Hilbert Space (RKHS). Mathematically, given two variables $X$ and $Y$, $\text{HSIC}(X,Y)$ is defined as follows:

where $K_{X}$ and $K_{Y}$ are two kernel functions for variables $X$ and $Y$, $X^{\prime}$ and $Y^{\prime}$ are two independent copies of $X$ and $Y$. Given the sampled instances ${(x_{i},y_{i})}_{i=1}^{n}$ from the batch training data, the $HSIC(X,Y)$ can be estimated as:

where $K_{X}$ and $K_{Y}$ are used kernel matrices (Gretton et al., 2005), with elements $K_{X_{ij}}=K_{X}(x_{i},x_{j})$ and $K_{Y_{ij}}=K_{Y}(y_{i},y_{j})$, $H=\mathbf{I}-\frac{1}{n}\mathbf{1}\mathbf{1}^{T}$ is the centering matrix, and $Tr(\cdot)$ denotes the trace of matrix. In practice, we adopt the widely used radial basis function (RBF) (Vert et al., 2004) as the kernel function:

where $\sigma$ is the parameter that controls the sharpness of RBF.

HSIC-based bottleneck learning. Given the original and denoised social graph structures $\mathbf{S}$ and $\mathbf{S^{\prime}}$, we minimize $HSIC(\mathbf{S^{\prime}};\mathbf{S})$ to replace the minimization of $I(\mathbf{S^{\prime}};\mathbf{S})$. However, social graphs are non-Euclidean data, making it difficult to measure dependency. In practice, we adopt Monte Carlo sampling (Shapiro, 2003) on all the node representations for calculation:

where $\mathcal{B}$ denotes the batch sampling users, $\mathbf{E^{\prime}}$ and $\mathbf{E}$ denote node representations, which are learned from recommenders $\mathcal{G}_{\theta,\phi}(U,V,\mathbf{S^{\prime}})$ and $\mathcal{G}_{\theta}(U,V,\mathbf{S})$. Thus, we can reduce the redundant social relations via the HSIC-based bottleneck regularization:

5. Instantiating the GBSR Framework

In this section, we instantiate our proposed GBSR with specific graph-based social recommender $\mathcal{G}_{\theta}(U,V,\mathbf{S})$. To avoid the effect of different message-passing mechanisms, we implement LightGCN-S as the backbone model (we also realize GBSR with other backbones, refer to the generality analysis). Firstly, we formulate the available data and denoised social structure as a graph $\mathcal{G}=\{U\cup V,\mathbf{A}\}$, where $U\cup V$ denotes the set of nodes, and $\mathbf{A}$ is the adjacent matrix defined as follows:

where $\mathbf{D}$ is the degree matrix of graph $\mathcal{G}$, $\mathbf{E}^{l+1}$ and $\mathbf{E}^{l}$ denote node embeddings in ${l+1}^{th}$ and ${l}^{th}$ graph convolution layer, respectively. When stacking $L$ graph convolution layers, the final node representations can be obtained with a readout operation:

After obtaining the learned node representations through GCNs, LightGCN-S infers the propensity that user $a$ interacts with item $i$ by an inner product: $\hat{r}_{ai}=<e_{a},e_{i}>$. All the above process are summarized as $\hat{r}_{ai}=\mathcal{G}_{\theta}(a,i,\mathbf{S})$.

Next, we give the illustration of graph-denoised social recommendation. We first use the initialized node embeddings to obtain user preference representations $\mathbf{P}=\mathbf{E}^{0}[:M]$, then achieve the denoised social structure $\mathbf{S^{\prime}}$ based on preference-guided social structure learning (section 3.2). Given the learned denoised social structure $\mathbf{S^{\prime}}$, we establish graph-denoised social recommender $\hat{r}_{ai}=\mathcal{G}_{\theta,\phi}(a,i,\mathbf{S^{\prime}})$. Then, we select the pairwise ranking loss (Rendle et al., 2009) to optimize model parameters:

where $\sigma(\cdot)$ is the sigmoid activation function, $\lambda$ is the regularization coefficient. $D_{a}=\{(i,j)|i\in R_{a}\!\wedge\!j\not\in R_{a}\}$ denotes the pairwise training data for user $a$. $R_{a}$ represents user $a$ interacted items on the training data. Combined with the HSIC-based bottleneck regularizer, we obtain the final optimization objective:

The overall learning process of GBSR is illustrated in Algorithm 1.

6. Model Discussion

In this section, we analyze the proposed GBSR from model complexity and model generalization.

As illustrated in Algorithm 1, the parameters of GBSR are composed of two parts: graph-based social recommender parameters $\theta$ and social denoising parameters $\phi$. Among them, $\theta=\mathbf{E}^{0}$ are the general parameters equipped for backbone models (such as LightGCN-S). $\phi$ are the parameters of MLPs, which are used to calculate the social edge confidence. Because $\phi$ are the shared parameters for all social edges, the additional parameters of GBSR are ignorable compared with backbone models.

6.2. Time Complexity

Compared with the backbone model (such as LightGCN-S), the additional time cost is social graph denoising and HSIC-bottleneck optimization. Social graph denoising is conducted on the observed social relations, which performs a sparse matrix. Besides, the time complexity of the HSIC-bottleneck regularizer lies in the number of the sampled nodes (refer to Eq.(13)). In practice, we adopt a mini-batch training strategy to reduce the time cost of bottleneck learning, and the additional time cost of GBSR is affordable. Besides, as we remove redundant social relations, the denoised yet informative social graph makes GBSR convergence much faster than the backbone model. Experiments also verify the efficiency of GBSR.

6.3. Model Generalization

The proposed GBSR is designed for social denoising under graph-based social recommendation scenarios. It does not depend on specific graph-based social recommenders, such as DiffNet++ (Wu et al., 2020a) and SocialLGN (Liao et al., 2022). Our proposed GBSR is a flexible denoising framework to enhance social recommendations, we also conduct experiments on four backbones to demonstrate the generalization. Besides the backbone model, the idea of introducing the information bottleneck principle to graph denoising can also be generalized for different recommendation scenarios.

Experiments

In this section, we conduct extensive experiments on three real-world datasets to validate the effectiveness of our proposed GBSR . We first introduce experimental settings, followed by recommendation performance comparisons. Finally, we give a detailed model investigation, including training efficiency, visualization of the denoised social graph, and parameter sensitivities.

We conduct empirical studies on three public datasets to verify the effectiveness of our proposed GBSR , including Yelp, Epinions, and Dianping (Yang et al., 2023b). All datasets involve user-user social links and user-item interactions. For the Yelp dataset, we follow the released version in (Yu et al., 2021a). For epinions and dianping datasets, we filter ratings less than 3 and keep the remaining ratings as positive feedback. After that, we sample 80% interactions as training data, and the remaining 20% as test data. The detailed statistics of all datasets are summarized in Table 1.

1.2. Baselines and Evaluation Metrics.

To evaluate the effectiveness of our proposed GBSR , we select state-of-the-art baselines for comparisons. Specifically, these baselines can be divided into two groups: graph-based social recommendation methods (He et al., 2020; Fan et al., 2019; Wu et al., 2020a) and social graph denoising methods (Zheng et al., 2020; Yu et al., 2020; Quan et al., 2023), which are list as follows:

LightGCN (He et al., 2020): is the SOTA graph-based collaborative filtering method, which simplifies GCNs by removing the redundant feature transformation and non-linear activation components for ID-based recommendation.

LightGCN-S: We extend LightGCN to graph-based social recommendation, that each user’s neighbors include their interacted items and linked social users. LightGCN-S is a basic and lightweight model, considering our proposed GBSR is a model-agnostic social graph denoising method, we select LightGCN-S as the backbone model.

GraphRec (Fan et al., 2019): is a classic graph-based social recommendation method, it incorporates user opinions and user two kinds of graphs for preference learning.

DiffNet++ (Wu et al., 2020a): is the SOTA graph-based social recommendation method, it recursively formulates user interest propagation and social influence diffusion process with a hierarchical attention mechanism.

SocialLGN (Liao et al., 2022): propagates user representations on both user-item interactions graph and user-user social graph with light graph convolutional layers, and fuses them for recommendation.

Rule-based: We follow (Quan et al., 2023) and remove unreliable social relations based on the similarity of the user-interacted items.

ESRF (Yu et al., 2020): proposes adversarial graph convolutional networks to enhance social recommendation, it generate alternative social neighbors and further perform neighbor denoising with adversarial training.

GDMSR (Quan et al., 2023): designs the robust preference-guided social denoising to enhance graph-based social recommendation, it only remains the informative social relations according to preference confidences.

As we focus on implicit recommendation scenarios, we employ two widely used ranking metrics: Recall@N and NDCG@N (Gunawardana and Shani, 2009; Steck, 2013). Specifically, Recall@N measures the percentage of the recalled positive samples for the Top-N ranking lists. Furthermore, NDCG@N assigns higher scores for those items in the top-ranked positions. In the evaluation stage, we adopt a full-ranking strategy that views all non-interacted items as candidates to avoid biased evaluation (Krichene and Rendle, 2020; Zhao et al., 2020). For each model, we repeat experiments in 5 times and report the average values.

1.3. Parameter Settings.

We implement our proposed GBSR and backbone with Tensorflow https://www.tensorflow.org. For all baselines, we follow the original settings and carefully fine-tune parameters for fair comparisons. For latent embedding based methods, we initialize their embeddings with a Gaussian distribution with a mean value of 0 and a standard variance of 0.01, and fix the embedding size to 64. For model optimization, we use Adam optimizer with a learning rate of 0.001 and a batch size of 2048. We follow the mainstream ranking-based methods (Rendle et al., 2009), and randomly select 1 non-interacted item as the negative sample for pairwise ranking optimization. We search the GCN layer in $$, the regularization parameter$\lambda$in$[0.0001,0.001,0.01]$. For the observation bias, we set$\epsilon=0.5$for all datasets. For information bottleneck constraint coefficient$\beta$, we use grid-search with different scales over three datasets, and report detailed analysis in experiments.

2. Recommendation Performances

As shown in Table 2, we compare our proposed GBSR with state-of-the-art methods on three benchmarks. For a fair comparison, all denoising methods are conducted on the LightGCN-S backbone. Given the empirical studies, we have the following observations:

Compared with LightGCN, graph-based social recommendation methods present slight improvements under most of the datasets, i.e., DiffNet++ obtains a 2.24% improvement on the NDCG@20 metric for Yelp dataset. However, this is not always the case, all social graph recommendations show a performance degradation on the Douban-Book dataset. While supported by social graphs, it is noteworthy that graph-based social recommendation methods do not consistently outperform LightGCN in terms of performance. These demonstrate that directly using social graphs may decrease recommendation performance, it’s necessary to remove redundant social relations to enhance recommendation.

Compared with directly using original social graphs, social denoising methods present better performances in most cases. This indicates that social noise is ubiquitous in real-world recommendation scenarios. All social denoising methods are implemented on LightGCN-S backbone, we find that GDMSR is the strongest baseline, which benefits from preference-guided social denoising and self-correcting curriculum learning. However, these social denoising methods don’t present large-margin improvements compared with the backbone model. The reason is that simple rule or assumption based denoising methods lack of theoretical guarantee, it’s hard to seek an effective trade-off between social denoising and recommendation accuracy.

Our proposed GBSR consistently outperforms all baselines under all experimental settings. Specifically, GBSR improves the strongest baseline $w.r.t$ NDCG@20 by 17.06%, 10% and 11.27% on Douban-Book, Yelp, and Epinions datasets, respectively. Compared with the backbone model, GBSR achieves impressive superiority over three benchmarks. These indicate that our proposed GBSR can significantly improve graph-based social recommendations, demonstrating the effectiveness of graph bottleneck learning to reduce redundant social relations. Compared with other social denoising methods, our GBSR can better obtain the trade-off between removing social relations and recommendation tasks.

2.2. Ablation study

We conduct ablation studies on three datasets to explore the effectiveness of each component of the proposed GBSR framework. As shown in Table 4, we compare GBSR with corresponding variants on Top-20 recommendation performances. GBSR-w/o HSIC denotes that remove the HSIC-based bottleneck regularization of GBSR , we only keep preference-guided social denoising module. From Table 4, we can find that GBSR-w/o HSIC performs worse in all cases, even worse than the backbone model. This indicates that simple social structure learning without HSIC-based bottleneck regularization is useless for recommendation tasks. Furthermore, under the constraint of the information bottleneck principle, the learned social structure is meaningful, which can effectively improve social recommendations on three datasets.

2.3. Generality study of GBSR

As we mentioned in the model discussion, the proposed GBSR is a model-agnostic social denoising framework. To better illustrate the generality of GBSR , we conduct experiments of GBSR on several graph-based social recommendation backbones. As shown in Table 3, we implement GBSR under four backbones, including GraphRec (Fan et al., 2019), DiffNet++(Wu et al., 2020a), SocialLGN (Liao et al., 2022), and LightGCN-S, and report their performances of Top-20 recommendation task. From Table 3, we observe that GBSR consistently outperforms each backbone by a large margin. For example, on the Yelp dataset, GBSR achieves 9.06%, 12.66%, 8.87%, and 11.21% improvements of NDCG@20 compared with GraphRec, DiffNet++, SocialLGN, and LightGCN-S, respectively. Similarly, GBSR also obtains 5.48%, 9.87%, 8.78%, and 10.39% improvements on the Recall@20 metric. Extensive experimental results show that our proposed GBSR has a good generalization ability, which can easily coupled with current graph-based social recommendation methods and further enhancement.

3. Investigation of GBSR

In this section, we further analyze GBSR from the following aspects: training efficiency, visualization of the denoised social graphs, and hyper-parameter sensitivity analysis.

To analyze the training efficiency of GBSR , we compare the convergence speed of GBSR and corresponding backbone (LightGCN-S). As shown in Figure 3, we compare the convergence process of both models. As the space limit, we only present the convergence process on Douban-Book and Yelp datasets. We set gcn layer to 3 and keep all experimental settings the same. According to these figures, we can observe that GBSR converges much faster than the backbone model. Particularly, GBSR reaches the best performances at the $82^{th}$, the $67^{th}$ epoch on Douban-Book and Yelp datasets. In contrast, LightGCN-S obtains the best results on $509^{th}$, and $261^{th}$ epoch, respectively. Empirical evidence shows that GBSR convergence 2-3 times faster than LightGCN-S.

3.2. Visualization and statistics of the denoised social graphs

Here we first present the visualization of the denoised social graph. As shown in Figure 4(a), we present the sampled ego-network from Douban-Book datasets. The red node denotes the center user of this ego-network, and the blue nodes denote social neighbors. The depth of the node color denotes the probability of edge dropping, where the darker the color, the lower the dropping probability. We can observe that user social neighbors perform different confidences of social relations. Besides, we analyze the statistics of the denoised social graphs. As shown in Figure 4(b), we plot the mean and variance values of social relation confidence on three datasets. We can observe that Douban-Book presents the lowest mean value of social confidence, which means that it has the most social noise over the three datasets. This also explains the results of Figure 1 that graph-based social recommendations show a performance decrease compared with LightGCN on the Douban-Book dataset. These results demonstrate that the proposed GBSR can effectively refine the observed social graph via information bottleneck, which provides informative social structures to enhance social recommendations.

3.3. Parameter Sensitivity Analysis.

In this part, we analyze the impact of different hyper-parameters of GBSR . There are two key parameters, bottleneck loss coefficient $\beta$ and RBF sharpness parameter $\sigma^{2}$. As both parameters determine the scale of bottleneck loss, we combine them to analyze the influence of recommendation results. As shown in Figure 5, we conduct careful grid-search of $(\beta,\sigma^{2})$ on three datasets. We can observe that GBSR reaches the best performance when $\beta=40,\sigma^{2}=2.5$ on Douban-Book, $\beta=2.0,\sigma^{2}=0.25$ on Yelp, and $\beta=3.0,\sigma^{2}=0.25$, respectively.

Related Works

With the emergence of social media, social recommendation has been an important technique and has attracted more and more research attention (Tang et al., 2013; Konstas et al., 2009; Ma et al., 2008; Jamali and Ester, 2010; Ma et al., 2011). Following the social homophily (McPherson et al., 2001) and social influence theory (Marsden and Friedkin, 1993), social recommendations are devoted to characterizing social relation effects on user preferences. Early efforts exploit social relations in a shallow form, such as co-factorization methods (Ma et al., 2008; Konstas et al., 2009) and regularization-based methods (Jamali and Ester, 2010; Ma et al., 2011; Jiang et al., 2014). For example, SoRec (Ma et al., 2008) jointly co-factorize the interaction and social matrices and then project interaction and social contexts into the same semantic space. (Ma et al., 2011) designs a social regularization term that assumes two socially connected users should be closer in preference space. Recently, with the great success of graph neural networks (Kipf and Welling, 2017; Veličković et al., 2018), graph-based social recommendations have been widely researched and achieved impressive process (Fan et al., 2019; Wu et al., 2019, 2020a; Liao et al., 2022; Yu et al., 2021b; Yang et al., 2023b). By formulating user-user social relations as a graph, graph-based social recommendations inject high-order social influences into user preference learning, vibrant the representation ability. For example, DiffNet models the high-order social influence diffusion process to enhance user representation (Wu et al., 2019), and DiffNet++ further improves it by combining both social influence diffusion and user-item interest propagation with a hierarchical attention mechanism (Wu et al., 2020a). Inspired by the architecture of LightGCN, (Liao et al., 2022) proposes SocialLGN to model user interaction and social behaviors. Instead of learning social graphs on Euclidean space, some works attempt to introduce hyperbolic learning for graph-based social recommendations (Wang et al., 2021c; Yang et al., 2023b). Despite the effectiveness of modeling high-order social influence to improve recommendation, these works are built on the clean social relation assumption. However, social graphs are inevitably noisy with redundant relations, and these graph-based social recommendation methods are usually far from satisfactory. Instead of directly using the original social graph, in this work, we propose a graph noising framework to improve social recommendation.

2. Recommendation Denoising

Recommendation denoising works mainly focus on implicit feedback, which aims to refine implicit feedback to build robust recommender systems (Wang et al., 2021a; Yang et al., 2021; Wang et al., 2023a; Gao et al., 2022; He et al., 2024). Most efforts are devoted to removing noise feedback, which is easily vulnerable to users’ unconscious behaviors and various biases. For example, (Wang et al., 2021a) proposes to drop noisy feedback based on the observation that noisy feedback has higher training loss, (Wang et al., 2023a) devises a bi-level optimization method to implement recommendation denoising. Besides, graph augmentation methods are proposed to realize recommendation denoising (Yang et al., 2021; Fan et al., 2023). Different from the above feedback-based denoising works, we focus on social denoising for recommendations. Social graphs are inevitably noisy with redundant relations, including unreliable relations and low preference-affinity relations (Quan et al., 2023; Sun, 2023). Early works employ statistics to identify unstable social relations (Ma et al., 2011; Pan et al., 2020), or model different user influences with attention mechanism (Sun et al., 2018; Wu et al., 2020a). Besides, fine-grained social leveraging (Fu et al., 2021) and adversarial learning based methods have been proposed (Yu et al., 2019, 2020). Recently, GDMSR (Quan et al., 2023) proposes a distilled social graph based on progressive preference-guided social denoising. Nevertheless, the above methods still face the challenge of lacking ground-truth. Whether rule-based or assumption-based social denoising is hard to guarantee the trade-off between social denoising and social recommendation. Distinguished by these denoising methods, we address the social denoising recommendation from a novel information bottleneck perspective, which seeks the denoised yet informative social structure to enhance recommendations.

3. Information Bottleneck and Applications

Information Bottleneck (IB) is an effective representation learning principle in machine learning tasks, that the optimal representation should satisfy the minimal yet efficient manner (Tishby et al., 2000; Tishby and Zaslavsky, 2015). In the era of deep learning, calculating high-dimensional variables’ mutual information (MI) is the key challenge for IB. The general solution is estimating the upper/lower bounds instead of directly calculating mutual information (Alemi et al., 2017; Cheng et al., 2020). Specifically, VIB (Alemi et al., 2017) leverages the variational technique to estimate the bounds of mutual information. Besides, MINE (Belghazi et al., 2018), InfoNCE (Oord et al., 2018) are proposed to estimate the lower bound of MI. In contrast, a few attempts propose to estimate the upper bound of MI (Cheng et al., 2020; Hledík et al., 2019). Besides optimizing the bounds of MI, HSIC-based methods (Ma et al., 2020; Wang et al., 2021b) are proposed to implement IB learning, which employs the Hilbert-Schmidt Independence Criterion (HSIC) to replace mutual information for optimization. HSIC measures the independence of two variables, which can approximate the mutual information objective (Gretton et al., 2005). IB principle has been successfully applied to many applications, such as image classification (Wang et al., 2023b), text understanding (Patrick et al., 2021), and graph learning (Wu et al., 2020c). In this work, we introduce the HSIC-based bottleneck to the graph-denoised social recommendation, aiming to filtering redundant social relations for robust recommendation.

Conclusion

In this paper, we investigate graph-denoised social recommendations and propose a novel Graph Bottlenecked Social Recommendation (GBSR) framework. Specifically, GBSR aims to learn the denoised yet informative social structure for recommendation tasks. To achieve this goal, we first design preference-guided social denoising, then optimize the denoising process via the information bottleneck principle. Particularly, we derive the lower bound of mutual information maximization and introduce HSIC regularization to replace mutual information minimization. Extensive experiments conducted on three benchmarks demonstrate the effectiveness of our proposed GBSR framework, i.e., over 10% improvements on Top-20 Recommendation. Moreover, GBSR is a model-agnostic framework, which can be flexibly coupled with various graph-based social recommenders. In the future, we will explore more potential of leveraging the IB principle to recommendation tasks, i.e., self-supervised recommendation, fairness-aware recommendation, and LLM-enhanced recommendation.

Acknowledgements

This work was supported in part by grants from the National Key Research and Development Program of China( Grant No.2021ZD0111802), and the National Natural Science Foundation of China( Grant No. U23B2031, 721881011).

References