# Social)GCF: Social Recommendation with Jacobi Polynomial-Based Graph Collaborative Filtering
## 1 Introduction  
In the age of information explosion, users need tools to accurately filter personally relevant information from the noisy data [1]. A recommender system is a powerful decision-making tool for users that addresses the information overload problem and has been widely applied in online services, e.g., e-commerce platforms (Amazon, BestBuy), and social platforms (X, Facebook). Recommender systems aim to provide lists of items that users will likely interact with (e.g., click or purchase) [2, 3]. Traditionally, recommender systems use collaborative filtering to extract effective user patterns and item patterns from past user-item interaction data. However, there remains a significant research gap in effectively utilizing the rich information available in social networks to improve the accuracy and personalization of recommendations, especially for new users or items with limited interaction history.  
Psychology [4] and life experiences show that people tend to have similar preferences within their social relationships and tend to be easily influenced by them. It indicates that a user's connections might provide additional assistance in deciding the preference of the user, and information about social relationships might enhance the performance of recommender systems if properly integrated. Recently, studies [5, 6] have realized the potential influence of social relationships in recommender systems. In the case wherethe recommender system utilizes users' social information for better recommendation performance, the system is called a social recommender system.  
By modeling users and items as nodes, and their interactions and social connections as edges, we can represent the user-item interaction data and social data using a user-item interaction graph and a social graph [7; 8]. Obviously, both graphs contain user information, and, to build reliable social recommendation systems, we need to extract and integrate user information from both [9; 10]. Generally, two frameworks can be adopted to compute the integrated representation [9]: bipartite graph strategy, where the two graphs are first propagated separately and the final user representations are computed later [7; 11; 12], and unified graph strategy, where the two graphs are combined into one graph and the final representation is obtained by propagating the unified graph [13; 14].  
In recent years, graph neural networks (GNNs), which were proposed to effectively learn node representations and obtain meaningful information from graphs, have been widely applied in graph-structured data [15; 16; 17; 18]. The core of GNNs lies in their ability to propagate node information through the network via node connections, allowing for the integration of local neighborhood features into the learning process. This capability enables GNNs to generalize and infer node and graph properties dynamically, thus addressing tasks that involve complex graph data more effectively than traditional neural network models.  
Due to the superiority of GNNs for graph data, GNNs have been widely applied in recommender systems [19; 20; 21; 22; 23], especially social recommendation systems [6; 24; 25]. Compared with previous non-GNN-based methods, GNN-based social recommendation methods have achieved better recommendation accuracy on real-world datasets.  
The paper proposes a new GNN-based social recommendation system called SocialJGCF. SocialJGCF adopts the bipartite graph approach, and the final user representation is an adaptive weighted average of the user representation from the interaction graph and social graph, while final item representation is computed by the propagation of the user-item interaction graph.  
The paper is structured as follows: Section 2 summarizes related work on recommendation systems. Section 3 gives a basic notation of the social recommendation problem and JGCF and presents the design of the SocialJGCF algorithm. Section 4 analyzes the experiment results of the proposed SocialJGCF. Section 5 concludes the paper.