## 3 Preliminaries
### Problem Definition  
Let \(\mathcal{U}\) denote the set of all users in the system, and \(\mathcal{P}\) represents the set of all available minor agricultural products. In real-world applications, both \(\mathcal{U}\) and \(\mathcal{P}\) typically exist at considerable scales, with datasets potentially encompassing hundreds of thousands of users and thousands of minor agricultural products. We define a mapping function \(\widehat{y_{u,p}}\) to quantify the preference level of user \(u\) for minor agricultural product \(p\), which can be formally expressed as follows:  
\[\widehat{y}:\mathcal{U}\times\mathcal{P}\rightarrow\mathbb{R}. \tag{1}\]  
Let R denote the set of non-negative real numbers with a total ordering relation within a specified range. The fundamental research objective of APGCN-CF is to identify the optimal minor agricultural product \(p^{*}\) for any given user u by maximizing the preference score \(\hat{y}(u,p)\), which can be formally expressed as follows:  
\[\forall u\in\mathcal{U},p^{*}=\operatorname*{arg\,max}_{p\in\mathcal{P}} \hat{y}(u,p). \tag{2}\]  
This formula defines the core objective of our APGCN-CF framework: to identify the optimal minor agricultural product for each user by maximizing their predicted preference score. It establishes a clear optimization target that guides the development of the entire recommendation architecture in subsequent sections. By framing the recommendation challenge as a preference maximization problem, we create a unified theoretical foundation that accommodates the integration of diverse information sources (user attributes, product  
\begin{table}
\begin{tabular}{c c} \hline
**Symbol** & **Description** \\ \hline \(\mathcal{U}\) & The set of all users in the system \\ \(\mathcal{P}\) & The set of all minor agricultural products in the system \\ \(\mathcal{G}\) & User–product interaction graph \\ \(v_{u}\) & Initial feature vector of user \(u\) \\ \(v_{p}\) & Initial feature vector of minor agricultural product \(p\) \\ \(e_{u,p}\) & Edge feature vector between user \(u\) and product \(p\) \\ \(h_{u}^{(k)}\) & Feature representation of user \(u\) after \(k\) layers of \\ \(h_{p}^{(k)}\) & Feature representation of minor agricultural product \(p\) \\ \(S(u,v)\) & After \(k\) layers of graph convolution \\ \(S(p,q)\) & Similarity measure between user \(u\) and user \(v\) \\ \(S(p,q)\) & Similarity measure between minor agricultural product \(p\) \\  & and product \(q\) \\ \(\mathcal{N}_{u}\) & The set of minor agricultural products that user \(u\) has \\  & interacted with \\ \(\mathcal{N}_{p}\) & The set of users who have interacted with minor \\  & agricultural product \(p\) \\ \(\hat{y}_{u,p}\) & Predicted preference score of user \(u\) for minor \\  & agricultural product \(p\) \\ \(r_{u,i}^{\text{true}}\) & True preference score of user \(u\) for item \(i\), as recorded in \\  & the dataset \\ \(r_{u,i}^{\text{pred}}\) & Predicted preference score of user \(u\) for item \(i\), obtained \\  & from the model \\ \(|E|\) & The total number of user–item interaction pairs in \\  & the dataset \\ \hline \end{tabular}
\end{table}
Table 1: Notations and descriptions.  
characteristics, and interaction patterns) while addressing the unique challenges of minor agricultural product recommendation, including seasonality, geographical constraints, and data sparsity. This formalized objective definition guides and facilitates the development, evaluation, and refinement of our recommendation framework, which is based on graph convolutional networks and collaborative filtering algorithms with mathematical precision, ensuring that the proposed framework maintains both theoretical rigor and practical applicability in rural e-commerce environments.