## 4 Conclusions  
This study presents a ChebyNet modeling method for fault diagnosis in rotating equipment under variable working conditions. The Path composition method is utilized to map complex data from variable working conditions into a non-Euclidean space, successfully revealing the intricate interactive relationships within the data. Symmetry processing enhancements have been incorporated into the ChebyNet framework, improving the symmetric transmission of information between graph nodes and ensuring the efficient and equitable integration of fault features. By combining the enhanced ChebyNet with a GAT, the method accurately captures both global features and local details of the data. Additionally, the incorporation of a node reconstruction task through self-supervised learning creates an effective synergy with the node classification task, enhancing the model's intrinsic understanding of the input data and significantly improving its generalization ability for processing data under variable working conditions. The experimental results indicate that the proposed method achieves an average diagnostic accuracy of 99.72%, surpassing other GNN models. Moreover, the method displays notable stability and generalization, emphasizing its robust performance and significant applicability in complex and variable industrial environments. This study enhances the application of GNNs in industrial contexts and introduces an innovative approach to tackling the challenges of fault diagnosis in rotating equipment under variable operating conditions.  
This study primarily focuses on a single data source and does not fully explore the potential of multimodal data fusion, which restricts the model's ability to comprehensively assess equipment status in complex and dynamic industrial environments. Looking ahead, we are committed to further exploring the application of multimodal data fusion within the GNN framework; investigating effective data preprocessing and feature extraction methods that fully preserve the uniqueness and complementarity of each data modality; exploring innovative data integration strategies to intelligently assign weights to different modal data to achieve efficient and accurate information fusion; and building a GNN model that can efficiently fuse multidimensional information from different sensors. Implementing the model for intelligent health management and maintenance of industrial equipment can yield more accurate and efficient solutions for industrial production, aiding enterprises in achieving cost reduction, enhanced efficiency, and sustainable development.  
[PERSON] proposed the method and wrote the paper; [PERSON] revised and improved the paper; [PERSON] and [PERSON] contributed equally to this paper; [PERSON] and [PERSON] conducted the experiment and analysis of results. All authors have read and agreed to the published version of the manuscript. This work is partially supported by the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2022B151520053), the National Natural Science Foundation of China (Grant No. 52175457), and the Guangdong Science and Technology Plan Project (Grant No. 2023A0505050151).  
**Institutional Review Board Statement:** Not applicable.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Accuracy** & **Precision** & **Recall** & **F1 Score** \\ \hline ChebyNet & 0.8720 & 0.8339 & 0.8723 & 0.8411 \\ S-ChebyNet & 0.9125 & 0.8691 & 0.9120 & 0.8829 \\ S-ChebyA & 0.9435 & 0.9651 & 0.9441 & 0.9402 \\ ChebyAT & 0.9975 & 0.9976 & 0.9975 & 0.9975 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Ablation experiment results.