Fig. 8: Architecture diagram of the graph neural network model based on the attention mechanism.

a Shows the input graph data of the model. The model first receives 1000 frames of network data at a 1% CNT concentration for training, providing solid initialization parameters for subsequent incremental training. This ensures that the model has good generalization capabilities when processing network data at other concentrations; b Illustrates the optimization mechanism of the model in the feature representation learning and message aggregation process. This module iteratively computes the relationships between nodes and their first-order neighbors using the multi-head attention mechanism, aggregating features with edge attributes. Subsequently, the TopKPooling layer selects and retains the most representative nodes based on their importance, enhancing feature extraction efficiency. Finally, the PairNorm normalization technique is applied to normalize node features, ensuring consistency in feature scales, improving the model’s stability and convergence speed, and further enhancing its performance in complex networks; c Presents the improved pooling strategy, which combines the advantages of global average pooling and global max pooling. First, global average pooling and global max pooling are applied to all nodes in each graph. Then, the results are concatenated to generate a more comprehensive graph-level embedding. This dual-pooling strategy captures both the global information of the graph structure and the significant features of key nodes, thereby improving the model’s representational power and generalization ability; d Demonstrates the workflow of the Estimator block. The input graph embedding is first processed with layer normalization and then undergoes a nonlinear feature transformation through a fully connected layer. Afterward, a dropout strategy is applied to enhance the model’s robustness, followed by another layer of normalization to optimize the feature distribution. Finally, after introducing non-linearity through a ReLU activation function, the output layer is used to make an accurate conductivity prediction. These steps progressively optimize the feature representation and ensure an effective mapping from high-dimensional graph embeddings to the target property space.