Incipient faults for analogue circuits in modern electronic systems are difficult to diagnose due to poor fault features. To address this issue, a method based on the attention weighted graph convolution network (Att-GCN) is proposed in this paper. The structural and data features of samples are jointly extracted to mine the effective characteristics from incipient faults. First, a wavelet packet energy transform and a probabilistic principal component analysis (ProbPCA) are employed to enhance the sample fault information. Then, the distance clustering method is deployed to construct the sample set into a non-European structure sample graph, where the structural features of fault samples are preserved. Second, an Att GCN, which combines the spatial-domain graph convolution network and improved self-attention mechanism, is constructed to extract the structural features and data features to obtain more effective fault information. Additionally, the multisample dropout method is introduced to reduce network overfitting in the training process. To assess the method's actual performance for fault diagnosis, experiments are carried out in the Sallen-Key bandpass filter circuit, the four-op-amp biquadratic filter circuit and the amplifier board circuit. The outcomes indicate that this method improves the incipient fault diagnosis accuracy for analogue circuits.