Improving connectivity and completeness are the most challenging aspects of small liver vessel segmentation. It is difficult for existing methods to obtain segmented liver vessel trees simultaneously with continuous geometry and detail in small vessels. We proposed a diffusion model-based method with a multi-scale graph attention guidance to break through the bottleneck to segment the liver vessels. Experiments show that the proposed method outperforms the other state-of-the-art methods used in this study on two public datasets of 3D-ircadb-01 and LiVS. Dice coefficient and Sensitivity are improved by at least 11.67% and 24.21% on 3D-ircadb-01 dataset, and are improved by at least 3.21% and 9.11% on LiVS dataset. Connectivity is also quantitatively evaluated in this study and our method performs best. The proposed method is reliable for small liver vessel segmentation.