## 4 Experiments
### Experimental Design
#### 4.4.2 Analysis of Experimental Results of Op-Trans
CCS + TransE only used the CCS negative-sampling optimization method for TransE, and Op-TransE used of Op-Trans lifting framework for the TransE model. We can observe that the Hits@10 of CCS + TransE is approximately 3.7% better than the TransE model. Op-TransE was based on CCS + TransE further considering the mapping properties of triplets in the embedding space, and the Hits@10 value of Op-TransE increased by about 5.1% compared with the TransE model. Table 8 summarizes the results of our comparison experiment on the FB15. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **MR** & **MRR** & **Hits@1** & **Hits@3** & **Hits@10** \\ \hline TransE & 251 & 0.779 & 0.709 & 0.820 & 0.892 \\ CCS + TransE & 237 & 0.792 & 0.728 & 0.822 & 0.925 \\ Op-TransE & 223 & 0.803 & 0.711 & 0.836 & 0.937 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Experimental results on the WN18 compared with the TransE model. Figure 10: Hits@10 values for the three methods on the dataset FB15K237,  indicates initial model, indicates CCS negative sample, indicates Op-Trans framework,  is denoted as various baseline models. Figure 9: Hits@10 values for the three methods on the dataset WN18 RR, indicates initial model, indicates CCS negative sample, indicates Op-Trans framework,  is denoted as various baseline models. On the FB15k, there are 592,213 positive triplets with 14,951 entities, and each entity corresponds to approximately 39.6 positive triplets on average. On the WN18, there are 151,442 positive triplets with 40,943 entities, and each entity corresponds approximately to about 3.7 positive triplets on average. Since the average number of positive triplets corresponding to each entity on WN18 is much less than that on FB15k, and the number of relationship types on WN18 is also much lower than that on FB15k, we think that the graphical information on WN18 may be simpler than that on FB15k. However, our method showed a better improvement on FB15k. This indicates that the Op-Trans framework performs better in the face of various complex entities and relation embedding, and has an advantage in complex knowledge graphs. To further evaluate the performance of Op-Trans and also verify that the model has better advantages for complex relationships, we continued to study the performance of Op-Trans in different relationship types on FB15K. Figure 11 shows the distribution of 1345 relations on FB15K, where the 1-to-1 type of simple relationship accounts for 24%, and the 1-to-N, N-to-1, and N-to-N types of complex relationship account for 23%, 29% and 24%, respectively. Table 9 shows the results of the comparison of the Hits@10 of four different types of relations on FB15K. From Table 9, we can observe that our proposed method has a great improvement on the effect of dealing with complex relationships compared with the TransE. For the 1-to-1 relation prediction, the head-entity forecast improved the Hits@10 by 15.1% and tail entities increased by 17.6%. For the 1-to-N relation type, the head entity Hits@10 prediction increased by 18.1%, and the tail entity increased by 17.8%. For the N-to-1 relations type, the head entity Hits@10 prediction increased by 26.9%, and the tail entity increased by 12.0%. For the more complex N-to-N relations type, the head entity Hits@10 prediction increased by 10.8%, and the tail entity prediction increased by 37%. It is worth noting that the Op-Trans outperforms other baseline models in most cases. In particular, Op-Trans can improve the performance of 1-to-N relation tail-entity prediction and N-to-1 relation head-entity prediction, which are difficult tasks for other models. Overall, our method offers significant advantages in dealing with complex relationships. Figure 11: Type distribution of 1345 relations on FB15K.