## 3 Our Proposed Method
### Case Study 2: Escherichia Coli Dataset  
Next, we tested our method on a time series gene expression dataset generated by the GNW tool. The settings used for GNW are shown in Table 1. We used four groups of datasets with sizes of 10, 50, 100, and 200. Each group contained three subgroups divided according to noise rates, and each subgroup had twenty different networks, for a total of 240 tested networks. After the dataset was generated, the expression values were clustered using the K-means method, with \(k\) equaling two or three depending on the distribution of the values. Based on the discretization dataset, MIDNI inferred the network structure and a set of regulatory rules for all genes. We analyzed its performance in terms of both the structural and dynamic accuracy. **Structural accuracy analysis. ** We compared the structural accuracy (see Equation (6)) of MIDNI with the DBN, MICRAT, and MIBNI methods on the _Escherichia coli_ dataset (Figure 4). As shown in the figure, the structural accuracy of MIDNI was lower than that of the other methods, particularly for the networks with the smallest size ( \(\mid\) V\(\mid\) = 10). In fact, the structural accuracy was affected by the number of positive predictions (=TP + FP in 2.4) of the inference method. Specifically, the structural accuracy is likely to become higher as the number of positive predictions decreases, as a GRN is a considerably sparse graph. Figure 5 shows the comparison results for the number of positive predictions among the inference methods. As shown in the figure, MIDNI predicted the existence of interactions significantly more often than the other methods. Therefore, it should be noted that its low structural accuracy represents a trade-off for the numerous positive predictions. **Dynamic accuracy analysis. ** It is well known that networks with different structures can produce the same underlying dynamics. Hence, it is essential to verify the network inference performance in terms of the dynamic accuracy. In particular, the dynamic accuracy is not affected by the number of positive predictions, unlike the structural accuracy. We compared the dynamic accuracy (see Equation (3)) of MIDNI, DBN, MICRAT, and MIBNI on the _Escherichia coli_ dataset (Figure 6). Figure 6a-c shows the results on the dataset with noise rates of 0%, 5%, and 10%, respectively. As shown in Figure 6a, MIDNI demonstrated considerably higher dynamic accuracy than the other methods regardless of the network size. This implies that MIDNI shows robust performance in terms of the dynamic accuracy against the network size. In addition, the dynamic accuracy of MIDNI was consistently higher than the other methods in the case of noisier datasets, as shown in Figure 6b,c. This implies that MIDNI is robust against noise in gene expression values. Figure 3: The effect of the number of incoming links on the inference result on the artificial dataset. The y-axis denotes the performance values and the x-axis represents the number of incoming links, specifying the difficulty of the inference problem. Finally, we examined the proportions of genes classified by two- or three-level discretization in each network (Figure 7). As shown in the figure, the proportions of the two- and three-level discretized genes were comparable to each other except for the networks with size 10. This result explains why the multi-level discretization should be considered to improve dynamic accuracy compared to the more commonly used binarization approach. Figure 4: Structural accuracy comparison on the _E. coli_ dataset. Figure 5: Comparison of the number of positive predictions on the _E. coli_ dataset. The x-axis denotes the size of the networks, while the y-axis denotes the average number of positive predictions for each target gene. Figure 6: _Cont._  
**Running time analysis. ** To compare the running time of MIDNI to the other methods, we examined the average running time over a total of 180 gene expression datasets on a PC with an AMD Ryzen 5 3400 G 3.7 GHz CPU and 16 GB of RAM (Figure 8).