## 2 Methodology
### Modified Frequency Degree Method (MFDM)  
The order of time series and reproduction are two key attributes in time series analysis. In order to better mine the features of the original time series and transform the time series into a network, [PERSON] et al. proposed a frequency degree algorithm [15]. The idea of the frequency degree algorithm is that the data with high reproduction frequency will correspond to the value with magnanimity. Specifically, the frequency degree algorithm divides the time series into equidistant intervals according to the amplitude and then connects each node to its nearest left and right neighbors in time order, and then it fully connects the nodes located in the same interval. This method can preserve the amplitude and time information of the original time series. In 2025, [PERSON] and [PERSON] proposed a modified frequency degree method [16] that further considers the positional information of the time series data points. Specifically, let \(N\) be the length of the time series and \(Q\) be the parameter to be input. We consider two cases: If \(N\) is divisible by \(Q\), then the data points of the time series are evenly distributed into \(Q\) intervals. If \(N\) is not divisible by \(Q\), then the data points of the time series are divided into \(Q+1\) intervals. The \(N\) divisible Q case: If \(N\) is evenly divided by \(Q\), these data points will be bisected into these intervals, \(d=\frac{N}{Q^{\prime}}\), the time series will be sorted from smallest to largest, and the time series data points will be divided according to [d: d: N]; then, the data points in the same interval will be connected with its two nearest neighbors, and the data points in the same interval will be fully connected. For convenience of illustration, we present a schematic diagram of the modified frequency degree method under the condition that it can be evenly divided in Figure 5.  
Figure 4: Example of the QG method for a time series with \(T=12,Q=4,k=1\). As shown in Figure 5, we select the time series 3, 6, 1.5, 9, 7, 8, 5, and 6, and \(Q=4\) with a length of 8. Firstly, the time series is sorted, and the points on the boundary line all belong to the following smaller interval. The data points are then divided into four intervals. The first-order time series is 1.5, 3, 5, 6, 6, 7, 8, and 9, and then the corresponding position of the time series is found according to \(d=\frac{8}{4}=2\); that is, [2:2:8] = 2, 4, 6, and 8 corresponds to the interval limits, so the divided interval ranges are \((-\infty,3],(3,6],(6,7],(7,9]\). So, the time series is 3, 6, 1.5, 9, 7, 8, 5, and 6, and the corresponding interval is [1, 2, 1, 4, 3, 4, 2, and 2]. Next, each data point is connected to its two nearest neighbors, and then the data points that are in the same interval are fully connected. If N does not divide \(Q\) exactly: If \(N\) does not divide \(Q\) exactly, these data points will be divided into \(Q+1\) intervals. We denote \(d=\frac{N}{Q}\), and the time series will be sorted from smallest to largest. The time series data points will be divided according to \(\lfloor d:d:N\rfloor\), and then the data points located in the same interval will be fully connected after each data point is connected with its two nearest neighbors. In Figure 6, we provide a schematic diagram of the modified frequency degree mapping algorithm when it is not evenly divided. As shown in Figure 6, we consider a time series of length 10: 0.27, 0.22, 0.76, 0.12, 0.82, 0.36, 0.7, 0.95, 0.25, and 0.82, with \(Q=4\). First, we sort the time series in ascending order: 0.12, 0.22, 0.25, 0.27, 0.36, 0.7, 0.76, 0.82, 0.82, and 0.95. Next, compute \(d\) as follows: \(d=\left\lfloor\frac{N}{Q}\right\rfloor=\left\lfloor\frac{10}{4}\right\rfloor=2\). Then, we divide the time series data points into intervals based on \([d:d:N]\), i.e., [2:2:10] = 2, 4, 6, 8, and 10. The intervals are as follows: \((-\infty,0.22]\), (0.22, 0.27], (0.27, 0.7], (0.7, 0.82], and (0.82, 0.95].