## 3 Analysis and Results
### Analysis of Coupled Dynamic Model  
The validity of the proposed method was first tested on signals generated using two Henon map unidirectional coupled subsystems with one as the driver subsystem 1 and the other as the responder subsystem 2. The equations of the system are expressed as follows:  
\[\left\{ \begin{array}{l}x1_{t+1}=1.4-x1_{t}^{2}+0.3\times y1_{t}\\ y1_{t+1}=x1_{t}\\ x2_{t+1}=1.4-(C\times x1_{t}+(1-C)\times x2_{t})\times x2_{t}+0.3\times y2_{t} \\ y2_{t+1}=x2_{t}\end{array} \right. \tag{8}\]  
The parameter \(C\) is a coupling parameter varying from 0 to 1. When \(C\) is 0, the two subsystems are entirely independent and there is no definite dynamical behavior between them. When \(C\) is 1, the two subsystems are completely synchronized and there is a definite dynamical relationship between them. The values of \(x1_{1}\), \(y1_{1}\), \(x2_{1}\) and \(y2_{1}\) are initialized randomly in the range from 0 to 1. Then, 50,000 points are calculated according to (8) and the first 20,000 are discarded as the transients. From the definition of CBTN, we can see that the unique parameter relevant to the CBTN is the embedding dimension \(d\). The parameter \(d\) defines the embedding spatial dimension of a given time series. Another noteworthy issue is the appropriate signal length in order to obtain reliable results. One fact is that the signal length is limited, and the other is that the calculation process can only be performed in one window. The values of these two parameters determine whether the results of the analysis can be described or not and whether it is possible to extract the deep relationships hidden between the two time series. Here, the determination process of these two parameters is explained by analyzing the unidirectional coupled Honen model with the deterministic coupling relationship. It was  
Figure 1: The directional weighted adjacency matrix constructed from the temporal adjacency relationship for symbol sets A and B. (**a**) The adjacency matrix for the symbolized set A; (**b**) The adjacency matrix for the symbolized set B.  
selected because the theoretical value of each expected return can be calculated. Based on this theoretical value, it can be evaluated whether the result obtained with a specific parameter setting can converge reliably and stably to the expected value or not. In addition, in order to highlight the impact of the CBTN on the analysis result, the CPE results for the same objects were used as a comparison. When the CPE was used, the embedded dimension was 5 and the delay time was 2. The appropriate signal length was determined by investigating the effect of the width of the analysis window on the results. For the time series \(x1\) and \(x2\) obtained using the unidirectional coupled Henon model under a certain coupling strength, the surrogate data \(x2^{Surrogate}\) were first calculated by surrogating \(x2\) using iAAFT (iterative amplitude-adjusted Fourier transform with five iterations) to mimic the random coupling state. Next, the sliding time window with a fixed moving step of 500 samples was chosen to segment the paired time series \(x1\) and \(x2\) and \(x1\) and \(x2^{Surrogate}\). The purpose of using the sliding time window is to enhance the effect of data analysis. The width of the sliding window width was increased from 200 to 5000 samples with a step of 200 samples. With each window width, the coupling strength between \(x1\) and \(x2\) and between \(x1\) and \(x2\) was calculated using the CBTN for individual sliding windows, obtaining the \(H_{NOTE}^{x1-x2^{Surrogate}}\) and \(H_{NOTE}^{x1-x2}\), respectively. The differences \(H_{NOTE}^{x1-x2^{Surrogate}}-H_{NOTE}^{x1-x2}\) were first calculated for individual windows and then averaged across windows as the measured difference. The same procedure was repeated for 30 times with each window width and then averaged across repetitions to obtain the average and standard deviation of the measured difference.